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Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws
Nonlinear Analysis 72 (2010) 635–650
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Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws John M. Hong a , Ying-Chin Su b,∗ a
Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
b
Department of Mathematics, Fu Jen Catholic University, Taipei County 24205, Taiwan
article
info
Article history: Received 4 May 2009 Accepted 2 July 2009 MSC: 35L60 35L65 35L67 Keywords: Hyperbolic systems of balance laws Riemann problem Boundary Riemann problem Lax’s method Generalized Glimm scheme
abstract In this paper we provide a generalized version of the Glimm scheme to establish the global existence of weak solutions to the initial-boundary value problem of 2 × 2 hyperbolic systems of conservation laws with source terms. We extend the methods in [J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. Dissertation, Stanford University, 1982; J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by ‘‘weaker than weak’’ solutions of the Riemann problem, J. Differential Equations 222 (2006) 515–549] to construct the approximate solutions of Riemann and boundary Riemann problems, which can be adopted as the building block of approximate solutions for our initial-boundary value problem. By extending the results in [J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697–715] and showing the weak convergence of residuals, we obtain stability and consistency of the scheme. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction We consider the following 2 × 2 nonlinear hyperbolic system of balance laws ut + f (a, u)x = a0 g (a, u),
(1.1)
where u = u(x, t ) = (v(x, t ), w(x, t )), f (a, u) = (f1 (a, u), f2 (a, u)), g (a, u) = (g1 (a, u), g2 (a, u)) are smooth functions of (a, u), a = a(x) is a Lipschitz continuous function of x which is of finite total variation and a0 = da . Following the lead by dx Isaacson–Temple [1] and LeFloch–Liu [2], we augment system (1.1) by adding the equation at = 0 and obtain the following 3 × 3 hyperbolic system of balance laws Ut + F (U )x = a0 G(U ),
(1.2)
where U = (a, v, w), F (U ) = (0, f (a, u)) and G(U ) = (0, g (a, u)). In this paper we investigate the global existence of weak solutions to the initial-boundary value problem of (1.2): Ut + F (U )x = a0 G(U ), U (x, 0) = U0 (x), v(0, t ) = vB (t ),
(
∗
(x, t ) ∈ (0, ∞) × (0, ∞), x ≥ 0, t ≥ 0,
Corresponding author. E-mail addresses: [email protected] (J.M. Hong), [email protected] (Y.-C. Su).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.07.003
(1.3)
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where U0 (x) ≡ (a(x), v0 (x), w0 (x)), vB (t ) are bounded functions with finite total variations and satisfy vB (0) = v0 (0). We assume that system (1.2) is strictly hyperbolic, that is, the eigenvalues λ0 = 0, λ1 and λ2 of Jacobian matrix DF (U ) are real and distinct. Moreover, each characteristic field of (1.2) is either genuinely nonlinear or linear degenerate [3]. We assume further that there is an open region Ω ⊂ R3 such that f , g satisfy
(A1 ) each component of ( ∂∂ uf )−1 (g − ∂ f1 (A2 ) ∂w (U ) 6= 0 for U ∈ Ω .
∂f )(U ) ∂a
is nonzero for U ∈ Ω ,
An important application of (1.1) is the one-dimensional compressible Euler equations in a variable area duct
ρt + (ρ u)x = −(a0 /a)ρ u, (ρ u)t + (ρ u2 + p(ρ))x = −(a0 /a)ρ u2 ,
(1.4)
where ρ , u and p(ρ) are the density, velocity and pressure of the fluid respectively, and a(x) represents the area of crosssection of the duct. We review some previous results to this topic. The entropy solutions to the Riemann problem of Ut + F (U )x = 0
(1.5)
was first studied by Lax [4,28]. The solutions are self-similar functions consisting of constant states separated by elementary waves (rarefaction waves, shock waves and contact discontinuities). To the Cauchy problem of (1.5), the global existence of weak solutions was established by Glimm [5] when initial data is uniformly bounded and of small total variation. For the initial-boundary value problem (IBVP for short), Nishida and Smoller [6] studied the piston and double piston problems and obtained the global existence of weak solutions. On the other hand, the IBVP of unsteady flow in gas dynamics was studied by Liu [7]. Moreover, the IBVP of (1.5) with arbitrary shape of boundary was first studied by Goodman [8]. He proved the global existence of weak solutions when the initial and boundary data satisfy the so-called smallness and non-degeneracy conditions. To the quasilinear hyperbolic system Ut + F (x, U )x = G(x, U ),
(1.6)
the global existence results for Cauchy problem was established by Liu [9]. We also remark that for the well-posedness theory for initial-boundary value problems, we refer the reader to [10–14]. On the other hand, the hyperbolic system of balance laws Ut + F (a(x), U )x = a0 G(a(x), U ),
(1.7)
was studied by Hong [15]. In [15], the global existence of weak solutions to Cauchy problem of (1.7) was obtained by the generalized Glimm scheme based on ‘‘weaker than weak’’ solutions of the Riemann problem. To the non-strictly hyperbolic (resonant) systems, we refer the readers to the results in [16–18,26,27]. Note that (1.7) can also be written as a system in non-conservative form. The global existence and stability of measure-valued solutions to non-conservative systems were studied by LeFloch [19,20], LeFloch–Liu [2] and Dal Maso–LeFloch–Murat [21]. To general quasilinear hyperbolic system Ut + F (U , x, t )x = G(U , x, t ),
(1.8)
the results for global existence of weak solutions to the Cauchy problem of (1.8) can be found in [22,23,29]. We notice that the results described above for the global existence of weak solutions to IBVPs are restricted to the hyperbolic systems without source terms. Recently, in the paper by Hong–Hsu–Su [24], the authors studied the following IBVP of quasilinear wave equation
utt − p(ρ(x), ux ) x = ρ(x)h(ρ(x), u, ux ), u(x, 0) = u0 (x), ut (x, 0) = w0 (x), ux (0, t ) = vB (t ),
(1.9)
or equivalently
vt − wx = 0, wt − p(ρ(x), v) x = ρ(x)h(ρ(x), u, v), w(x, 0) = w0 (x), v(0, t ) = vB (t ), u(x, 0) = u0 (x),
(1.10)
where v = ux and w = ut . By Glimm scheme and the perturbation technique for approximate solutions near the boundary, they obtained the global existence of weak solutions to (1.10) under the condition that the first and second derivatives of source h(ρ, u, v) to be uniformly bounded. In this paper, the generalized Glimm scheme we use enables us to relax the conditions of f and g given in [24] to establish the global existence results. More precisely, we only require condition (A2 ) on f to construct the approximate solutions near the boundary. Furthermore, since the solutions constructed for the boundary Riemann problem are exact solutions, we do not need to impose any condition on the second derivatives of source term. Now we give the definition for the weak solutions of IBVP (1.3), and state the main theorem of this paper.
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Definition 1.1. Let E := [0, ∞) × [0, ∞) and U := (a, v, w). Also for given test function φ ∈ C0∞ (E ), we define the residual R(U , φ, E ) by R(U , φ, E ) :=
Z Z
∞
Z
(U φt + F (U )φx + a G(U )φ)dxdt + 0
U0 (x)φ(x, 0)dx +
F (U (0, t ))φ(0, t )dt . 0
0
E
∞
Z
Then we say a measurable function U is a weak solution of initial-boundary value problem (1.3) if R(U , φ, E ) = 0 for all φ ∈ C0∞ (E ). Main theorem. Consider the following initial-boundary value problem ut + f (a, u)x = a0 g (a, u), (a, u)(x, 0) = (a(x), u0 (x)), v(0, t ) = vB (t ),
(
(x, t ) ∈ (0, ∞) × (0, ∞), (1.11)
where u = (v, w), a = a(x) is a Lipschitz continuous function with small total variation, and f = (f1 , f2 ), g = (g1 , g2 ) are smooth functions of a, u satisfying (A1 ) and (A2 ). Also u0 (x), vB (t ) ∈ L∞ have small total variations. Let U := (a, u) and let {Uθε,∆x } denote a sequence of approximate solutions constructed by a generalized Glimm scheme, then there exists a null set Nε in probability space Φ and a sequence {∆xi } such that for any θ ∈ Φ \ Nε , the measurable function U (x, t ) := limε,∆xi →0 Uθε,∆xi is a weak solution of (1.11). The paper is organized as follows. In Section 2 we study the perturbed Riemann problem of (1.3) away from the boundary x = 0 and review the results in [15,25] for the approximate solutions to a perturbed Riemann problem. In Section 3 we extend the results of [4] to construct the weak solutions to a boundary Riemann problem. In Section 4 we construct the generalized Glimm scheme and study its stability which leads to the compactness of an approximate solution. In Section 5, we establish the global existence results for IBVP (1.3) by showing the weak convergence of the residual given in Definition 1.1. 2. Riemann and perturbed Riemann problems In this section, for given positive x0 away from zero, we study the following Riemann problem Ut + F (U )x = a0 G(U ),
U (x, 0) = U0 (x) =
UL , UR ,
x < x0 , x > x0 ,
(2.1)
where U = (a, v, w), F (U ) = (0, f1 (U ), f2 (U )), G(U ) = (0, g1 (U ), g2 (U )), and UL = (aL , vL , wL ), UR = (aR , vR , wR ) ∈ Ω are two nearby constant states. Note that the source term a0 G(U ) is not defined in the sense of distributions, cf. [15]. To overcome this difficulty, we choose 0 < ε 1 and re-formulate the source term to study the following corresponding perturbed Riemann problem (PRP for short):
( ε
ε
ε 0
ε
Ut + F (U )x = (a ) G(U ),
ε
ε
U (x, 0) = U0 (x) =
UL ,
φ ε (x), UR ,
x < x0 − ε, |x − x0 | ≤ ε, x > x0 + ε.
(2.2)
Here aε (x) can be expressed as aL ,
( ε
a (x) =
ηε (x), aR ,
x < x0 − ε, |x − x0 | ≤ ε, x > x 0 + ε0 ,
(2.3)
where ηε (x) is a monotone function connecting aL , aR at x = x0 − ε and x = x0 + ε respectively, and φ ε (x) = (ηε (x), v0ε (x), w0ε (x)) is also a monotone function connecting UL , UR at x = x0 −ε and x = x0 +ε respectively. By the results of [15,24], the stability of U ε can be obtained with respect to the set of monotone functions {ηε (x)}, {φ ε (x)}, which allows us to define the measurable solution of (2.1) by the ε -limit of the weak solutions of (2.2). Now, we construct the approximate solutions of (2.2). Since the system in (2.1) is strictly hyperbolic in Ω , the eigenvalues {λi (U )|U ∈ Ω , i = 0, 1, 2} of Jacobian matrix DF (U ) are real and distinct. Without loss of generality, we assume that
−∞ < λ1 (U ) < λ0 (U ) = 0 < λ2 (U ) < ∞,
U ∈ Ω.
Let {(0, ri ) |i = 1, 2} be the right eigenvectors of DF (U ) corresponding to {λi |i = 1, 2} and (1, r ∗ )T be the right eigenvector of DF (U ) corresponding to λ0 . By assumption we see that each characteristic field is either genuinely non-linear or linear degenerate (in particular, the characteristic field with respect to λ0 ). Next, we define the following regions T
ΩL (ε) := {(x, t )|x < x0 − ε, t ≥ 0}, ΩR (ε) := {(x, t )|x > x0 + ε, t ≥ 0}, Ωε (ε) := {(x, t )||x − x0 | ≤ ε, t ≥ 0}.
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Fig. 1. Solution of the perturbed Riemann problem (2.2), (2.3).
Then, due to the vanishing of (aε )0 in ΩL (ε) and ΩR (ε), the approximate solution of (2.2) in those regions satisfies the following homogeneous conservation laws uεt + f (aL , uε )x = 0 ε
ε
ut + f (aR , u )x = 0
in ΩL (ε),
(2.4)
in ΩR (ε).
(2.5)
Therefore, by the results of [4], the weak solution in ΩL (ε), ΩR (ε) consists of constant states separated by elementary waves (rarefaction waves, shocks and contact discontinuities) which can be expressed as the integral curves of {(0, ri )T |i = 1, 2} in the phase plane. In addition, the solution curve of each characteristic field starting at U∗ can be parameterized as U1 (σ1 ; U∗ ) = U∗ + σ1 R1 (U∗ ) + O(σ12 ),
(2.6)
U2 (σ2 ; U∗ ) = U∗ + σ2 R2 (U∗ ) + O(σ22 ),
where Ri (U∗ ) := (0, ri (U∗ ))T and Ui (σi ; U∗ ) is at least C 2 in σi for i = 1, 2. Next, we study the solution of (2.2) in Ωε (ε). Due to the appearance of the characteristic field of λ0 , we can construct the time-independent solutions (standing waves) of (2.2) in Ωε (ε). Let u1 denote the constant state of solution near x = x0 − ε in ΩL (ε), then the solution of (2.2) in Ωε (ε) is solved by
(
daε
df
(aε , uε ) = g (aε , uε ), dx dx uε (x0 − ε) = u1 .
(2.7)
Next, by (A1 ), the monotonicity of aε and re-scaling technique, we can rewrite (2.7) into
duε
∂f ε ∂u da uε (aL ) = u1 .
−1
=
g−
∂f (U ε ), ∂a
(2.8)
We define the following vector
∂f R0 (U ) = (1, r0 (U )) := 1, ∂u T
−1
!T ∂f g− (U ) . ∂a
(2.9)
Then, by (2.8) and (2.9), we can express the solution Usε = (aε , uε (aε )) of (2.2) in Ωε (ε) as an integral curve in phase space, which can be parameterized as Usε (σ0 ; U1 ) = U1 + σ0 R0 (U1 ) + O(σ02 ), ε
(2.10) ε
where U1 := (aL , u1 ). Note that u may not match the initial data on {(x, 0)||x − x0 | ≤ ε}. So u constructed above can only be treated as an approximate solution of (2.2). However, by the results of [25], the effect caused by the jump of uε on {(x, 0)||x − x0 | ≤ ε} can be neglected when ε → 0. To complete the construction of approximate solution of (2.2), we need to determine the constant states on the left of x = x0 − ε and on the right of x = x0 + ε (U1 and U2 in Fig. 1). It can be achieved by Lax’s method [25]. We have the following theorem. Theorem 2.1. Given 0 < ε 1, we consider the perturbed Riemann problem (2.2) and (2.3) where f , g satisfy condition (A1 ) and x0 is away from 0. Suppose that UL , UR ∈ Ω and |UL − UR | is sufficiently small. Then (2.2) and (2.3) admits a unique approximate solution uε constructed by the process described above. The approximate solution consists of at most four constant states separated by either shocks, rarefaction waves and a smooth standing wave (Fig. 1). Next, following the work in [25], we give a definition to the measurable solution of the Riemann problem (2.1). Definition 2.2. Let {U ε (x, t )|0 < ε 1} be a sequence of approximate solutions of (2.2) and (2.3). Then, the function U (x, t ) := limε→0 U ε (x, t ) is called a measurable solution of (2.1).
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Fig. 2. Measurable solution to the Riemann problem (2.1).
By letting ε → 0 to the sequence of approximate solutions of (2.2) and (2.3) in Theorem 2.1, we obtain the following theorem. Corollary 2.3. The Riemann problem (2.1) has a unique self-similar measurable solution consisting of at most four constant states separated by either shocks, rarefaction waves or a standing wave discontinuity (Fig. 2).
3. Boundary Riemann problem In this section we study the measurable solutions to the following boundary Riemann problem (BRP for short):
U + F (U )x = a0 G(U ), t U (x, 0) = UR , a(0, t ) = a(0) := aB , v(0, t ) = vB ,
(x, t ) ∈ (0, ∞) × (0, ∞), x > 0,
(3.1)
t > 0,
where U = (a, v, w)T , UR = (aR , vR , wR )T , F = (0, f1 , f2 )T and G = (0, g1 , g2 )T . To overcome the same difficulty appeared in Section 2, we reformulate the source term in (3.1). Therefore, for 0 < ε 1, we study the corresponding perturbed boundary Riemann problem (PBRP for short):
ε U + F (U ε )x = (aε )0 G(U ε ), tε U (x, 0) = UR , ε ε U (x, 0) = φ (x), a(0, t ) = aB , v(0, t ) = vB ,
(x, t ) ∈ (0, ∞) × (0, ∞), x > ε, 0 ≤ x ≤ ε, t > 0,
(3.2)
where ε
a (x) =
ηε (x), aR ,
0 ≤ x ≤ ε, x > ε,
(3.3)
and ηε (x) is a monotone function connecting aB , aR at x = 0 and x = ε respectively. Also φ ε (x) = (ηε (x), v0ε (x), w0ε (x)) is also a monotone function connecting states U∗ := (aB , vB , wB ) at x = 0 (with wB needed to be decided by Lax’s method) and UR at x = ε . Then, in view of Definition 2.2, the measurable solution of BRP (3.1) is defined as U (x, t ) := limε→0 U ε (x, t ) where U ε (x, t ) is a solution of (3.2). The main topic of this section is to use Lax’s method to prove the existence of weak solutions to (3.2) and (3.3). To start, we notice that the second characteristic field is either genuinely nonlinear or linearly degenerate. Then, by the vanishing of the source term in x > ε , we can use the elementary wave of homogeneous system to construct the solution of (3.2) in x > ε . Given a state U2 and a sufficiently small parameter σ2 , we define the mapping T2 : R × Ω → R3 by T2 (σ2 ; U2 ) = UR ,
(3.4)
where UR is the state which can be connected by a 2-wave curve starting at U2 with (signed) wave strength σ2 (Section 2). Note that the 2-wave curve can be parameterized as a C 2 curve starting at U2 : UR = U2 + σ2 R2 (U2 ) + O(σ22 ).
(3.5)
In addition, we see that T2 is at least C in arguments σ2 , U2 . Next, we study the time-independent solution corresponding to λ0 (U ) = 0. Followed by the results in Section 2, given a state UB ≡ (aB , vB , wB ) nearby U2 , we define the mapping T0ε : R × Ω → R3 by 2
T0ε (σ0 ; UB ) = U2 ,
(3.6)
where U2 is the state which can be connected by a 0-wave curve starting at UB with (signed) wave strength σ0 . Also, the 0-wave curve can be parameterized as U2 = UB + σ0 R0 (UB ) + O(σ02 ).
(3.7)
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Now, we define the composite mapping TBε : R × Ω → R3 by TBε (σ2 , σ0 ; UB ) := T2 (σ2 ; T0ε (σ0 ; UB )).
(3.8)
ε
Note that TB is at least a C 2 mapping of σ0 , σ2 . Then by (3.4)–(3.8) we obtain TBε (σ2 , σ0 ; UB ) = UB + σ0 R0 (UB ) + σ2 R2 (UB ) + O(1)(σ0 + σ2 )2
= UR .
(3.9)
In addition, we have TBε (0, 0; UB ) = UB ,
∂ TBε = R0 (UB ), ∂σ0 (0,0;UB )
∂ TBε = R2 (UB ). ∂σ2 (0,0;UB )
(3.10)
Next, we consider the mapping H given by H (σ0 , σ2 ) := [UR ]1,2 − [UB ]1,2 ,
(3.11)
where [·]1,2 denotes the 2-vector consisting of the first and second components of vector [·]. Then, by (3.9)–(3.11) we obtain that H (0, 0) = (0, 0) and the Jacobian matrix of H at (0, 0) is DH (0, 0) = [(R0 (UB ), R2 (UB ))]1,2 =
1 [r0 (UB )]1
0 . [r2 (UB )]1
(3.12)
To apply Lax’s method, we need the following lemma. Lemma 3.1. Under condition ( A2 ), the Jacobian matrix DH in (3.12) has rank 2, which means that it is non-singular. Proof. Assume that DH (0, 0) is singular, i.e., [r2 (UB )]1 = 0, so λ2 (UB )[r2 (UB )]1 = 0. Next, since the right eigenvector r2 (UB ) of Df is a nonzero vector, we have [r2 (UB )]2 6= 0. On the other hand, by Df (UB ) · r2 (UB ) = λ2 (UB )r2 (UB ), ∂f
it implies ∂w1 (UB )[r2 (UB )]2 = λ2 (UB )[r2 (UB )]1 = 0. Then, by Condition (A2 ) we have the contradiction. The proof is complete. Thus, by Lemma 3.1 and the inverse function theorem, there exist neighborhoods N0 of (0, 0) and NB of [UB ]1,2 such that H is a diffeomorphism from N0 to NB . It follows that we can solve for σ0 , σ2 in terms of UR and UB = (aB , vB , wB ) when |(aB , vB ) − [UR ]1,2 | is sufficiently small. Next, we decide the value of wB when UR = (aR , vR , wR ) ∈ NB is given. In view of the third equation of (3.9), we have
wR = wB + σ0 R30 (UB ) + σ2 R32 (UB ) + O(1)(σ0 + σ2 )2 ≡ η(wB ),
(3.13)
where R30 , R32 are the third component of R0 , R2 respectively, and σ0 , σ2 are determined by previous analysis. We see that dη dw B
= 1 + σ0
dR30 dw B
+ σ2
dR32 dwB
+ O(1)(σ0 + σ2 )2 ,
(3.14)
dη
which implies that dw |σ0 =σ2 =0 = 1. Then, by inverse function theorem there exists a unique wB satisfying (3.13) when B σ0 , σ2 are sufficiently small. Therefore, we obtain the following theorem regarding the existence of solutions to the PBRP (3.2). Theorem 3.2. Consider the PBRP (3.2) where UR ∈ Ω ⊂ R3 and |(aB , vB ) − (aR , vR )| is sufficiently small. Then, under conditions (A1 ), (A2 ), there exists a solution U ε (x, t ) of (3.2) consisting of at most three constant states separated by a 2-wave issued from (ε, 0) and a smooth standing wave in 0 ≤ x ≤ ε . In addition, there is a unique wB such that U ε (0, t ) = (aB , vB , wB ) for t > 0 (Fig. 3). We notice that the structure of U ε (x, t ) of (3.2) depends on the choice of aε . However, the wave curves of U ε (x, t ) on the phase plane are uniquely determined. It follows that the constant states in U ε (x, t ) are independent of the choice of aε . Therefore, by letting ε → 0 in U ε (x, t ), we obtain the existence and uniqueness of measurable solution to (3.1). Corollary 3.3. The BRP (3.1) has a unique measurable solution U (x, t ) := limε→0 U ε (x, t ) consisting of at most three constant states separated by a 2-wave and a standing wave discontinuity. Furthermore, there exists a unique constant wB such that U (0, t ) = (aB , vB , wB ) for t > 0.
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641
Fig. 3. Solution of the PBRP in the phase plane.
4. Generalized Glimm scheme and its stability In this section we construct the generalized Glimm scheme for IBVP (1.3) and study the stability of scheme. To start, we partition quarter plane {(x, t ) ∈ [0, ∞) × [0, ∞)} into xk = k∆x,
ti = i∆t , i, k = 0, 1, 2, . . . ,
where ∆x and ∆t are sufficiently small. Next, given 0 < ε 1, we assume that ∆x, ∆t satisfy the generalized C-F-L condition
∆x/∆t > (1 − ε)−1 sup{|λi (U )||i = 1, 2}.
(4.1)
Ω
Condition (4.1) ensures that waves in the approximate solutions cannot interact in the same time level. We also define
∆˜x := (1 + ε)−1 ∆x.
(4.2)
Next, we approximate the initial and boundary data in (1.3) as follows.
(I)0 If x ≥ x1 and k > 0 is even, then initial data U0ε (x) is chosen by 0 xk−1 ≤ x < xk − ε ∆˜x, Uk−1 , ε U0 (x) = Ψk0 (x), |x − xk | ≤ ε ∆˜x, 0 Uk+1 , xk + ε ∆˜x < x ≤ xk+1 ,
(4.3)
where Uk0−1 := U0 (xk−1 ) and Ψk0 (x) = (aε0 (x), v0ε (x), w0ε (x)) is a smooth monotone function connecting Uk0−1 and Uk0+1 . (II)0 If 0 ≤ x ≤ x1 , then the initial and boundary data are chosen by U0ε (x) =
ΨB0 (x), U10 ,
v (t ) = vB (∆t /2), 0 B
0 ≤ x ≤ ε ∆˜x, ε ∆˜x < x ≤ x1 ,
(4.4)
0 < t < t1 ,
where U10 is given in Case (I)0 and ΨB0 (x) is a smooth monotone function connecting (aB , vB (0), w0 (0)) and U10 . Then, for such initial and boundary data we construct the approximate solutions to a set of PRPs and a PBRP in the 0-th time step {(x, t ) ∈ [0, ∞) × [0, ∆t )}. Let U ε,0 (x, t ) denote the approximate solution obtained by solving PRPs and the PBRP with (4.3) and (4.4) in the 0-th time step. Then initial data U1ε (x, t1 ) and boundary data vB1 (t ) in the first time step are chosen by the random choice method described in the following.
(I)1 For x ≥ x1 , U1ε (x, t1 ) is chosen by 1 xk−1 ≤ x < xk − ε ∆˜x, Uk−1 , ε 1 U1 (x, t1 ) = Ψk (x), |x − xk | ≤ ε ∆˜x, 1 Uk+1 , xk + ε ∆˜x < x ≤ xk+1 ,
k = 2, 4, 6, . . . ,
(4.5)
where Uk1−1 := U ε,0 (xk−1 + θ1 ∆˜x, t1 ), θ1 ∈ (−1, 1) is a random number, and Ψk1 (x) = (aε1 (x), v1ε (x), w1ε (x)) is a smooth monotone function connecting Uk1−1 and Uk1+1 .
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(II)1 If 0 ≤ x ≤ x1 , then the initial and boundary data are chosen by 1 ΨB (x), 0 ≤ x ≤ ε ∆˜x, U1ε (x, t1 ) = U11 , ε ∆˜x < x ≤ x1 , vB1 (t ) = vB (3∆t /2),
(4.6)
t1 < t < t2 ,
where ΨB1 (x) is a smooth monotone function connecting (aB , vB (t1 ), w ε,0 (0, t1− )) and U11 . Again, we solve for a set of PRPs and a PBRP in the first time step with (4.5) and (4.6). We repeat this process for each time step, and let U ε,i−1 (x, t ) denote the approximate solution by this process in the (i − 1)-th time step, then the initial and boundary data in the i-th time step will be chosen as follows:
(I)i For x ≥ x1 , Uiε (x, ti ) is chosen by i Uk−1 , xk−1 ≤ x < xk − ε ∆˜x, Uiε (x, ti ) = Ψki (x), |x − xk | ≤ ε ∆˜x, i Uk+1 , xk + ε ∆˜x < x ≤ xk+1 ,
k = 2, 4, 6, . . . ,
(4.7)
where Uki −1 = U ε,i−1 (xk−1 + θi ∆˜x, ti ), θi ∈ (−1, 1), i ∈ N, is a random number, and Ψki (x) = (aεi (x), viε (x), wiε (x)) is a smooth monotone function connecting Uki −1 , Uki +1 . (II)i If 0 ≤ x ≤ x1 , then we chose
ΨBi (x), 0 ≤ x ≤ ε ∆˜x, U1i , ε∆˜x < x ≤ x1 , 1 ∆t , ti < t < ti+1 , vBi (t ) = vB i+
Uiε (x, ti ) =
(4.8)
2
(x) connects (aB , vB (ti ), wε,i−1 (0, ti− )) and U1i . We then obtain an approximate solution Uθε,∆x (x, t ) to IBVP where (1.3) by a generalized Glimm scheme with a random sequence θ := {θ1 , θ2 , . . .} in x ≥ 0, t ≥ 0. Similarly, we can construct another approximate solution, denoted by Uθ ,∆x , to (1.3) by taking ε → 0 in Uθε,∆x . ΨBi
To obtain the compactness of the approximate solutions, we study the stability of this scheme. By the results of [5], we need to show that the total variation of approximate solution stays bounded uniformly. To achieve this, we study the wave interaction estimates. First we consider the interaction of waves near the boundary. Here we emphasize that, the 0-th wave field is characteristic to the boundary. So the interaction of waves is more complicated than the case in [8]. However, we show in the following that the results in [8] can be extended to our problem. First, we state the following lemma which is the extension of results in [8]. Lemma 4.1. (a) (Elementary wave interaction) Let U be a constant state which is connected to constant state V by an i-wave with strength αi , and V can be connected to another constant state W by a j-wave with strength βj . On the other hand, suppose that U can also be connected to V 0 by a j-wave with strength βj and V 0 can be connected to W 0 by an i-wave with strength αi . Then there exists a continuous function η(αi , βj ) such that W − W 0 = η(αi , βj )αi βj .
(4.9)
(b) (Combining waves of the same family) Let U be connected to V by αi and V connected to W by βi , waves of the same family. If U is connected to W 0 by an i-wave of strength αi + βi , then there is a continuous function ζ (αi , βi ) such that W − W0 =
0,
ζ (αi , βi )αi βi ,
if αi and βi are both rarefaction waves or both smooth standing waves, otherwise.
(4.10)
Since the standing waves are also C 2 (by the wave construction in the previous section), the proof of Lemma 4.1 is similar to the one given in [8], and so we omit it. Now, we study wave interactions near the boundary. Here we adopt the notations in [3], and the wave strengths of elementary waves and standing waves are defined as in [15,4,3]. Such as let (UL , UR ) denote the solution of the perturbed Riemann problem [UL , UR ] consisting of constant states UL ≡ U0 , U1 , U2 , U3 ≡ UR with the parametrization Tk (εk ; Uk ) = Uk+1 , then the solution (UL , UR ) can be written as
(UL , UR ) ≡ (U0 , U1 , U2 , U3 )/(ε0 , ε1 , ε2 ). The parameter εi is called the wave strength of the i-wave that connects the states Ui and Ui+1 . We assume that the solution of the PBRP on the k-th time step is (Uk , UM ) ≡ (Uk , U2 , UM )/(α0k , α2k ) where Uk := (aB , vk , wk ), and the 1-wave with strength β1 in the solution of nearby PRP on the k-th time step is (UM , UR ) ≡ (UM , UR )/(β1 ). Also, let the solution of the
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Fig. 4. Approximate solutions of the PRPs and PBRPs.
Fig. 5. Interactions of waves with two interchanges of waves.
PBRP on the (k + 1)-th time step be (Uk+1 , UR ) ≡ (Uk+1 , U20 , UR )/(γ0k+1 , γ2k+1 ) where Uk+1 := (aB , vk+1 , wk+1 ), see Fig. 4. Then, the method of obtaining the wave interaction estimates near the boundary is divided into the following steps. Step I. By the results in Section 3, we construct the solution of the PBRP [U¯ k , UR ] where U¯ k := (aB , vk , w ¯ k ) with w ¯ k uniquely decided. Let (U¯ k , UR ) ≡ (U¯ k , U¯ 2 , UR )/(γ0k , γ2k ) be the solution of [U¯ k , UR ]. Step II. We estimate the difference of total wave strengths between two solutions (U¯ k , UR ) and (Uk+1 , UR ). Step III. Similarly, we estimate the difference of total wave strengths between (U¯ k , UR ) and (Uk , UR ). Then, by the triangle inequality we can obtain the wave interaction estimates near the boundary. To start at Step I, by the results in Section 3, we are able to construct the solution (U¯ k , UR ) ≡ (U¯ k , U¯ 2 , UR )/(γ0k , γ2k ) of [U¯ k , UR ]. Next, to achieve the goal of Step II, we use the differentiability of H given in (3.11) to obtain
|γjk+1 − γjk | ≤ C |vk+1 − vk |,
j = 0, 2.
(4.11)
For Step III, we notice that the map UR = Ti (τi ; UL )
is a smooth function of (τi ; UL ),
(4.12)
where UR is the state which can be connected to UL on the right by an i-wave, i 6= 0, with (signed) strength τi (Section 3). Then, by the results of [8], we first study the case where β1 interacts with α2k , α0k in order (see Fig. 5). We observe that 0 the states U10 , UM and UR0 in Fig. 5 can be determined by Lemma 4.1-(a) with wave strengths β1 , α0k , and α2k . Furthermore, following by (4.9) and (4.12) and the triangle inequality, we obtain that
|UR0 − UR | ≤ C (|α0k β1 | + |α2k β1 |),
(4.13)
where UR is connected to Uk on the right by β1 , α and α On the other hand, it is easy to see that k 0
0
k 2.
|[U10 ]1,2 − (aB , vk )| ≤ |U10 − Uk | ≤ C |β1 |,
(4.14)
where U1 := (aB , vk , wk ) is connected to Uk on the right by β1 , and [U1 ]1,2 stands for the first two components of U1 . Next, we solve the PBRP [(aB , vk , ∗), U10 ]. Again, by the results in Section 3, the solution of [(aB , vk , ∗), U10 ] exists. Let (Uk0 , U10 ) ≡ (UK0 , U10 )/(γ20 ) denote the solution of [Uk0 , U10 ] where Uk0 ≡ (aB , vk , wk0 ) for some wk0 . Then we study the case that γ20 interchanges with α0k and then combines with α2k (see Fig. 6). By the differentiability of H and (4.14), it follows that γ20 satisfies 0
0
0
0
0
|γ20 | ≤ C |(aB , vk ) − [U10 ]1,2 | ≤ C |β1 |.
(4.15)
Therefore, by (4.9), (4.10) and (4.12) and the triangle inequality, we obtain
|UR00 − UR0 | ≤ C (|α0k γ20 | + |α2k γ20 |) ≤ C (|γ20 |),
(4.16)
where UR is connected to Uk on the right by waves α and γ2 + α Here we used the fact that wave strengths α and α2k are bounded. It follows from (4.15) and (4.16) that 00
|UR00 − UR0 | ≤ C |β1 |.
0
k 0
0
k 2.
k 0
(4.17)
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Fig. 6. Interchange and combination of waves.
Fig. 7. Approximate solution of IBVP and two mesh curves Ji , Ji+1 .
In view of (U¯ k , UR ) ≡ (U¯ k , U¯ 2 , UR )/(γ0k , γ2k ) in Step I, and by (4.13) and (4.17) and the differentiability of H, we obtain that {γjk | j = 0, 2} satisfies
|γjk − (γj0 + αjk )| ≤ C |[UR ]1,2 − [UR00 ]1,2 | ≤ C (|UR − UR0 | + |UR0 − UR00 |) ≤ C (|α0k β1 | + |α2k β1 | + |β1 |),
(4.18)
where γ00 := 0. Finally, by (4.11), (4.15) and (4.18), for j = 0, 2 we have
|γjk+1 − (αjk + β1 )| ≤ |γjk − (γj0 + αjk )| + |γjk+1 − γjk | + |γj0 | + |β1 | ≤ C (|α0k β1 | + |α2k β1 | + |β1 | + |vk+1 − vk |). Therefore, by previous analysis, we have the following theorem regarding the wave interaction estimates. Theorem 4.2. Let (Uk , UM ) ≡ (Uk , U2 , UM )/(α0k , α2k ), (Uk+1 , UR ) ≡ (Uk+1 , U20 , UR )/(γ0k+1 , γ2k+1 ) be the solutions of the PBRPs in Uθε,∆x (x, t ) on the k-th and (k + 1)th time steps respectively. Also, let (UM , UR ) ≡ (UM , UR )/(β1 ) be the 1-wave in the solution of the PRP next to (Uk , UM ) on the k-th time step. Then there exists a constant C such that
|γjk+1 − (αjk + β1 )| ≤ C (|α0k β1 | + |α2k β1 | + |β1 | + |vk+1 − vk |),
j = 0, 2.
(4.19)
Next, we study the wave interaction estimates for the waves away from the boundary. First, we refer the readers to [3] for the definitions of mesh points, mesh curves, immediate successors and approaching of waves in the Glimm scheme. Given a diamond region ∆, which is enclosed by mesh curves I, J and is away from the boundary, we let {α ≡ (α1 , α0 , α2 ), β ≡ (β1 , β0 , β2 )} be the incoming waves of ∆. Note that α , β are waves from two different PRPs across the lower-half of ∆. On the other hand, we let γ ≡ (γ1 , γ0 , γ2 ) be the outgoing waves of ∆, which means that γ is an entire perturbed Riemann solution across the upper-half of ∆. Then, by the results of [5,15], we have
γi = αi + βi + O(1)|α||β|,
i = 0, 1, 2.
(4.20)
When the waves approach, by (4.20), we obtain that
γi = αi + βi + O(1)D(α, β), as |α| + |β| → 0, i = 0, 1, 2. P Here D(α, β) = |αi ||βj | whose sum is over all pairs of i-wave and j-wave that are approaching.
(4.21)
By Theorem 4.2 and (4.20) and (4.21), we are able to show that the total variation of approximate solution Uθε,∆x (x, t ) is uniformly bounded, which will proceed as follows. Let {θi : i = 0, 1, 2, . . .} be a equi-distributed random sequence of numbers in (−1, 1) with θ0 = 0. Also let pk−1,i = (xk−1 + θi ∆˜x, ti ), i = 0, 1, 2, . . ., k = 2, 4, 6, . . ., p0,n = (0, (n + 21 )∆t ), n = 0, 1, 2, . . ., be the mesh points. Then we can connect {pk,i } to get a set of mesh curves and diamond regions. We notice that, every mesh curve in our case consists of some portion of the boundary x = 0, moreover, the diamond regions near the boundary x = 0 are triangle regions, see Fig. 7.
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Next, due to the terms |β1 |, |vk+1 − vk | appear in the RHS of (4.19), we define the following functionals on any mesh curve Ji : L(Ji ) :=
X
|α0 | +
X
|α2 | + K1
X
|vk − vk−1 | + K1
X
|α1 |,
(4.22)
k∈Wi
X {|αj ||αj0 | : αj , αj0 cross Ji and approach}, (4.23) P where constant K1 > 1 will be decided later. Here |αj |, j = 0, 1, 2 are the sum over all αj crossing Ji , and Wi := {k : (0, (k + 1/2)∆t ) ∈ Ji }. Note that the functional L in (4.22) can be considered as an equivalent norm for the total variations of Uθε,∆x (x, t ) on mesh curves. Next, we define the Glimm functional of Uθε,∆x (x, t ) on the mesh curve Ji by Q (Ji ) :=
F (Ji ) := L(Ji ) + K2 Q (Ji ),
(4.24)
where K2 > 1 is a sufficiently large constant. Then, by the results in [5], the non-increasing of the Glimm functional F ε ε implies the global boundedness of Uθ, ∆x (x, t ) and T .V .[Uθ,∆x (x, t )]. So we show the non-increasing of the Glimm functional by induction, which is given as follows. Let J0 be the mesh curve located between t = 0 and t = t1 for x > 0, and includes the half-ray {x = 0, t ≥ ∆t /2}. Since T .V .[U0 ], T .V .[vB ] are sufficiently small, we have F (J0 ) ≤ L(J0 ) + K2 L(J0 )2
≤ O(1)(K1 + K12 K2 (T .V .[U0 ] + T .V .[vB ]))(T .V .[U0 ] + T .V .[vB ]) ≤ O(1)K1 (T .V .[U0 ] + T .V .[vB ]).
(4.25)
Next, suppose that F (Ji ) is bounded and let Ji+1 is an immediate successor of Ji . Then, there exists a diamond or triangle region Γii+1 enclosed by Ji+1 , Ji . In the case where Γii+1 is a diamond region, we have already obtained the non-increasing of the Glimm functional by the results in [15], so we only consider the case where Γii+1 is a triangle region where Ji+1 , Ji only differ on mesh point p0,i−1 (Fig. 6). Followed by (4.22)–(4.24) and Theorem 4.2, we obtain F (Ji+1 ) − F (Ji ) = |γ0 | + |γ2 | − |α0 | − |α2 | − K1 |β1 | − K1 |vi − vi−1 |
+ K2
nX
|αj0 |(|γ0 | + |γ2 | − |α0 | − |α2 |) −
X
o |αj0 ||β1 | − K2 (|α0 ||β1 | + |α2 ||β1 |),
(4.26)
where the sums in (4.26) are overall approaching waves {αj0 } which are on the right of β1 . It follows form (4.19) and (4.26) that F (Ji+1 ) − F (Ji ) ≤ O(1)C (|α0 β1 | + |α2 β1 | + |β1 | + |vi − vi−1 |) − K1 |β1 | − K1 |vi − vi−1 |
X X + K2 O(1)C |αj0 |(|α0 β1 | + |α2 β1 | + |β1 | + |vi − vi−1 |) − |αj0 ||β1 | − K2 (|α0 ||β1 | + |α2 ||β1 |) ≤ (−K1 + O(1)C + O(1)CK2 F (Ji ))(|β1 | + |vi − vi−1 |) + (−K2 + O(1)C + O(1)CK2 F (Ji ))(|α0 β1 | + |α2 β1 |) ≤ (−K1 + O(1)C + O(1)CK2 F (J0 ))(|β1 | + |vi − vi−1 |) + (−K2 + O(1)C + O(1)CK2 F (J0 ))(|α0 β1 | + |α2 β1 |) ≤ 0, if we choose constants K1 , K2 ≥ O(1)2C and K2 F (J0 ) ≤ 1. We show the non-increasing of the Glimm functional F . We notice that Uθε,∆x and Uθ,∆x have the same value of the Glimm functional on each mesh curve since the structure of wave curves in both solutions is the same. Therefore, by the results of [5,15,24], it leads to the following theorem regarding the compactness of approximate solutions. Theorem 4.3. Suppose that the total variations of U0 (x), vB (t ) are small. Then approximate solution Uθε,∆x of (1.3) by the generalized Glimm scheme satisfies ε (1) T .V .[Uθ, ∆x (x, t )] ≤ C1 (T .V .[U0 ] + T .V .[vB ]), where constant C1 is independent of θ , ∆x and ε . ε ε (2) T .V .[Uθ, ∆x (x, tj )] + supx Uθ,∆x (x, tj ) ≤ C2 (T .V .[U0 ] + T .V .[vB ]), where constant C2 is independent of θ , ∆x and ε . ε ε ∗ ∗∗ (3) kUθ,∆x (·, t ) − Uθ,∆x (·, t )kL1 ≤ C3 (|t ∗ − t ∗∗ | + ∆t ), where constant C3 is independent of θ , ∆x and ε . ε Letting ε → 0 to {Uθ, ∆x } in Theorem 4.3, we obtain the following result.
Corollary 4.4. Suppose that the total variations of U0 (x), vB (t ) are sufficiently small. Let Uθε,∆x (x, t ) be the approximate solution of (1.3) by the generalized Glimm scheme, and Uθ,∆x (x, t ) := limε→0 Uθε,∆x (x, t ). Then (1) T .V .[Uθ,∆x (x, t )] ≤ C1 (T .V .[U0 ] + T .V .[vB ]), where constant C1 is independent of θ and ∆x. (2) T .V .[Uθ,∆x (x, tj )] + supx Uθ,∆x (x, tj ) ≤ C2 (T .V .[U0 ] + T .V .[vB ]), where constant C2 is independent of θ and ∆x.
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(3) kUθ ,∆x (·, t ∗ ) − Uθ,∆x (·, t ∗∗ )kL1 ≤ K3 (|t ∗ − t ∗∗ | + ∆t ), where constant K3 is independent of θ and ∆x. Thus, by Theorem 4.3, Corollary 4.4 and the results in [5,3], we obtain the convergence of {Uθε,∆x } and {Uθ ,∆x }. ε Theorem 4.5. Given 0 < ε 1. Let {Uθ, ∆x } be a sequence of approximate solutions to (1.3) by the generalized Glimm scheme ε and Uθ,∆x := limε→0 Uθ,∆x . Then there exists a subsequence of {Uθε,∆x } (resp., {Uθ ,∆x }), still denoted by {Uθε,∆x } (resp., {Uθ ,∆x }), which converges to some measurable function U ε (x, t ) := (aε (x), v ε (x, t ), w ε (x, t )) (resp., U (x, t ) := (a(x), v(x, t ), w(x, t ))) in L1loc . In addition, for every continuous function F , we have F (Uθε,∆x ) → F (U ε (x, t )) (resp., F (Uθ ,∆xi ) → F (U (x, t ))) in L1loc .
Furthermore, U ε (x, t ) → U (x, t ), F (U ε (x, t )) → F (U (x, t )) in L1loc as ε → 0.
Proof. We only show kU ε − U kL1 → 0 as ε → 0. For the other proof, we refer the readers to [3]. We see that loc
ε kUθ, ∆x
− Uθ,∆x kL1 = O(1)
O(1)(∆t )−1
X
loc
m=1,3,...
O(1)(∆t )−1
X
Z
tn+1 tn
n =0
xm+1
X Z
tn
n=0
= O(1)(ε∆x)
tn+1
Z
xm−1
! |Uθε,∆x
− Uθ ,∆x |dx dt
(osc [Uθε,∆x (x, t )|[xm−1 ,xm+1 ] ])dt
X
m=1,3,...
= O(1)(ε∆x)(TV [U0 ] + TV [vB ]), where O(1) is independent of θ , ∆x and ε . Therefore, by the fact that kU ε − Uθε,∆x kL1 , kUθ ,∆x − U kL1 → 0 as ∆x → 0 and loc
the triangle inequality, we obtain kU ε − U kL1 → 0 as ∆x, ε → 0. The proof is complete. loc
loc
By the results in Section 3, we notice that ε i |vθ, ∆x (0, t ) − vB (t )| ≤ |vB (t ) − vB (t )| ≤ O (1)∆x, ε for any t > 0, i ∈ N ∪ {0}. This implies that vθ, ∆x will match boundary data vB (t ) as ∆x tends to 0.
5. Weak convergence of the residuals In Section 4 we obtained the compactness of approximate solutions of (1.3) by showing the stability of the generalized Glimm scheme. In this section we establish the global existence of weak solutions to (1.3) by showing the weak convergence ε of the residual. To start, let {Uθ, ∆x } denote a sequence of approximate solutions of (1.3) by the generalized Glimm scheme, ε ∞ and U (x, t ) denote the limit of Uθ, ∆x as ε , ∆x approach 0. Then, for E := [0, ∞) × [0, ∞) and test function φ ∈ C0 (E ), we recall the residual R(U , φ, E ) given in Definition 1.1: R(U , φ, E ) :=
Z Z
(U φt + F (U )φx + a0 G(U )φ)dxdt +
∞
Z
E
U0 (x)φ(x, 0)dx +
0
Z
∞
F (U (0, t ))φ(0, t )dt .
0
ε We wish to show that R(Uθ, ∆x , φ, E ) approaches 0 in the weak sense as ε , ∆x tend to 0. Then, following by the results of [24], we obtain the global existence of weak solutions to problem (1.3). We define [
Γi := {(x, t )|0 ≤ x ≤ x1 , ti ≤ t < ti+1 },
Γi ,
Γ :=
i∈N∪{0}
ε RΓi (Uθ, ∆x , φ) :=
Z Z Γi
ε,i
ε,i
ε,i
ε,i
Uθ,∆x φt + F (Uθ,∆x )φx + (aθ ,∆x )0 G(Uθ ,∆x )φ dxdt ,
ε,i Uθ,∆x
ε ε = Uθ, ∆x |Γi . Then, by the construction of Uθ ,∆x and the divergence theorem, we obtain t =t − x=x1 t =t − Z ε∆˜x Z x1 Z t− i+1 i+1 i+1 ε,i ε,i ε, i Γi ε R (Uθ,∆x , φ) = (Uθ,∆x φ) dx + (Uθ ,∆x φ) dx + F (Uθ ,∆x )φ dt . + 0 ε ∆˜x t t = ti i
where φ ∈ C0 (E ) and ∞
t = ti
(5.1)
x =0
In addition, we see that ε∆˜x
Z 0
t =t − i+1
ε,i (Uθ, ∆x φ)
Z
ε ∆˜x
dx = 0
+
t = ti
Z
ε ∆˜x
+ 0
Z ≤ 0
ε,i ε,i − [(Uθ, ∆x φ)(x, ti+1 ) − (Uθ ,∆x φ)(x, ti )]dx
ε ∆˜x
ε,i ε,i + [(Uθ, ∆x φ)(x, ti ) − (Uθ ,∆x φ)(x, ti )]dx
ε,i ε,i − [(Uθ, ∆x φ)(x, ti+1 ) − (Uθ ,∆x φ)(x, ti )]dx
+ O(1)kφk∞ (ε∆˜x) osc [Uθε,,∆i−x1 |[0,x2 ] ] + osc [vB (t )|[ti − 1 ∆t ,ti ] ] + osc [Uθε,,∆i x |[0,x1 ] ] , 2
(5.2)
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647
where osc [U ] denotes the oscillation of U. Summing all RΓi and by (5.1) and (5.2), we obtain ε RΓ (Uθ, ∆x , φ) = −
∞ Z X
x1 0
i =1
− ti+1
∞ Z X
+
ε,i bUθ, ∆x c(x)φ(x, ti )dx −
ti
i =0
x=x1
ε,i F (Uθ,∆x )φ
x1
Z 0
ε,0
Uθ ,∆x (x, 0)φ(x, 0)dx
dt + O(1)εkφk∞ ,
(5.3)
x =0
ε,i
− ε ε where bUθ,∆x c(x) := Uθ, ∆x (x, ti ) − Uθ,∆x (x, ti ). ε Next, we estimate the residual for {Uθ,∆x } away from the boundary. Set Γ 0 := E \ Γ . Then by a similar calculation, it leads to ∞ Z X
ε RΓ (Uθ, ∆x , φ) = − 0
∞ x1
i=1
− ti+1
∞ Z X
−
ε,i bUθ, ∆x c(x)φ(x, ti )dx −
ti
i=0
∞
Z
x1
ε,0
Uθ ,∆x (x, 0)φ(x, 0)dx
ε,i
F (Uθ,∆x (x1 , t ))φ(x1 , t )dt + O(1)εkφk∞ T .V .[U0 ].
(5.4)
Thus, by (5.3) and (5.4) and the generalized C-F-L condition (4.1), we obtain ε R(Uθ, ∆x , φ, E )
=R
Γ
ε (Uθ, ∆x , φ)
+R
Γ0
ε (Uθ, ∆x , φ)
∞
Z
U0 (x)φ(x, 0)dx +
+
∞
Z
0
F (U (0, t ))φ(0, t )dt 0
= O(1)εkφk∞ + O(1)∆xkφk∞ (T .V .[U0 ] + T .V .[vB ]) −
∞ Z X 0
i =1
∞ Z X
−
i =0
− ti+1
ti
ε,i F (Uθ,∆x (0, t ))φ(0, t )dt
∞
bUθε,,∆i x c(x)φ(x, ti )dx
∞
Z
F (U (0, t ))φ(0, t )dt
+ 0
= O(1)εkφk∞ + O(1)∆xkφk∞ (T .V .[U0 ] + T .V .[vB ]) −
∞ Z X i =1
∞ 0
bUθε,,∆i x c(x)φ(x, ti )dx.
(5.5)
Now we show that the last term on the RHS of (5.5) converges in the order of ε 1/2 as ∆x → 0 in the probability space Φ of random choice numbers. To start, let J (θ , ∆x, φ) := i
∞
Z 0
i bUθ, ∆x c(x)φ(x, ti )dx,
and Jεi (θ , ∆x, φ) :=
∞
Z 0
ε,i bUθ, ∆x c(x)φ(x, ti )dx,
− i where bUθ, ∆x c(x) := Uθ,∆x (x, ti ) − Uθ,∆x (x, ti ) and i is a nonnegative integer. Then, for i 6= j we claim that
hJ i (θ , ∆x, φ), J j (θ , ∆x, φ)i ≡
Z Z
J i (θ , ∆x, φ)J j (θ , ∆x, φ)dθj
Y
dθl
l6=j
= 0.
(5.6)
Here h, i is the L inner product with respect to random number θ ∈ Φ , and φ is a function with compact support and is piecewise constant in each segment [xm−1 , xm+1 ] × {ti }, m is positive and odd. Indeed, we have 2
Z Φ
J j (θ , ∆x, φ)dθj =
Z
xm+1
Z
X Φ m=1,3,5,...
m=1,3,5,...
= 0,
Φ
Uθ,∆x (xm + θj ∆x, tj− ) − Uθ ,∆x (x, tj− ) dxdθj
xm−1
X nZ Z
=
xm+1
xm−1
Uθ,∆x (xm + θj ∆x, tj )dxdθj − −
Z
xm+1 xm−1
Z Φ
Uθ ,∆x (xm + θj ∆x, tj− )dθj dx
o (5.7)
which is enough to show that (5.6) holds. We notice that (5.6) is just an extension of Glimm’s results from the Cauchy problem to IBVP. Next, we have the following estimates for Jεi .
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J.M. Hong, Y.-C. Su / Nonlinear Analysis 72 (2010) 635–650
Lemma 5.1. For i, j = 1, 2, 3, . . ., we have
(i) |Jεi (θ ε , ∆x, φ)| = O(1)kφk∞ ∆x(T .V .[U0 ] + T .V .[vB ]), X |Jεi (θ ε , ∆x, φ)| = O(1)kφk∞ r (φ)(T .V .[U0 ] + T .V .[vB ]),
(5.8) (5.9)
i
where r (φ) is the diameter of support (φ). (ii) Given φ which has compact support and is piecewise constant on each segment [xm−1 , xm+1 ] × {ti }, m = 1, 3, 5, . . ., we have
hJεi (θ ε , ∆x, φ), Jεj (θ ε , ∆x, φ)i = O(1)(kφk∞ )2 · ε(∆x)2 ,
i 6= j.
(5.10)
Proof. First we show (i). For simplification, we let Jεi = Jεi (θ ε , ∆x, φ) and J j = J j (θ , ∆x, φ) and so on. Then for i ∈ N, we obtain
Z |Jε | =
∞
i
0
ε,i bUθ, ∆x c(x)φ(x, ti )dx
≤ kφk∞
x1
nZ 0
− ε ε |Uθ, ∆x (x, ti ) − Uθ,∆x (x, ti )|dx +
(
X Z m=2,4,...
xm+1 xm−1
|Uθε,∆x (x, ti ) − Uθε,∆x (x, ti− )|dx
o
) X
≡ kφk∞ I1 +
Im
.
m=2,4,...
To estimate I1 and Im for m = 2, 4, . . ., we observe that x1
Z
ε ε ˜ − |Uθ, ∆x (x, ti ) − Uθ,∆x (x1 + θi ∆x, ti )|dx +
I1 ≤ 0
x1
Z 0
− ε ε ˜ − |Uθ, ∆x (x1 + θi ∆x, ti ) − Uθ,∆x (x, ti )|dx
≤ ε∆˜x osc [Uθ,∆x (x, ti− )|[0,x2 ] ] + osc [vB (t )|[ti − 1 ∆t ,ti ] ] + (∆x)osc [Uθ ,∆x (x, ti− )|[0,x2 ] ], 2
and xm+1
Z Im ≤
xm−1
ε ε ˜ − |Uθ, ∆x (x, ti ) − Uθ,∆x (xm−1 + θi ∆x, ti )|dx +
Z
xm+1 xm−1
− ε |Uθε,∆x (xm−1 + θi ∆˜x, ti− ) − Uθ, ∆x (x, ti )|dx
≤ (2ε ∆˜x + ∆˜x)osc [Uθ,∆x (x, ti− )|[xm−1 ,xm+2 ] ] + (2∆x)osc [Uθ,∆x (x, ti− )|[xm−2 ,xm+1 ] ]. It follows that
|Jεi | = O(1)kφk∞ ∆x(T .V .[U0 ] + T .V .[vB ]) + (ε ∆˜x)osc [vB (t )|[ti − 1 ∆t ,ti ] ] , 2
= O(1)kφk∞ ∆x(T .V .[U0 ] + T .V .[vB ]). It consequently leads to
X
|Jεi | ≤ O(1)kφk∞ r (φ)(∆t )−1 (∆x)(T .V .[U0 ] + T .V .[vB ])
i
≤ O(1)kφk∞ r (φ)(T .V .[U0 ] + T .V .[vB ]), where r (φ) = diameter of support (φ). We prove (i). To show (ii), we first observe that
hJε , J i = i
j
Z
i
Jε
Z
J j dθj
Y
dθl = 0,
(5.11)
l6=j
for i 6= j since θ ∈Φ J j (θ , ∆x, φ)dθj = 0 in (5.7). Next, we have the following estimate that j
R
Z Z Φ
θj ∈Φ
|Jεj − J j |dθj
Y l6=j
dθl ≤
Z Z Φ
θj ∈ Φ
≤ kφk∞
∞
Z 0
Z Z Φ
Y ε,j j |bUθ, dθl ∆x c(x) − bUθ,∆x c(x)||φ(x, tj )|dx dθj l6=j
X Z
xm+1
θj ∈Φ m=1,3,... xm−1
ε |Uθ ,∆x (x, tj ) − Uθ,∆x (x, tj )|
Y − − ε + |Uθ, ( x , t ) − U ( x , t )| dxd θ dθl θ, ∆ x j ∆x j j l6=j
J.M. Hong, Y.-C. Su / Nonlinear Analysis 72 (2010) 635–650
≤ kφk∞
X
649
(2ε ∆˜x) T .V .[Uθε,∆x (x, tj )|[xm−1 ,xm+1 ] ] + T .V .[Uθ ,∆x (x, tj− )|[xm−1 ,xm+1 ] ]
m=1,3,...
≤ kφk∞ (2ε ∆˜x) T .V .[Uθε,∆x (·, tj )] + T .V .[Uθ ,∆x (·, tj− )]
≤ 4kφk∞ (ε∆x)(T .V .[U0 ] + T .V .[vB ]).
(5.12)
Therefore, by (5.11) and (5.12), for i 6= j we have
hJεi , Jεj i = hJεi , Jεj − J j i Z Z Y dθl ≤ kJεi k∞ |Jεj − J j |dθj l6=j
≤ O(1)(kφk∞ ) · ε(∆x) . 2
We complete the proof.
2
Thus, by Lemma 5.1 and the results in [15], we obtain the following theorem. ε Theorem 5.2 (Cf. [15]). Given 0 < ε 1, and let {Uθ, ∆x } be the sequence of approximate solutions to initial-boundary value problem (1.3) by the generalized Glimm scheme. Then, for 0 < ε 1, we can find a null set Nε ⊂ Φ and a subsequence {∆xi } → 0 such that for any θ ∈ Φ \ Nε and φ ∈ C0∞ , we have
X
Jεi (θ , ∆xi , φ) = O(1)ε 1/2
as ∆xi → 0.
i
According to Theorem 5.2, the last term on the RHS of (5.5) is of order ε 1/2 as subsequence ∆xi → 0 and θ ∈ Φ \ Nε . Now, we show the residual of U (x, t ) in Theorem 4.5 is equal to 0, which means that U (x, t ) is a global weak solution of (1.3). First, by the Lebesgue bounded convergence theorem, the residual of U (x, t ) can be estimated by R(U , φ, E ) =
ε lim R(Uθ, ∆xi , φ, E ) +
ε,∆xi →0
Z Z lim
ε,∆xi →0
t >0
a0 G(U ) − (aεθ ,∆xi )0 G(Uθε,∆xi ) dxdt ,
≡ K1 + K2 .
(5.13)
Therefore, by Theorem 5.2. and the results in [15], we obtain that K1 , K2 approach to 0 as ε , ∆xi → 0. We then establish the global existence of weak solutions of (1.3), which is stated in the following theorem. Main theorem. Consider the following initial-boundary value problem
a = 0, t ut + f (a, u)x = a0 g (a, u), (a, u)(x, 0) = (a(x), u0 (x)), v(0, t ) = vB (t ),
(x, t ) ∈ (0, ∞) × (0, ∞),
(5.14)
where u = (v, w) and f = (f1 , f2 ), g = (g1 , g2 ) are smooth functions of a, u satisfying (A1 ) and (A2 ). Also a(x) is a Lipschitz continuous function with small total variation, and u0 (x), vB (t ) ∈ L∞ have small total variations. Let U := (a, u), and {Uθε,∆x } be the sequence of approximate solutions constructed by the generalized Glimm scheme, then there exists a null set Nε in probability space Φ and a sequence {∆xi } such that for any θ ∈ Φ \ Nε , the measurable function U (x, t ) := limε,∆xi →0 Uθε,∆xi is a weak solution of (5.14). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
E. Isaacson, B. Temple, Nonlinear resonant in inhomogenous systems of conservation laws, Contemp. Math. 108 (1990). P.G. LeFloch, T.-P. Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. 5 (1993) 261–280. J. Smoller, Shock Waves and Reaction–Diffusion Equations, 2nd ed., Springer-Verlag, Berlin, New York, 1994. P.D. Lax, Hyperbolic system of conservation laws, II, Comm. Pure Appl. Math. 10 (1957) 537–566. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697–715. T. Nishida, J. Smoller, Mixed problems for nonlinear conservation laws, J. Differential Equations 23 (1977) 244–269. T.P. Liu, Initial-boundary value problems for gas dynamics, Arch. Ration. Mech. Anal. 64 (1977) 137–168. J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. dissertation, Stanford University, 1982. T.P. Liu, Quasilinear hyperbolic systems, Comm. Math. Phys. 68 (1979) 141–172. D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl. 4 (1997) 1–42. D. Amadori, R.M. Colombo, Continuous dependence for 2 × 2 conservation laws with boundary, J. Differential Equations 138 (1997) 229–266. D. Amadori, L. Gosse, G. Guerra, Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal. 162 (2002) 327–366. [13] D. Amadori, G. Guerra, Uniqueness and continuous dependence for systems of balance laws with dissipation, Nonlinear Anal. TMA 49 (2002) 987–1014. [14] C. Donadello, A. Marson, Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws, Nonlinear Differential Equations Appl. 14 (2007) 569–592.
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[15] J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by weaker than weak solutions of the Riemann problem, J. Differential Equations 222 (2006) 515–549. [16] J.M. Hong, B. Temple, A bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law, SIAM J. Appl. Math. 64 (3) (2004) 819–857. [17] E. Isaacson, B. Temple, Convergence of 2 × 2 Godunov method for a general resonant nonlinear balance laws, SIAM J. Appl. Math. 55 (1995) 625–640. [18] B. Temple, Global solution of the Cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws, Adv. Appl. Math. 3 (1982) 335–375. [19] P.G. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form, Comm. Partial Differential Equations 13 (1988) 669–727. [20] P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form, Institute for Mathematics and its Applications, Minneapolis, Preprint 593 (1989). [21] G. Dal Maso, P.G. LeFloch, F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995) 483–548. [22] C.M. Dafermos, L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and dissipation, Indiana U. Math. J. 31 (1982) 471–491. [23] J. Hong, P.G. LeFloch, A version of Glimm method based on generalized Riemann problems, J. Portugal Math. 64 (2007) 199–236. [24] J.M. Hong, C.-H. Hsu, Y.-C. Su, Global solutions for initial-boundary value problem of quasilinear wave equations, J. Differential Equations 245 (2008) 223–248. [25] Y. Chang, J.M. Hong, C.-H. Hsu, Globally Lipschitz continuous solutions to a class of quasilinear wave equations, J. Differential Equations 236 (2007) 504–531. [26] P. Goatin, P.G. LeFloch, The Riemann problem for a class of resonant nonlinear systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 881–902. [27] J.M. Hong, B. Temple, The generic solution of the Riemann problem in a neighborhood of a point of resonance for systems of nonlinear balance laws, Methods Appl. Anal. 10 (2) (2003) 279–294. [28] P.D. Lax, Hyperbolic system of conservation laws and mathematical theory of shock waves, in: Conf. Board Math. Sci., vol. 11, SIAM, Philadelpia, PA, 1973. [29] M. Luskin, B. Temple, The existence of a global weak solution to the non-linear waterhammer problem, Comm. Pure Appl. Math. 35 (1982) 697–735.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Generalized Glimm scheme to the initial boundary value problem of hyperbolic systems of balance laws John M. Hong a , Ying-Chin Su b,∗ a
Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
b
Department of Mathematics, Fu Jen Catholic University, Taipei County 24205, Taiwan
article
info
Article history: Received 4 May 2009 Accepted 2 July 2009 MSC: 35L60 35L65 35L67 Keywords: Hyperbolic systems of balance laws Riemann problem Boundary Riemann problem Lax’s method Generalized Glimm scheme
abstract In this paper we provide a generalized version of the Glimm scheme to establish the global existence of weak solutions to the initial-boundary value problem of 2 × 2 hyperbolic systems of conservation laws with source terms. We extend the methods in [J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. Dissertation, Stanford University, 1982; J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by ‘‘weaker than weak’’ solutions of the Riemann problem, J. Differential Equations 222 (2006) 515–549] to construct the approximate solutions of Riemann and boundary Riemann problems, which can be adopted as the building block of approximate solutions for our initial-boundary value problem. By extending the results in [J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697–715] and showing the weak convergence of residuals, we obtain stability and consistency of the scheme. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction We consider the following 2 × 2 nonlinear hyperbolic system of balance laws ut + f (a, u)x = a0 g (a, u),
(1.1)
where u = u(x, t ) = (v(x, t ), w(x, t )), f (a, u) = (f1 (a, u), f2 (a, u)), g (a, u) = (g1 (a, u), g2 (a, u)) are smooth functions of (a, u), a = a(x) is a Lipschitz continuous function of x which is of finite total variation and a0 = da . Following the lead by dx Isaacson–Temple [1] and LeFloch–Liu [2], we augment system (1.1) by adding the equation at = 0 and obtain the following 3 × 3 hyperbolic system of balance laws Ut + F (U )x = a0 G(U ),
(1.2)
where U = (a, v, w), F (U ) = (0, f (a, u)) and G(U ) = (0, g (a, u)). In this paper we investigate the global existence of weak solutions to the initial-boundary value problem of (1.2): Ut + F (U )x = a0 G(U ), U (x, 0) = U0 (x), v(0, t ) = vB (t ),
(
∗
(x, t ) ∈ (0, ∞) × (0, ∞), x ≥ 0, t ≥ 0,
Corresponding author. E-mail addresses: [email protected] (J.M. Hong), [email protected] (Y.-C. Su).
0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.07.003
(1.3)
636
J.M. Hong, Y.-C. Su / Nonlinear Analysis 72 (2010) 635–650
where U0 (x) ≡ (a(x), v0 (x), w0 (x)), vB (t ) are bounded functions with finite total variations and satisfy vB (0) = v0 (0). We assume that system (1.2) is strictly hyperbolic, that is, the eigenvalues λ0 = 0, λ1 and λ2 of Jacobian matrix DF (U ) are real and distinct. Moreover, each characteristic field of (1.2) is either genuinely nonlinear or linear degenerate [3]. We assume further that there is an open region Ω ⊂ R3 such that f , g satisfy
(A1 ) each component of ( ∂∂ uf )−1 (g − ∂ f1 (A2 ) ∂w (U ) 6= 0 for U ∈ Ω .
∂f )(U ) ∂a
is nonzero for U ∈ Ω ,
An important application of (1.1) is the one-dimensional compressible Euler equations in a variable area duct
ρt + (ρ u)x = −(a0 /a)ρ u, (ρ u)t + (ρ u2 + p(ρ))x = −(a0 /a)ρ u2 ,
(1.4)
where ρ , u and p(ρ) are the density, velocity and pressure of the fluid respectively, and a(x) represents the area of crosssection of the duct. We review some previous results to this topic. The entropy solutions to the Riemann problem of Ut + F (U )x = 0
(1.5)
was first studied by Lax [4,28]. The solutions are self-similar functions consisting of constant states separated by elementary waves (rarefaction waves, shock waves and contact discontinuities). To the Cauchy problem of (1.5), the global existence of weak solutions was established by Glimm [5] when initial data is uniformly bounded and of small total variation. For the initial-boundary value problem (IBVP for short), Nishida and Smoller [6] studied the piston and double piston problems and obtained the global existence of weak solutions. On the other hand, the IBVP of unsteady flow in gas dynamics was studied by Liu [7]. Moreover, the IBVP of (1.5) with arbitrary shape of boundary was first studied by Goodman [8]. He proved the global existence of weak solutions when the initial and boundary data satisfy the so-called smallness and non-degeneracy conditions. To the quasilinear hyperbolic system Ut + F (x, U )x = G(x, U ),
(1.6)
the global existence results for Cauchy problem was established by Liu [9]. We also remark that for the well-posedness theory for initial-boundary value problems, we refer the reader to [10–14]. On the other hand, the hyperbolic system of balance laws Ut + F (a(x), U )x = a0 G(a(x), U ),
(1.7)
was studied by Hong [15]. In [15], the global existence of weak solutions to Cauchy problem of (1.7) was obtained by the generalized Glimm scheme based on ‘‘weaker than weak’’ solutions of the Riemann problem. To the non-strictly hyperbolic (resonant) systems, we refer the readers to the results in [16–18,26,27]. Note that (1.7) can also be written as a system in non-conservative form. The global existence and stability of measure-valued solutions to non-conservative systems were studied by LeFloch [19,20], LeFloch–Liu [2] and Dal Maso–LeFloch–Murat [21]. To general quasilinear hyperbolic system Ut + F (U , x, t )x = G(U , x, t ),
(1.8)
the results for global existence of weak solutions to the Cauchy problem of (1.8) can be found in [22,23,29]. We notice that the results described above for the global existence of weak solutions to IBVPs are restricted to the hyperbolic systems without source terms. Recently, in the paper by Hong–Hsu–Su [24], the authors studied the following IBVP of quasilinear wave equation
utt − p(ρ(x), ux ) x = ρ(x)h(ρ(x), u, ux ), u(x, 0) = u0 (x), ut (x, 0) = w0 (x), ux (0, t ) = vB (t ),
(1.9)
or equivalently
vt − wx = 0, wt − p(ρ(x), v) x = ρ(x)h(ρ(x), u, v), w(x, 0) = w0 (x), v(0, t ) = vB (t ), u(x, 0) = u0 (x),
(1.10)
where v = ux and w = ut . By Glimm scheme and the perturbation technique for approximate solutions near the boundary, they obtained the global existence of weak solutions to (1.10) under the condition that the first and second derivatives of source h(ρ, u, v) to be uniformly bounded. In this paper, the generalized Glimm scheme we use enables us to relax the conditions of f and g given in [24] to establish the global existence results. More precisely, we only require condition (A2 ) on f to construct the approximate solutions near the boundary. Furthermore, since the solutions constructed for the boundary Riemann problem are exact solutions, we do not need to impose any condition on the second derivatives of source term. Now we give the definition for the weak solutions of IBVP (1.3), and state the main theorem of this paper.
J.M. Hong, Y.-C. Su / Nonlinear Analysis 72 (2010) 635–650
637
Definition 1.1. Let E := [0, ∞) × [0, ∞) and U := (a, v, w). Also for given test function φ ∈ C0∞ (E ), we define the residual R(U , φ, E ) by R(U , φ, E ) :=
Z Z
∞
Z
(U φt + F (U )φx + a G(U )φ)dxdt + 0
U0 (x)φ(x, 0)dx +
F (U (0, t ))φ(0, t )dt . 0
0
E
∞
Z
Then we say a measurable function U is a weak solution of initial-boundary value problem (1.3) if R(U , φ, E ) = 0 for all φ ∈ C0∞ (E ). Main theorem. Consider the following initial-boundary value problem ut + f (a, u)x = a0 g (a, u), (a, u)(x, 0) = (a(x), u0 (x)), v(0, t ) = vB (t ),
(
(x, t ) ∈ (0, ∞) × (0, ∞), (1.11)
where u = (v, w), a = a(x) is a Lipschitz continuous function with small total variation, and f = (f1 , f2 ), g = (g1 , g2 ) are smooth functions of a, u satisfying (A1 ) and (A2 ). Also u0 (x), vB (t ) ∈ L∞ have small total variations. Let U := (a, u) and let {Uθε,∆x } denote a sequence of approximate solutions constructed by a generalized Glimm scheme, then there exists a null set Nε in probability space Φ and a sequence {∆xi } such that for any θ ∈ Φ \ Nε , the measurable function U (x, t ) := limε,∆xi →0 Uθε,∆xi is a weak solution of (1.11). The paper is organized as follows. In Section 2 we study the perturbed Riemann problem of (1.3) away from the boundary x = 0 and review the results in [15,25] for the approximate solutions to a perturbed Riemann problem. In Section 3 we extend the results of [4] to construct the weak solutions to a boundary Riemann problem. In Section 4 we construct the generalized Glimm scheme and study its stability which leads to the compactness of an approximate solution. In Section 5, we establish the global existence results for IBVP (1.3) by showing the weak convergence of the residual given in Definition 1.1. 2. Riemann and perturbed Riemann problems In this section, for given positive x0 away from zero, we study the following Riemann problem Ut + F (U )x = a0 G(U ),
U (x, 0) = U0 (x) =
UL , UR ,
x < x0 , x > x0 ,
(2.1)
where U = (a, v, w), F (U ) = (0, f1 (U ), f2 (U )), G(U ) = (0, g1 (U ), g2 (U )), and UL = (aL , vL , wL ), UR = (aR , vR , wR ) ∈ Ω are two nearby constant states. Note that the source term a0 G(U ) is not defined in the sense of distributions, cf. [15]. To overcome this difficulty, we choose 0 < ε 1 and re-formulate the source term to study the following corresponding perturbed Riemann problem (PRP for short):
( ε
ε
ε 0
ε
Ut + F (U )x = (a ) G(U ),
ε
ε
U (x, 0) = U0 (x) =
UL ,
φ ε (x), UR ,
x < x0 − ε, |x − x0 | ≤ ε, x > x0 + ε.
(2.2)
Here aε (x) can be expressed as aL ,
( ε
a (x) =
ηε (x), aR ,
x < x0 − ε, |x − x0 | ≤ ε, x > x 0 + ε0 ,
(2.3)
where ηε (x) is a monotone function connecting aL , aR at x = x0 − ε and x = x0 + ε respectively, and φ ε (x) = (ηε (x), v0ε (x), w0ε (x)) is also a monotone function connecting UL , UR at x = x0 −ε and x = x0 +ε respectively. By the results of [15,24], the stability of U ε can be obtained with respect to the set of monotone functions {ηε (x)}, {φ ε (x)}, which allows us to define the measurable solution of (2.1) by the ε -limit of the weak solutions of (2.2). Now, we construct the approximate solutions of (2.2). Since the system in (2.1) is strictly hyperbolic in Ω , the eigenvalues {λi (U )|U ∈ Ω , i = 0, 1, 2} of Jacobian matrix DF (U ) are real and distinct. Without loss of generality, we assume that
−∞ < λ1 (U ) < λ0 (U ) = 0 < λ2 (U ) < ∞,
U ∈ Ω.
Let {(0, ri ) |i = 1, 2} be the right eigenvectors of DF (U ) corresponding to {λi |i = 1, 2} and (1, r ∗ )T be the right eigenvector of DF (U ) corresponding to λ0 . By assumption we see that each characteristic field is either genuinely non-linear or linear degenerate (in particular, the characteristic field with respect to λ0 ). Next, we define the following regions T
ΩL (ε) := {(x, t )|x < x0 − ε, t ≥ 0}, ΩR (ε) := {(x, t )|x > x0 + ε, t ≥ 0}, Ωε (ε) := {(x, t )||x − x0 | ≤ ε, t ≥ 0}.
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J.M. Hong, Y.-C. Su / Nonlinear Analysis 72 (2010) 635–650
Fig. 1. Solution of the perturbed Riemann problem (2.2), (2.3).
Then, due to the vanishing of (aε )0 in ΩL (ε) and ΩR (ε), the approximate solution of (2.2) in those regions satisfies the following homogeneous conservation laws uεt + f (aL , uε )x = 0 ε
ε
ut + f (aR , u )x = 0
in ΩL (ε),
(2.4)
in ΩR (ε).
(2.5)
Therefore, by the results of [4], the weak solution in ΩL (ε), ΩR (ε) consists of constant states separated by elementary waves (rarefaction waves, shocks and contact discontinuities) which can be expressed as the integral curves of {(0, ri )T |i = 1, 2} in the phase plane. In addition, the solution curve of each characteristic field starting at U∗ can be parameterized as U1 (σ1 ; U∗ ) = U∗ + σ1 R1 (U∗ ) + O(σ12 ),
(2.6)
U2 (σ2 ; U∗ ) = U∗ + σ2 R2 (U∗ ) + O(σ22 ),
where Ri (U∗ ) := (0, ri (U∗ ))T and Ui (σi ; U∗ ) is at least C 2 in σi for i = 1, 2. Next, we study the solution of (2.2) in Ωε (ε). Due to the appearance of the characteristic field of λ0 , we can construct the time-independent solutions (standing waves) of (2.2) in Ωε (ε). Let u1 denote the constant state of solution near x = x0 − ε in ΩL (ε), then the solution of (2.2) in Ωε (ε) is solved by
(
daε
df
(aε , uε ) = g (aε , uε ), dx dx uε (x0 − ε) = u1 .
(2.7)
Next, by (A1 ), the monotonicity of aε and re-scaling technique, we can rewrite (2.7) into
duε
∂f ε ∂u da uε (aL ) = u1 .
−1
=
g−
∂f (U ε ), ∂a
(2.8)
We define the following vector
∂f R0 (U ) = (1, r0 (U )) := 1, ∂u T
−1
!T ∂f g− (U ) . ∂a
(2.9)
Then, by (2.8) and (2.9), we can express the solution Usε = (aε , uε (aε )) of (2.2) in Ωε (ε) as an integral curve in phase space, which can be parameterized as Usε (σ0 ; U1 ) = U1 + σ0 R0 (U1 ) + O(σ02 ), ε
(2.10) ε
where U1 := (aL , u1 ). Note that u may not match the initial data on {(x, 0)||x − x0 | ≤ ε}. So u constructed above can only be treated as an approximate solution of (2.2). However, by the results of [25], the effect caused by the jump of uε on {(x, 0)||x − x0 | ≤ ε} can be neglected when ε → 0. To complete the construction of approximate solution of (2.2), we need to determine the constant states on the left of x = x0 − ε and on the right of x = x0 + ε (U1 and U2 in Fig. 1). It can be achieved by Lax’s method [25]. We have the following theorem. Theorem 2.1. Given 0 < ε 1, we consider the perturbed Riemann problem (2.2) and (2.3) where f , g satisfy condition (A1 ) and x0 is away from 0. Suppose that UL , UR ∈ Ω and |UL − UR | is sufficiently small. Then (2.2) and (2.3) admits a unique approximate solution uε constructed by the process described above. The approximate solution consists of at most four constant states separated by either shocks, rarefaction waves and a smooth standing wave (Fig. 1). Next, following the work in [25], we give a definition to the measurable solution of the Riemann problem (2.1). Definition 2.2. Let {U ε (x, t )|0 < ε 1} be a sequence of approximate solutions of (2.2) and (2.3). Then, the function U (x, t ) := limε→0 U ε (x, t ) is called a measurable solution of (2.1).
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Fig. 2. Measurable solution to the Riemann problem (2.1).
By letting ε → 0 to the sequence of approximate solutions of (2.2) and (2.3) in Theorem 2.1, we obtain the following theorem. Corollary 2.3. The Riemann problem (2.1) has a unique self-similar measurable solution consisting of at most four constant states separated by either shocks, rarefaction waves or a standing wave discontinuity (Fig. 2).
3. Boundary Riemann problem In this section we study the measurable solutions to the following boundary Riemann problem (BRP for short):
U + F (U )x = a0 G(U ), t U (x, 0) = UR , a(0, t ) = a(0) := aB , v(0, t ) = vB ,
(x, t ) ∈ (0, ∞) × (0, ∞), x > 0,
(3.1)
t > 0,
where U = (a, v, w)T , UR = (aR , vR , wR )T , F = (0, f1 , f2 )T and G = (0, g1 , g2 )T . To overcome the same difficulty appeared in Section 2, we reformulate the source term in (3.1). Therefore, for 0 < ε 1, we study the corresponding perturbed boundary Riemann problem (PBRP for short):
ε U + F (U ε )x = (aε )0 G(U ε ), tε U (x, 0) = UR , ε ε U (x, 0) = φ (x), a(0, t ) = aB , v(0, t ) = vB ,
(x, t ) ∈ (0, ∞) × (0, ∞), x > ε, 0 ≤ x ≤ ε, t > 0,
(3.2)
where ε
a (x) =
ηε (x), aR ,
0 ≤ x ≤ ε, x > ε,
(3.3)
and ηε (x) is a monotone function connecting aB , aR at x = 0 and x = ε respectively. Also φ ε (x) = (ηε (x), v0ε (x), w0ε (x)) is also a monotone function connecting states U∗ := (aB , vB , wB ) at x = 0 (with wB needed to be decided by Lax’s method) and UR at x = ε . Then, in view of Definition 2.2, the measurable solution of BRP (3.1) is defined as U (x, t ) := limε→0 U ε (x, t ) where U ε (x, t ) is a solution of (3.2). The main topic of this section is to use Lax’s method to prove the existence of weak solutions to (3.2) and (3.3). To start, we notice that the second characteristic field is either genuinely nonlinear or linearly degenerate. Then, by the vanishing of the source term in x > ε , we can use the elementary wave of homogeneous system to construct the solution of (3.2) in x > ε . Given a state U2 and a sufficiently small parameter σ2 , we define the mapping T2 : R × Ω → R3 by T2 (σ2 ; U2 ) = UR ,
(3.4)
where UR is the state which can be connected by a 2-wave curve starting at U2 with (signed) wave strength σ2 (Section 2). Note that the 2-wave curve can be parameterized as a C 2 curve starting at U2 : UR = U2 + σ2 R2 (U2 ) + O(σ22 ).
(3.5)
In addition, we see that T2 is at least C in arguments σ2 , U2 . Next, we study the time-independent solution corresponding to λ0 (U ) = 0. Followed by the results in Section 2, given a state UB ≡ (aB , vB , wB ) nearby U2 , we define the mapping T0ε : R × Ω → R3 by 2
T0ε (σ0 ; UB ) = U2 ,
(3.6)
where U2 is the state which can be connected by a 0-wave curve starting at UB with (signed) wave strength σ0 . Also, the 0-wave curve can be parameterized as U2 = UB + σ0 R0 (UB ) + O(σ02 ).
(3.7)
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Now, we define the composite mapping TBε : R × Ω → R3 by TBε (σ2 , σ0 ; UB ) := T2 (σ2 ; T0ε (σ0 ; UB )).
(3.8)
ε
Note that TB is at least a C 2 mapping of σ0 , σ2 . Then by (3.4)–(3.8) we obtain TBε (σ2 , σ0 ; UB ) = UB + σ0 R0 (UB ) + σ2 R2 (UB ) + O(1)(σ0 + σ2 )2
= UR .
(3.9)
In addition, we have TBε (0, 0; UB ) = UB ,
∂ TBε = R0 (UB ), ∂σ0 (0,0;UB )
∂ TBε = R2 (UB ). ∂σ2 (0,0;UB )
(3.10)
Next, we consider the mapping H given by H (σ0 , σ2 ) := [UR ]1,2 − [UB ]1,2 ,
(3.11)
where [·]1,2 denotes the 2-vector consisting of the first and second components of vector [·]. Then, by (3.9)–(3.11) we obtain that H (0, 0) = (0, 0) and the Jacobian matrix of H at (0, 0) is DH (0, 0) = [(R0 (UB ), R2 (UB ))]1,2 =
1 [r0 (UB )]1
0 . [r2 (UB )]1
(3.12)
To apply Lax’s method, we need the following lemma. Lemma 3.1. Under condition ( A2 ), the Jacobian matrix DH in (3.12) has rank 2, which means that it is non-singular. Proof. Assume that DH (0, 0) is singular, i.e., [r2 (UB )]1 = 0, so λ2 (UB )[r2 (UB )]1 = 0. Next, since the right eigenvector r2 (UB ) of Df is a nonzero vector, we have [r2 (UB )]2 6= 0. On the other hand, by Df (UB ) · r2 (UB ) = λ2 (UB )r2 (UB ), ∂f
it implies ∂w1 (UB )[r2 (UB )]2 = λ2 (UB )[r2 (UB )]1 = 0. Then, by Condition (A2 ) we have the contradiction. The proof is complete. Thus, by Lemma 3.1 and the inverse function theorem, there exist neighborhoods N0 of (0, 0) and NB of [UB ]1,2 such that H is a diffeomorphism from N0 to NB . It follows that we can solve for σ0 , σ2 in terms of UR and UB = (aB , vB , wB ) when |(aB , vB ) − [UR ]1,2 | is sufficiently small. Next, we decide the value of wB when UR = (aR , vR , wR ) ∈ NB is given. In view of the third equation of (3.9), we have
wR = wB + σ0 R30 (UB ) + σ2 R32 (UB ) + O(1)(σ0 + σ2 )2 ≡ η(wB ),
(3.13)
where R30 , R32 are the third component of R0 , R2 respectively, and σ0 , σ2 are determined by previous analysis. We see that dη dw B
= 1 + σ0
dR30 dw B
+ σ2
dR32 dwB
+ O(1)(σ0 + σ2 )2 ,
(3.14)
dη
which implies that dw |σ0 =σ2 =0 = 1. Then, by inverse function theorem there exists a unique wB satisfying (3.13) when B σ0 , σ2 are sufficiently small. Therefore, we obtain the following theorem regarding the existence of solutions to the PBRP (3.2). Theorem 3.2. Consider the PBRP (3.2) where UR ∈ Ω ⊂ R3 and |(aB , vB ) − (aR , vR )| is sufficiently small. Then, under conditions (A1 ), (A2 ), there exists a solution U ε (x, t ) of (3.2) consisting of at most three constant states separated by a 2-wave issued from (ε, 0) and a smooth standing wave in 0 ≤ x ≤ ε . In addition, there is a unique wB such that U ε (0, t ) = (aB , vB , wB ) for t > 0 (Fig. 3). We notice that the structure of U ε (x, t ) of (3.2) depends on the choice of aε . However, the wave curves of U ε (x, t ) on the phase plane are uniquely determined. It follows that the constant states in U ε (x, t ) are independent of the choice of aε . Therefore, by letting ε → 0 in U ε (x, t ), we obtain the existence and uniqueness of measurable solution to (3.1). Corollary 3.3. The BRP (3.1) has a unique measurable solution U (x, t ) := limε→0 U ε (x, t ) consisting of at most three constant states separated by a 2-wave and a standing wave discontinuity. Furthermore, there exists a unique constant wB such that U (0, t ) = (aB , vB , wB ) for t > 0.
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Fig. 3. Solution of the PBRP in the phase plane.
4. Generalized Glimm scheme and its stability In this section we construct the generalized Glimm scheme for IBVP (1.3) and study the stability of scheme. To start, we partition quarter plane {(x, t ) ∈ [0, ∞) × [0, ∞)} into xk = k∆x,
ti = i∆t , i, k = 0, 1, 2, . . . ,
where ∆x and ∆t are sufficiently small. Next, given 0 < ε 1, we assume that ∆x, ∆t satisfy the generalized C-F-L condition
∆x/∆t > (1 − ε)−1 sup{|λi (U )||i = 1, 2}.
(4.1)
Ω
Condition (4.1) ensures that waves in the approximate solutions cannot interact in the same time level. We also define
∆˜x := (1 + ε)−1 ∆x.
(4.2)
Next, we approximate the initial and boundary data in (1.3) as follows.
(I)0 If x ≥ x1 and k > 0 is even, then initial data U0ε (x) is chosen by 0 xk−1 ≤ x < xk − ε ∆˜x, Uk−1 , ε U0 (x) = Ψk0 (x), |x − xk | ≤ ε ∆˜x, 0 Uk+1 , xk + ε ∆˜x < x ≤ xk+1 ,
(4.3)
where Uk0−1 := U0 (xk−1 ) and Ψk0 (x) = (aε0 (x), v0ε (x), w0ε (x)) is a smooth monotone function connecting Uk0−1 and Uk0+1 . (II)0 If 0 ≤ x ≤ x1 , then the initial and boundary data are chosen by U0ε (x) =
ΨB0 (x), U10 ,
v (t ) = vB (∆t /2), 0 B
0 ≤ x ≤ ε ∆˜x, ε ∆˜x < x ≤ x1 ,
(4.4)
0 < t < t1 ,
where U10 is given in Case (I)0 and ΨB0 (x) is a smooth monotone function connecting (aB , vB (0), w0 (0)) and U10 . Then, for such initial and boundary data we construct the approximate solutions to a set of PRPs and a PBRP in the 0-th time step {(x, t ) ∈ [0, ∞) × [0, ∆t )}. Let U ε,0 (x, t ) denote the approximate solution obtained by solving PRPs and the PBRP with (4.3) and (4.4) in the 0-th time step. Then initial data U1ε (x, t1 ) and boundary data vB1 (t ) in the first time step are chosen by the random choice method described in the following.
(I)1 For x ≥ x1 , U1ε (x, t1 ) is chosen by 1 xk−1 ≤ x < xk − ε ∆˜x, Uk−1 , ε 1 U1 (x, t1 ) = Ψk (x), |x − xk | ≤ ε ∆˜x, 1 Uk+1 , xk + ε ∆˜x < x ≤ xk+1 ,
k = 2, 4, 6, . . . ,
(4.5)
where Uk1−1 := U ε,0 (xk−1 + θ1 ∆˜x, t1 ), θ1 ∈ (−1, 1) is a random number, and Ψk1 (x) = (aε1 (x), v1ε (x), w1ε (x)) is a smooth monotone function connecting Uk1−1 and Uk1+1 .
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(II)1 If 0 ≤ x ≤ x1 , then the initial and boundary data are chosen by 1 ΨB (x), 0 ≤ x ≤ ε ∆˜x, U1ε (x, t1 ) = U11 , ε ∆˜x < x ≤ x1 , vB1 (t ) = vB (3∆t /2),
(4.6)
t1 < t < t2 ,
where ΨB1 (x) is a smooth monotone function connecting (aB , vB (t1 ), w ε,0 (0, t1− )) and U11 . Again, we solve for a set of PRPs and a PBRP in the first time step with (4.5) and (4.6). We repeat this process for each time step, and let U ε,i−1 (x, t ) denote the approximate solution by this process in the (i − 1)-th time step, then the initial and boundary data in the i-th time step will be chosen as follows:
(I)i For x ≥ x1 , Uiε (x, ti ) is chosen by i Uk−1 , xk−1 ≤ x < xk − ε ∆˜x, Uiε (x, ti ) = Ψki (x), |x − xk | ≤ ε ∆˜x, i Uk+1 , xk + ε ∆˜x < x ≤ xk+1 ,
k = 2, 4, 6, . . . ,
(4.7)
where Uki −1 = U ε,i−1 (xk−1 + θi ∆˜x, ti ), θi ∈ (−1, 1), i ∈ N, is a random number, and Ψki (x) = (aεi (x), viε (x), wiε (x)) is a smooth monotone function connecting Uki −1 , Uki +1 . (II)i If 0 ≤ x ≤ x1 , then we chose
ΨBi (x), 0 ≤ x ≤ ε ∆˜x, U1i , ε∆˜x < x ≤ x1 , 1 ∆t , ti < t < ti+1 , vBi (t ) = vB i+
Uiε (x, ti ) =
(4.8)
2
(x) connects (aB , vB (ti ), wε,i−1 (0, ti− )) and U1i . We then obtain an approximate solution Uθε,∆x (x, t ) to IBVP where (1.3) by a generalized Glimm scheme with a random sequence θ := {θ1 , θ2 , . . .} in x ≥ 0, t ≥ 0. Similarly, we can construct another approximate solution, denoted by Uθ ,∆x , to (1.3) by taking ε → 0 in Uθε,∆x . ΨBi
To obtain the compactness of the approximate solutions, we study the stability of this scheme. By the results of [5], we need to show that the total variation of approximate solution stays bounded uniformly. To achieve this, we study the wave interaction estimates. First we consider the interaction of waves near the boundary. Here we emphasize that, the 0-th wave field is characteristic to the boundary. So the interaction of waves is more complicated than the case in [8]. However, we show in the following that the results in [8] can be extended to our problem. First, we state the following lemma which is the extension of results in [8]. Lemma 4.1. (a) (Elementary wave interaction) Let U be a constant state which is connected to constant state V by an i-wave with strength αi , and V can be connected to another constant state W by a j-wave with strength βj . On the other hand, suppose that U can also be connected to V 0 by a j-wave with strength βj and V 0 can be connected to W 0 by an i-wave with strength αi . Then there exists a continuous function η(αi , βj ) such that W − W 0 = η(αi , βj )αi βj .
(4.9)
(b) (Combining waves of the same family) Let U be connected to V by αi and V connected to W by βi , waves of the same family. If U is connected to W 0 by an i-wave of strength αi + βi , then there is a continuous function ζ (αi , βi ) such that W − W0 =
0,
ζ (αi , βi )αi βi ,
if αi and βi are both rarefaction waves or both smooth standing waves, otherwise.
(4.10)
Since the standing waves are also C 2 (by the wave construction in the previous section), the proof of Lemma 4.1 is similar to the one given in [8], and so we omit it. Now, we study wave interactions near the boundary. Here we adopt the notations in [3], and the wave strengths of elementary waves and standing waves are defined as in [15,4,3]. Such as let (UL , UR ) denote the solution of the perturbed Riemann problem [UL , UR ] consisting of constant states UL ≡ U0 , U1 , U2 , U3 ≡ UR with the parametrization Tk (εk ; Uk ) = Uk+1 , then the solution (UL , UR ) can be written as
(UL , UR ) ≡ (U0 , U1 , U2 , U3 )/(ε0 , ε1 , ε2 ). The parameter εi is called the wave strength of the i-wave that connects the states Ui and Ui+1 . We assume that the solution of the PBRP on the k-th time step is (Uk , UM ) ≡ (Uk , U2 , UM )/(α0k , α2k ) where Uk := (aB , vk , wk ), and the 1-wave with strength β1 in the solution of nearby PRP on the k-th time step is (UM , UR ) ≡ (UM , UR )/(β1 ). Also, let the solution of the
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Fig. 4. Approximate solutions of the PRPs and PBRPs.
Fig. 5. Interactions of waves with two interchanges of waves.
PBRP on the (k + 1)-th time step be (Uk+1 , UR ) ≡ (Uk+1 , U20 , UR )/(γ0k+1 , γ2k+1 ) where Uk+1 := (aB , vk+1 , wk+1 ), see Fig. 4. Then, the method of obtaining the wave interaction estimates near the boundary is divided into the following steps. Step I. By the results in Section 3, we construct the solution of the PBRP [U¯ k , UR ] where U¯ k := (aB , vk , w ¯ k ) with w ¯ k uniquely decided. Let (U¯ k , UR ) ≡ (U¯ k , U¯ 2 , UR )/(γ0k , γ2k ) be the solution of [U¯ k , UR ]. Step II. We estimate the difference of total wave strengths between two solutions (U¯ k , UR ) and (Uk+1 , UR ). Step III. Similarly, we estimate the difference of total wave strengths between (U¯ k , UR ) and (Uk , UR ). Then, by the triangle inequality we can obtain the wave interaction estimates near the boundary. To start at Step I, by the results in Section 3, we are able to construct the solution (U¯ k , UR ) ≡ (U¯ k , U¯ 2 , UR )/(γ0k , γ2k ) of [U¯ k , UR ]. Next, to achieve the goal of Step II, we use the differentiability of H given in (3.11) to obtain
|γjk+1 − γjk | ≤ C |vk+1 − vk |,
j = 0, 2.
(4.11)
For Step III, we notice that the map UR = Ti (τi ; UL )
is a smooth function of (τi ; UL ),
(4.12)
where UR is the state which can be connected to UL on the right by an i-wave, i 6= 0, with (signed) strength τi (Section 3). Then, by the results of [8], we first study the case where β1 interacts with α2k , α0k in order (see Fig. 5). We observe that 0 the states U10 , UM and UR0 in Fig. 5 can be determined by Lemma 4.1-(a) with wave strengths β1 , α0k , and α2k . Furthermore, following by (4.9) and (4.12) and the triangle inequality, we obtain that
|UR0 − UR | ≤ C (|α0k β1 | + |α2k β1 |),
(4.13)
where UR is connected to Uk on the right by β1 , α and α On the other hand, it is easy to see that k 0
0
k 2.
|[U10 ]1,2 − (aB , vk )| ≤ |U10 − Uk | ≤ C |β1 |,
(4.14)
where U1 := (aB , vk , wk ) is connected to Uk on the right by β1 , and [U1 ]1,2 stands for the first two components of U1 . Next, we solve the PBRP [(aB , vk , ∗), U10 ]. Again, by the results in Section 3, the solution of [(aB , vk , ∗), U10 ] exists. Let (Uk0 , U10 ) ≡ (UK0 , U10 )/(γ20 ) denote the solution of [Uk0 , U10 ] where Uk0 ≡ (aB , vk , wk0 ) for some wk0 . Then we study the case that γ20 interchanges with α0k and then combines with α2k (see Fig. 6). By the differentiability of H and (4.14), it follows that γ20 satisfies 0
0
0
0
0
|γ20 | ≤ C |(aB , vk ) − [U10 ]1,2 | ≤ C |β1 |.
(4.15)
Therefore, by (4.9), (4.10) and (4.12) and the triangle inequality, we obtain
|UR00 − UR0 | ≤ C (|α0k γ20 | + |α2k γ20 |) ≤ C (|γ20 |),
(4.16)
where UR is connected to Uk on the right by waves α and γ2 + α Here we used the fact that wave strengths α and α2k are bounded. It follows from (4.15) and (4.16) that 00
|UR00 − UR0 | ≤ C |β1 |.
0
k 0
0
k 2.
k 0
(4.17)
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Fig. 6. Interchange and combination of waves.
Fig. 7. Approximate solution of IBVP and two mesh curves Ji , Ji+1 .
In view of (U¯ k , UR ) ≡ (U¯ k , U¯ 2 , UR )/(γ0k , γ2k ) in Step I, and by (4.13) and (4.17) and the differentiability of H, we obtain that {γjk | j = 0, 2} satisfies
|γjk − (γj0 + αjk )| ≤ C |[UR ]1,2 − [UR00 ]1,2 | ≤ C (|UR − UR0 | + |UR0 − UR00 |) ≤ C (|α0k β1 | + |α2k β1 | + |β1 |),
(4.18)
where γ00 := 0. Finally, by (4.11), (4.15) and (4.18), for j = 0, 2 we have
|γjk+1 − (αjk + β1 )| ≤ |γjk − (γj0 + αjk )| + |γjk+1 − γjk | + |γj0 | + |β1 | ≤ C (|α0k β1 | + |α2k β1 | + |β1 | + |vk+1 − vk |). Therefore, by previous analysis, we have the following theorem regarding the wave interaction estimates. Theorem 4.2. Let (Uk , UM ) ≡ (Uk , U2 , UM )/(α0k , α2k ), (Uk+1 , UR ) ≡ (Uk+1 , U20 , UR )/(γ0k+1 , γ2k+1 ) be the solutions of the PBRPs in Uθε,∆x (x, t ) on the k-th and (k + 1)th time steps respectively. Also, let (UM , UR ) ≡ (UM , UR )/(β1 ) be the 1-wave in the solution of the PRP next to (Uk , UM ) on the k-th time step. Then there exists a constant C such that
|γjk+1 − (αjk + β1 )| ≤ C (|α0k β1 | + |α2k β1 | + |β1 | + |vk+1 − vk |),
j = 0, 2.
(4.19)
Next, we study the wave interaction estimates for the waves away from the boundary. First, we refer the readers to [3] for the definitions of mesh points, mesh curves, immediate successors and approaching of waves in the Glimm scheme. Given a diamond region ∆, which is enclosed by mesh curves I, J and is away from the boundary, we let {α ≡ (α1 , α0 , α2 ), β ≡ (β1 , β0 , β2 )} be the incoming waves of ∆. Note that α , β are waves from two different PRPs across the lower-half of ∆. On the other hand, we let γ ≡ (γ1 , γ0 , γ2 ) be the outgoing waves of ∆, which means that γ is an entire perturbed Riemann solution across the upper-half of ∆. Then, by the results of [5,15], we have
γi = αi + βi + O(1)|α||β|,
i = 0, 1, 2.
(4.20)
When the waves approach, by (4.20), we obtain that
γi = αi + βi + O(1)D(α, β), as |α| + |β| → 0, i = 0, 1, 2. P Here D(α, β) = |αi ||βj | whose sum is over all pairs of i-wave and j-wave that are approaching.
(4.21)
By Theorem 4.2 and (4.20) and (4.21), we are able to show that the total variation of approximate solution Uθε,∆x (x, t ) is uniformly bounded, which will proceed as follows. Let {θi : i = 0, 1, 2, . . .} be a equi-distributed random sequence of numbers in (−1, 1) with θ0 = 0. Also let pk−1,i = (xk−1 + θi ∆˜x, ti ), i = 0, 1, 2, . . ., k = 2, 4, 6, . . ., p0,n = (0, (n + 21 )∆t ), n = 0, 1, 2, . . ., be the mesh points. Then we can connect {pk,i } to get a set of mesh curves and diamond regions. We notice that, every mesh curve in our case consists of some portion of the boundary x = 0, moreover, the diamond regions near the boundary x = 0 are triangle regions, see Fig. 7.
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Next, due to the terms |β1 |, |vk+1 − vk | appear in the RHS of (4.19), we define the following functionals on any mesh curve Ji : L(Ji ) :=
X
|α0 | +
X
|α2 | + K1
X
|vk − vk−1 | + K1
X
|α1 |,
(4.22)
k∈Wi
X {|αj ||αj0 | : αj , αj0 cross Ji and approach}, (4.23) P where constant K1 > 1 will be decided later. Here |αj |, j = 0, 1, 2 are the sum over all αj crossing Ji , and Wi := {k : (0, (k + 1/2)∆t ) ∈ Ji }. Note that the functional L in (4.22) can be considered as an equivalent norm for the total variations of Uθε,∆x (x, t ) on mesh curves. Next, we define the Glimm functional of Uθε,∆x (x, t ) on the mesh curve Ji by Q (Ji ) :=
F (Ji ) := L(Ji ) + K2 Q (Ji ),
(4.24)
where K2 > 1 is a sufficiently large constant. Then, by the results in [5], the non-increasing of the Glimm functional F ε ε implies the global boundedness of Uθ, ∆x (x, t ) and T .V .[Uθ,∆x (x, t )]. So we show the non-increasing of the Glimm functional by induction, which is given as follows. Let J0 be the mesh curve located between t = 0 and t = t1 for x > 0, and includes the half-ray {x = 0, t ≥ ∆t /2}. Since T .V .[U0 ], T .V .[vB ] are sufficiently small, we have F (J0 ) ≤ L(J0 ) + K2 L(J0 )2
≤ O(1)(K1 + K12 K2 (T .V .[U0 ] + T .V .[vB ]))(T .V .[U0 ] + T .V .[vB ]) ≤ O(1)K1 (T .V .[U0 ] + T .V .[vB ]).
(4.25)
Next, suppose that F (Ji ) is bounded and let Ji+1 is an immediate successor of Ji . Then, there exists a diamond or triangle region Γii+1 enclosed by Ji+1 , Ji . In the case where Γii+1 is a diamond region, we have already obtained the non-increasing of the Glimm functional by the results in [15], so we only consider the case where Γii+1 is a triangle region where Ji+1 , Ji only differ on mesh point p0,i−1 (Fig. 6). Followed by (4.22)–(4.24) and Theorem 4.2, we obtain F (Ji+1 ) − F (Ji ) = |γ0 | + |γ2 | − |α0 | − |α2 | − K1 |β1 | − K1 |vi − vi−1 |
+ K2
nX
|αj0 |(|γ0 | + |γ2 | − |α0 | − |α2 |) −
X
o |αj0 ||β1 | − K2 (|α0 ||β1 | + |α2 ||β1 |),
(4.26)
where the sums in (4.26) are overall approaching waves {αj0 } which are on the right of β1 . It follows form (4.19) and (4.26) that F (Ji+1 ) − F (Ji ) ≤ O(1)C (|α0 β1 | + |α2 β1 | + |β1 | + |vi − vi−1 |) − K1 |β1 | − K1 |vi − vi−1 |
X X + K2 O(1)C |αj0 |(|α0 β1 | + |α2 β1 | + |β1 | + |vi − vi−1 |) − |αj0 ||β1 | − K2 (|α0 ||β1 | + |α2 ||β1 |) ≤ (−K1 + O(1)C + O(1)CK2 F (Ji ))(|β1 | + |vi − vi−1 |) + (−K2 + O(1)C + O(1)CK2 F (Ji ))(|α0 β1 | + |α2 β1 |) ≤ (−K1 + O(1)C + O(1)CK2 F (J0 ))(|β1 | + |vi − vi−1 |) + (−K2 + O(1)C + O(1)CK2 F (J0 ))(|α0 β1 | + |α2 β1 |) ≤ 0, if we choose constants K1 , K2 ≥ O(1)2C and K2 F (J0 ) ≤ 1. We show the non-increasing of the Glimm functional F . We notice that Uθε,∆x and Uθ,∆x have the same value of the Glimm functional on each mesh curve since the structure of wave curves in both solutions is the same. Therefore, by the results of [5,15,24], it leads to the following theorem regarding the compactness of approximate solutions. Theorem 4.3. Suppose that the total variations of U0 (x), vB (t ) are small. Then approximate solution Uθε,∆x of (1.3) by the generalized Glimm scheme satisfies ε (1) T .V .[Uθ, ∆x (x, t )] ≤ C1 (T .V .[U0 ] + T .V .[vB ]), where constant C1 is independent of θ , ∆x and ε . ε ε (2) T .V .[Uθ, ∆x (x, tj )] + supx Uθ,∆x (x, tj ) ≤ C2 (T .V .[U0 ] + T .V .[vB ]), where constant C2 is independent of θ , ∆x and ε . ε ε ∗ ∗∗ (3) kUθ,∆x (·, t ) − Uθ,∆x (·, t )kL1 ≤ C3 (|t ∗ − t ∗∗ | + ∆t ), where constant C3 is independent of θ , ∆x and ε . ε Letting ε → 0 to {Uθ, ∆x } in Theorem 4.3, we obtain the following result.
Corollary 4.4. Suppose that the total variations of U0 (x), vB (t ) are sufficiently small. Let Uθε,∆x (x, t ) be the approximate solution of (1.3) by the generalized Glimm scheme, and Uθ,∆x (x, t ) := limε→0 Uθε,∆x (x, t ). Then (1) T .V .[Uθ,∆x (x, t )] ≤ C1 (T .V .[U0 ] + T .V .[vB ]), where constant C1 is independent of θ and ∆x. (2) T .V .[Uθ,∆x (x, tj )] + supx Uθ,∆x (x, tj ) ≤ C2 (T .V .[U0 ] + T .V .[vB ]), where constant C2 is independent of θ and ∆x.
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J.M. Hong, Y.-C. Su / Nonlinear Analysis 72 (2010) 635–650
(3) kUθ ,∆x (·, t ∗ ) − Uθ,∆x (·, t ∗∗ )kL1 ≤ K3 (|t ∗ − t ∗∗ | + ∆t ), where constant K3 is independent of θ and ∆x. Thus, by Theorem 4.3, Corollary 4.4 and the results in [5,3], we obtain the convergence of {Uθε,∆x } and {Uθ ,∆x }. ε Theorem 4.5. Given 0 < ε 1. Let {Uθ, ∆x } be a sequence of approximate solutions to (1.3) by the generalized Glimm scheme ε and Uθ,∆x := limε→0 Uθ,∆x . Then there exists a subsequence of {Uθε,∆x } (resp., {Uθ ,∆x }), still denoted by {Uθε,∆x } (resp., {Uθ ,∆x }), which converges to some measurable function U ε (x, t ) := (aε (x), v ε (x, t ), w ε (x, t )) (resp., U (x, t ) := (a(x), v(x, t ), w(x, t ))) in L1loc . In addition, for every continuous function F , we have F (Uθε,∆x ) → F (U ε (x, t )) (resp., F (Uθ ,∆xi ) → F (U (x, t ))) in L1loc .
Furthermore, U ε (x, t ) → U (x, t ), F (U ε (x, t )) → F (U (x, t )) in L1loc as ε → 0.
Proof. We only show kU ε − U kL1 → 0 as ε → 0. For the other proof, we refer the readers to [3]. We see that loc
ε kUθ, ∆x
− Uθ,∆x kL1 = O(1)
O(1)(∆t )−1
X
loc
m=1,3,...
O(1)(∆t )−1
X
Z
tn+1 tn
n =0
xm+1
X Z
tn
n=0
= O(1)(ε∆x)
tn+1
Z
xm−1
! |Uθε,∆x
− Uθ ,∆x |dx dt
(osc [Uθε,∆x (x, t )|[xm−1 ,xm+1 ] ])dt
X
m=1,3,...
= O(1)(ε∆x)(TV [U0 ] + TV [vB ]), where O(1) is independent of θ , ∆x and ε . Therefore, by the fact that kU ε − Uθε,∆x kL1 , kUθ ,∆x − U kL1 → 0 as ∆x → 0 and loc
the triangle inequality, we obtain kU ε − U kL1 → 0 as ∆x, ε → 0. The proof is complete. loc
loc
By the results in Section 3, we notice that ε i |vθ, ∆x (0, t ) − vB (t )| ≤ |vB (t ) − vB (t )| ≤ O (1)∆x, ε for any t > 0, i ∈ N ∪ {0}. This implies that vθ, ∆x will match boundary data vB (t ) as ∆x tends to 0.
5. Weak convergence of the residuals In Section 4 we obtained the compactness of approximate solutions of (1.3) by showing the stability of the generalized Glimm scheme. In this section we establish the global existence of weak solutions to (1.3) by showing the weak convergence ε of the residual. To start, let {Uθ, ∆x } denote a sequence of approximate solutions of (1.3) by the generalized Glimm scheme, ε ∞ and U (x, t ) denote the limit of Uθ, ∆x as ε , ∆x approach 0. Then, for E := [0, ∞) × [0, ∞) and test function φ ∈ C0 (E ), we recall the residual R(U , φ, E ) given in Definition 1.1: R(U , φ, E ) :=
Z Z
(U φt + F (U )φx + a0 G(U )φ)dxdt +
∞
Z
E
U0 (x)φ(x, 0)dx +
0
Z
∞
F (U (0, t ))φ(0, t )dt .
0
ε We wish to show that R(Uθ, ∆x , φ, E ) approaches 0 in the weak sense as ε , ∆x tend to 0. Then, following by the results of [24], we obtain the global existence of weak solutions to problem (1.3). We define [
Γi := {(x, t )|0 ≤ x ≤ x1 , ti ≤ t < ti+1 },
Γi ,
Γ :=
i∈N∪{0}
ε RΓi (Uθ, ∆x , φ) :=
Z Z Γi
ε,i
ε,i
ε,i
ε,i
Uθ,∆x φt + F (Uθ,∆x )φx + (aθ ,∆x )0 G(Uθ ,∆x )φ dxdt ,
ε,i Uθ,∆x
ε ε = Uθ, ∆x |Γi . Then, by the construction of Uθ ,∆x and the divergence theorem, we obtain t =t − x=x1 t =t − Z ε∆˜x Z x1 Z t− i+1 i+1 i+1 ε,i ε,i ε, i Γi ε R (Uθ,∆x , φ) = (Uθ,∆x φ) dx + (Uθ ,∆x φ) dx + F (Uθ ,∆x )φ dt . + 0 ε ∆˜x t t = ti i
where φ ∈ C0 (E ) and ∞
t = ti
(5.1)
x =0
In addition, we see that ε∆˜x
Z 0
t =t − i+1
ε,i (Uθ, ∆x φ)
Z
ε ∆˜x
dx = 0
+
t = ti
Z
ε ∆˜x
+ 0
Z ≤ 0
ε,i ε,i − [(Uθ, ∆x φ)(x, ti+1 ) − (Uθ ,∆x φ)(x, ti )]dx
ε ∆˜x
ε,i ε,i + [(Uθ, ∆x φ)(x, ti ) − (Uθ ,∆x φ)(x, ti )]dx
ε,i ε,i − [(Uθ, ∆x φ)(x, ti+1 ) − (Uθ ,∆x φ)(x, ti )]dx
+ O(1)kφk∞ (ε∆˜x) osc [Uθε,,∆i−x1 |[0,x2 ] ] + osc [vB (t )|[ti − 1 ∆t ,ti ] ] + osc [Uθε,,∆i x |[0,x1 ] ] , 2
(5.2)
J.M. Hong, Y.-C. Su / Nonlinear Analysis 72 (2010) 635–650
647
where osc [U ] denotes the oscillation of U. Summing all RΓi and by (5.1) and (5.2), we obtain ε RΓ (Uθ, ∆x , φ) = −
∞ Z X
x1 0
i =1
− ti+1
∞ Z X
+
ε,i bUθ, ∆x c(x)φ(x, ti )dx −
ti
i =0
x=x1
ε,i F (Uθ,∆x )φ
x1
Z 0
ε,0
Uθ ,∆x (x, 0)φ(x, 0)dx
dt + O(1)εkφk∞ ,
(5.3)
x =0
ε,i
− ε ε where bUθ,∆x c(x) := Uθ, ∆x (x, ti ) − Uθ,∆x (x, ti ). ε Next, we estimate the residual for {Uθ,∆x } away from the boundary. Set Γ 0 := E \ Γ . Then by a similar calculation, it leads to ∞ Z X
ε RΓ (Uθ, ∆x , φ) = − 0
∞ x1
i=1
− ti+1
∞ Z X
−
ε,i bUθ, ∆x c(x)φ(x, ti )dx −
ti
i=0
∞
Z
x1
ε,0
Uθ ,∆x (x, 0)φ(x, 0)dx
ε,i
F (Uθ,∆x (x1 , t ))φ(x1 , t )dt + O(1)εkφk∞ T .V .[U0 ].
(5.4)
Thus, by (5.3) and (5.4) and the generalized C-F-L condition (4.1), we obtain ε R(Uθ, ∆x , φ, E )
=R
Γ
ε (Uθ, ∆x , φ)
+R
Γ0
ε (Uθ, ∆x , φ)
∞
Z
U0 (x)φ(x, 0)dx +
+
∞
Z
0
F (U (0, t ))φ(0, t )dt 0
= O(1)εkφk∞ + O(1)∆xkφk∞ (T .V .[U0 ] + T .V .[vB ]) −
∞ Z X 0
i =1
∞ Z X
−
i =0
− ti+1
ti
ε,i F (Uθ,∆x (0, t ))φ(0, t )dt
∞
bUθε,,∆i x c(x)φ(x, ti )dx
∞
Z
F (U (0, t ))φ(0, t )dt
+ 0
= O(1)εkφk∞ + O(1)∆xkφk∞ (T .V .[U0 ] + T .V .[vB ]) −
∞ Z X i =1
∞ 0
bUθε,,∆i x c(x)φ(x, ti )dx.
(5.5)
Now we show that the last term on the RHS of (5.5) converges in the order of ε 1/2 as ∆x → 0 in the probability space Φ of random choice numbers. To start, let J (θ , ∆x, φ) := i
∞
Z 0
i bUθ, ∆x c(x)φ(x, ti )dx,
and Jεi (θ , ∆x, φ) :=
∞
Z 0
ε,i bUθ, ∆x c(x)φ(x, ti )dx,
− i where bUθ, ∆x c(x) := Uθ,∆x (x, ti ) − Uθ,∆x (x, ti ) and i is a nonnegative integer. Then, for i 6= j we claim that
hJ i (θ , ∆x, φ), J j (θ , ∆x, φ)i ≡
Z Z
J i (θ , ∆x, φ)J j (θ , ∆x, φ)dθj
Y
dθl
l6=j
= 0.
(5.6)
Here h, i is the L inner product with respect to random number θ ∈ Φ , and φ is a function with compact support and is piecewise constant in each segment [xm−1 , xm+1 ] × {ti }, m is positive and odd. Indeed, we have 2
Z Φ
J j (θ , ∆x, φ)dθj =
Z
xm+1
Z
X Φ m=1,3,5,...
m=1,3,5,...
= 0,
Φ
Uθ,∆x (xm + θj ∆x, tj− ) − Uθ ,∆x (x, tj− ) dxdθj
xm−1
X nZ Z
=
xm+1
xm−1
Uθ,∆x (xm + θj ∆x, tj )dxdθj − −
Z
xm+1 xm−1
Z Φ
Uθ ,∆x (xm + θj ∆x, tj− )dθj dx
o (5.7)
which is enough to show that (5.6) holds. We notice that (5.6) is just an extension of Glimm’s results from the Cauchy problem to IBVP. Next, we have the following estimates for Jεi .
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J.M. Hong, Y.-C. Su / Nonlinear Analysis 72 (2010) 635–650
Lemma 5.1. For i, j = 1, 2, 3, . . ., we have
(i) |Jεi (θ ε , ∆x, φ)| = O(1)kφk∞ ∆x(T .V .[U0 ] + T .V .[vB ]), X |Jεi (θ ε , ∆x, φ)| = O(1)kφk∞ r (φ)(T .V .[U0 ] + T .V .[vB ]),
(5.8) (5.9)
i
where r (φ) is the diameter of support (φ). (ii) Given φ which has compact support and is piecewise constant on each segment [xm−1 , xm+1 ] × {ti }, m = 1, 3, 5, . . ., we have
hJεi (θ ε , ∆x, φ), Jεj (θ ε , ∆x, φ)i = O(1)(kφk∞ )2 · ε(∆x)2 ,
i 6= j.
(5.10)
Proof. First we show (i). For simplification, we let Jεi = Jεi (θ ε , ∆x, φ) and J j = J j (θ , ∆x, φ) and so on. Then for i ∈ N, we obtain
Z |Jε | =
∞
i
0
ε,i bUθ, ∆x c(x)φ(x, ti )dx
≤ kφk∞
x1
nZ 0
− ε ε |Uθ, ∆x (x, ti ) − Uθ,∆x (x, ti )|dx +
(
X Z m=2,4,...
xm+1 xm−1
|Uθε,∆x (x, ti ) − Uθε,∆x (x, ti− )|dx
o
) X
≡ kφk∞ I1 +
Im
.
m=2,4,...
To estimate I1 and Im for m = 2, 4, . . ., we observe that x1
Z
ε ε ˜ − |Uθ, ∆x (x, ti ) − Uθ,∆x (x1 + θi ∆x, ti )|dx +
I1 ≤ 0
x1
Z 0
− ε ε ˜ − |Uθ, ∆x (x1 + θi ∆x, ti ) − Uθ,∆x (x, ti )|dx
≤ ε∆˜x osc [Uθ,∆x (x, ti− )|[0,x2 ] ] + osc [vB (t )|[ti − 1 ∆t ,ti ] ] + (∆x)osc [Uθ ,∆x (x, ti− )|[0,x2 ] ], 2
and xm+1
Z Im ≤
xm−1
ε ε ˜ − |Uθ, ∆x (x, ti ) − Uθ,∆x (xm−1 + θi ∆x, ti )|dx +
Z
xm+1 xm−1
− ε |Uθε,∆x (xm−1 + θi ∆˜x, ti− ) − Uθ, ∆x (x, ti )|dx
≤ (2ε ∆˜x + ∆˜x)osc [Uθ,∆x (x, ti− )|[xm−1 ,xm+2 ] ] + (2∆x)osc [Uθ,∆x (x, ti− )|[xm−2 ,xm+1 ] ]. It follows that
|Jεi | = O(1)kφk∞ ∆x(T .V .[U0 ] + T .V .[vB ]) + (ε ∆˜x)osc [vB (t )|[ti − 1 ∆t ,ti ] ] , 2
= O(1)kφk∞ ∆x(T .V .[U0 ] + T .V .[vB ]). It consequently leads to
X
|Jεi | ≤ O(1)kφk∞ r (φ)(∆t )−1 (∆x)(T .V .[U0 ] + T .V .[vB ])
i
≤ O(1)kφk∞ r (φ)(T .V .[U0 ] + T .V .[vB ]), where r (φ) = diameter of support (φ). We prove (i). To show (ii), we first observe that
hJε , J i = i
j
Z
i
Jε
Z
J j dθj
Y
dθl = 0,
(5.11)
l6=j
for i 6= j since θ ∈Φ J j (θ , ∆x, φ)dθj = 0 in (5.7). Next, we have the following estimate that j
R
Z Z Φ
θj ∈Φ
|Jεj − J j |dθj
Y l6=j
dθl ≤
Z Z Φ
θj ∈ Φ
≤ kφk∞
∞
Z 0
Z Z Φ
Y ε,j j |bUθ, dθl ∆x c(x) − bUθ,∆x c(x)||φ(x, tj )|dx dθj l6=j
X Z
xm+1
θj ∈Φ m=1,3,... xm−1
ε |Uθ ,∆x (x, tj ) − Uθ,∆x (x, tj )|
Y − − ε + |Uθ, ( x , t ) − U ( x , t )| dxd θ dθl θ, ∆ x j ∆x j j l6=j
J.M. Hong, Y.-C. Su / Nonlinear Analysis 72 (2010) 635–650
≤ kφk∞
X
649
(2ε ∆˜x) T .V .[Uθε,∆x (x, tj )|[xm−1 ,xm+1 ] ] + T .V .[Uθ ,∆x (x, tj− )|[xm−1 ,xm+1 ] ]
m=1,3,...
≤ kφk∞ (2ε ∆˜x) T .V .[Uθε,∆x (·, tj )] + T .V .[Uθ ,∆x (·, tj− )]
≤ 4kφk∞ (ε∆x)(T .V .[U0 ] + T .V .[vB ]).
(5.12)
Therefore, by (5.11) and (5.12), for i 6= j we have
hJεi , Jεj i = hJεi , Jεj − J j i Z Z Y dθl ≤ kJεi k∞ |Jεj − J j |dθj l6=j
≤ O(1)(kφk∞ ) · ε(∆x) . 2
We complete the proof.
2
Thus, by Lemma 5.1 and the results in [15], we obtain the following theorem. ε Theorem 5.2 (Cf. [15]). Given 0 < ε 1, and let {Uθ, ∆x } be the sequence of approximate solutions to initial-boundary value problem (1.3) by the generalized Glimm scheme. Then, for 0 < ε 1, we can find a null set Nε ⊂ Φ and a subsequence {∆xi } → 0 such that for any θ ∈ Φ \ Nε and φ ∈ C0∞ , we have
X
Jεi (θ , ∆xi , φ) = O(1)ε 1/2
as ∆xi → 0.
i
According to Theorem 5.2, the last term on the RHS of (5.5) is of order ε 1/2 as subsequence ∆xi → 0 and θ ∈ Φ \ Nε . Now, we show the residual of U (x, t ) in Theorem 4.5 is equal to 0, which means that U (x, t ) is a global weak solution of (1.3). First, by the Lebesgue bounded convergence theorem, the residual of U (x, t ) can be estimated by R(U , φ, E ) =
ε lim R(Uθ, ∆xi , φ, E ) +
ε,∆xi →0
Z Z lim
ε,∆xi →0
t >0
a0 G(U ) − (aεθ ,∆xi )0 G(Uθε,∆xi ) dxdt ,
≡ K1 + K2 .
(5.13)
Therefore, by Theorem 5.2. and the results in [15], we obtain that K1 , K2 approach to 0 as ε , ∆xi → 0. We then establish the global existence of weak solutions of (1.3), which is stated in the following theorem. Main theorem. Consider the following initial-boundary value problem
a = 0, t ut + f (a, u)x = a0 g (a, u), (a, u)(x, 0) = (a(x), u0 (x)), v(0, t ) = vB (t ),
(x, t ) ∈ (0, ∞) × (0, ∞),
(5.14)
where u = (v, w) and f = (f1 , f2 ), g = (g1 , g2 ) are smooth functions of a, u satisfying (A1 ) and (A2 ). Also a(x) is a Lipschitz continuous function with small total variation, and u0 (x), vB (t ) ∈ L∞ have small total variations. Let U := (a, u), and {Uθε,∆x } be the sequence of approximate solutions constructed by the generalized Glimm scheme, then there exists a null set Nε in probability space Φ and a sequence {∆xi } such that for any θ ∈ Φ \ Nε , the measurable function U (x, t ) := limε,∆xi →0 Uθε,∆xi is a weak solution of (5.14). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
E. Isaacson, B. Temple, Nonlinear resonant in inhomogenous systems of conservation laws, Contemp. Math. 108 (1990). P.G. LeFloch, T.-P. Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. 5 (1993) 261–280. J. Smoller, Shock Waves and Reaction–Diffusion Equations, 2nd ed., Springer-Verlag, Berlin, New York, 1994. P.D. Lax, Hyperbolic system of conservation laws, II, Comm. Pure Appl. Math. 10 (1957) 537–566. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697–715. T. Nishida, J. Smoller, Mixed problems for nonlinear conservation laws, J. Differential Equations 23 (1977) 244–269. T.P. Liu, Initial-boundary value problems for gas dynamics, Arch. Ration. Mech. Anal. 64 (1977) 137–168. J.B. Goodman, Initial boundary value problem for hyperbolic systems of conservation laws, Ph.D. dissertation, Stanford University, 1982. T.P. Liu, Quasilinear hyperbolic systems, Comm. Math. Phys. 68 (1979) 141–172. D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl. 4 (1997) 1–42. D. Amadori, R.M. Colombo, Continuous dependence for 2 × 2 conservation laws with boundary, J. Differential Equations 138 (1997) 229–266. D. Amadori, L. Gosse, G. Guerra, Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws, Arch. Ration. Mech. Anal. 162 (2002) 327–366. [13] D. Amadori, G. Guerra, Uniqueness and continuous dependence for systems of balance laws with dissipation, Nonlinear Anal. TMA 49 (2002) 987–1014. [14] C. Donadello, A. Marson, Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws, Nonlinear Differential Equations Appl. 14 (2007) 569–592.
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[15] J.M. Hong, An extension of Glimm’s method to inhomogeneous strictly hyperbolic systems of conservation laws by weaker than weak solutions of the Riemann problem, J. Differential Equations 222 (2006) 515–549. [16] J.M. Hong, B. Temple, A bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law, SIAM J. Appl. Math. 64 (3) (2004) 819–857. [17] E. Isaacson, B. Temple, Convergence of 2 × 2 Godunov method for a general resonant nonlinear balance laws, SIAM J. Appl. Math. 55 (1995) 625–640. [18] B. Temple, Global solution of the Cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws, Adv. Appl. Math. 3 (1982) 335–375. [19] P.G. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form, Comm. Partial Differential Equations 13 (1988) 669–727. [20] P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form, Institute for Mathematics and its Applications, Minneapolis, Preprint 593 (1989). [21] G. Dal Maso, P.G. LeFloch, F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995) 483–548. [22] C.M. Dafermos, L. Hsiao, Hyperbolic systems of balance laws with inhomogeneity and dissipation, Indiana U. Math. J. 31 (1982) 471–491. [23] J. Hong, P.G. LeFloch, A version of Glimm method based on generalized Riemann problems, J. Portugal Math. 64 (2007) 199–236. [24] J.M. Hong, C.-H. Hsu, Y.-C. Su, Global solutions for initial-boundary value problem of quasilinear wave equations, J. Differential Equations 245 (2008) 223–248. [25] Y. Chang, J.M. Hong, C.-H. Hsu, Globally Lipschitz continuous solutions to a class of quasilinear wave equations, J. Differential Equations 236 (2007) 504–531. [26] P. Goatin, P.G. LeFloch, The Riemann problem for a class of resonant nonlinear systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004) 881–902. [27] J.M. Hong, B. Temple, The generic solution of the Riemann problem in a neighborhood of a point of resonance for systems of nonlinear balance laws, Methods Appl. Anal. 10 (2) (2003) 279–294. [28] P.D. Lax, Hyperbolic system of conservation laws and mathematical theory of shock waves, in: Conf. Board Math. Sci., vol. 11, SIAM, Philadelpia, PA, 1973. [29] M. Luskin, B. Temple, The existence of a global weak solution to the non-linear waterhammer problem, Comm. Pure Appl. Math. 35 (1982) 697–735.