Global structure stability of Riemann solutions for linearly degenerate hyperbolic conservation laws under small BV perturbations of the initial data

Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

Contents lists available at ScienceDirect

Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

Global structure stability of Riemann solutions for linearly degenerate hyperbolic conservation laws under small BV perturbations of the initial dataI Zhi-Qiang Shao ∗ Department of Mathematics, Fuzhou University, Fuzhou 350002, China

article

abstract

info

In this paper, we study the global structure stability of the Riemann solution u = U ( xt ) for general n × n quasilinear hyperbolic systems of conservation laws under a small BV perturbation of the Riemann initial data. We prove the global existence and uniqueness of piecewise C 1 solution containing only n contact discontinuities to a class of the generalized Riemann problem, which can be regarded as a small BV perturbation of the corresponding Riemann problem, for general n × n linearly degenerate quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to the one of the self-similar solution u = U ( xt ) to the corresponding Riemann problem. Our result indicates that this kind of Riemann solution u = U ( xt ) mentioned above for general n × n quasilinear hyperbolic systems of conservation laws possesses a global nonlinear structure stability under a small BV perturbation of the Riemann initial data. Some applications to quasilinear hyperbolic systems of conservation laws arising in physics, particularly to the system describing the motion of the relativistic string in Minkowski space R1+n , are also given. © 2010 Elsevier Ltd. All rights reserved.

Article history: Received 7 November 2009 Accepted 10 February 2010 Keywords: Generalized Riemann problem Quasilinear hyperbolic system of conservation laws Riemann solution Contact discontinuity Global structure stability

1. Introduction and main result Consider the following quasilinear hyperbolic system of conservation laws:

∂t u + ∂x f (u) = 0,

x ∈ R, t > 0,

(1.1)

where u = (u1 , . . . , un ) is the unknown vector-valued function of (t , x), f : R → R is a given C vector function of u. It is assumed that system (1.1) is strictly hyperbolic, i.e., for any given u on the domain under consideration, the Jacobian A(u) = ∇ f (u) has n real distinct eigenvalues T

n

λ1 (u) < λ2 (u) < · · · < λn (u).

n

3

(1.2)

Let li (u) = (li1 (u), . . . , lin (u)) (resp. ri (u) = (ri1 (u), . . . , rin (u)) ) be a left (resp. right) eigenvector corresponding to λi (u) (i = 1, . . . , n): T

li (u)A(u) = λi (u)li (u)

(resp. A(u)ri (u) = λi (u)ri (u)).

(1.3)

I Supported by the National Natural Science Foundation of China (No. 70371025), the Scientific Research Foundation of the Ministry of Education of China (No. 02JA790014), the Natural Science Foundation of Fujian Province of China (No. 2009J01006) and the Science and Technology Developmental Foundation of Fuzhou University (No. 2004-XQ-16). ∗ Tel.: +86 0591 83852790. E-mail address: [email protected].

1468-1218/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2010.02.009

3792

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

We have det |lij (u)| 6= 0 (equivalently, det |rij (u)| 6= 0).

(1.4)

Without loss of generality, we may assume that on the domain under consideration li (u)rj (u) ≡ δij

(i, j = 1, . . . , n)

(1.5)

and riT (u)ri (u) ≡ 1 (i = 1, . . . , n),

(1.6)

where δij stands for the Kronecker symbol. Clearly, all λi (u), lij (u) and rij (u) (i, j = 1, . . . , n) have the same regularity as A(u), i.e., C 2 regularity. We also assume that on the domain under consideration, each characteristic field is linearly degenerate in the sense of Lax (cf. [1]):

∇λi (u)ri (u) ≡ 0.

(1.7)

We are interested in the generalized Riemann problem for system (1.1), which is a Cauchy problem with a piecewise C 1 initial data of the form:

 t =0:u=

u− 0 (x), u+ 0 (x),

x ≤ 0, x ≥ 0,

(1.8)

+ 1 where u− 0 (x) and u0 (x) are C vector functions defined for x ≤ 0 and x ≥ 0 respectively with + u− 0 (0) 6= u0 (0).

(1.9)

Problems (1.1) and (1.8) may be regarded as a perturbation of the the corresponding Riemann problem (1.1) and t =0:u=

 b u− , b u+ ,

x ≤ 0, x ≥ 0,

(1.10)

in which

b u± = u± 0 (0).

(1.11)

Let

θ = |b u− − b u+ |.

(1.12)

When θ > 0 is suitably small, by Lax [1], the Riemann problem (1.1) and (1.10) admits a unique self-similar solution composed of n + 1 constant states b u(0) = b u− ,b u(1) , . . . ,b u(n−1) ,b u(n) = b u+ separated by contact discontinuities, i.e., u=U

x t

 (0) u , x ≤b λ1 t , b = b u(j) , b λj t ≤ x ≤ b λj+1 t (j = 1, . . . , n − 1),  (n) b u , x ≥b λn t ,

(1.13)

where x = b λj t stands for the jth contact discontinuity (j = 1, . . . , n). As in [2], this kind of solution is simply called Lax’s Riemann solution of the system (1.1). For the self-similar solution of the Riemann problem of general quasilinear hyperbolic systems of conservation laws, the local nonlinear structure stability has been proved by Li and Yu [3] for one-dimensional case, and by Majda [4] for multidimensional case. If the system (1.1) is strictly hyperbolic and linearly degenerate, Li and Kong [5] proved the global structure stability of the self-similar solution with small amplitude under perturbation (1.8) satisfying (1.10). In this case the self-similar solution contains only n contact discontinuities. Precisely speaking, they obtained the following well-known result. Theorem 1.1. Suppose that system (1.1) is strictly hyperbolic and all characteristic fields are linearly degenerate. Suppose + 1 furthermore that u− 0 (x) and u0 (x) are all C vector functions on x ≤ 0 and on x ≥ 0 respectively, satisfying that there exists a constant µ > 0 such that 0

0

− + 1+µ θ , sup{(1 + |x|)1+µ (|u− (|u+ 0 (x)| + |u0 (x)|)} < ∞. 0 (x)| + |u0 (x)|)} + sup{(1 + |x|) x≤0

(1.14)

x≥0

Suppose finally that f (u) is a C 3 vector function and (1.9) holds. Then there exists θ0 > 0 so small that for any given θ ∈ (0, θ0 ], the generalized Riemann problem (1.1) and (1.8) admits a unique global piecewise C 1 solution u = u(t , x)

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

3793

containing only n contact discontinuities with small amplitude x = xi (
t ) (i = 1, . . . , n) on t ≥ 0. This solution possesses a global structure similar to that of the self-similar solution u = U xt of the corresponding Riemann problem (1.1) and (1.10). Precisely speaking,

 (0) u (t , x), u = u(t , x) = u(i) (t , x),  (n) u (t , x),

x ≤ x1 (t ), xi (t ) ≤ x ≤ xi+1 (t ) x ≥ xn (t ),

(i = 1, . . . , n − 1),

(1.15)

where u(i) (t , x) (i = 0, 1, . . . , n) are all C 1 solutions to system (1.1) on each corresponding domains respectively and for i = 1, . . . , n, u(i−1) (t , x) and u(i) (t , x) are connected to each other by the ith contact discontinuity x = xi (t ). Moreover, x0i (0) = b λi

(i = 1, . . . , n)

(1.16)

and u(i) (0, 0) = b u(i)

(i = 0, 1, . . . , n).

(1.17)

Remark 1.1. Recently, under the assumption that Lax’s Riemann solution of the system (1.1) only contains non-degenerate shocks and contact discontinuities but no centered rarefaction waves and other weak discontinuities, Kong [2,6] proved the global structure stability of this kind of Lax’s Riemann solution with small amplitude, Shao [7,8] also studied that the global structure stability of this kind of Lax’s Riemann solution with small amplitude in a half space. However, it is well known that the BV space is a suitable framework for one-dimensional Cauchy problem for the hyperbolic systems of conservation laws (see [9,10]), the result in Bressan [11] suggests that one may achieve global smoothness even if the C 1 norm of the initial data is large. So the following question arises naturally: can we obtain the global existence and uniqueness of piecewise C 1 solution containing only contact discontinuities to a class of the generalized Riemann problem, which can be regarded as a small BV perturbation of the corresponding Riemann problem, for system (1.1) with the following piecewise C 1 initial data:

 b u− + u− (x), t =0:u= b u+ + u+ (x),

x ≤ 0, x ≥ 0,

(1.18)

where u± (x) ∈ C 1 with both bounded C 1 norm and small bounded variation, such that

ku− (x)kC 1 , ku+ (x)kC 1 ≤ M ,

(1.19)

for some M > 0 bounded but possibly large, and also such that +∞

Z

|u0+ (x)|dx, 0

Z

0

|u0− (x)|dx ≤ ε,

(1.20)

−∞

for some ε > 0 sufficiently small? Here, it is important to mention that the global existence of weak solutions to a strictly hyperbolic system of conservation laws in one space dimension when the initial data is a small BV perturbation of a solvable Riemann problem has been proved by Schochet [12], unfortunately his method is not useful to show that the solutions are still contact discontinuities. An analogous result on stability of contact discontinuities under perturbations of small bounded variation is stated by Corli and Sable-Tougeron [13]. In this paper we exploit to some extent the ideas of Bressan [11], we will develop the method of using continuous Glimm’s functional to solve this problem globally and to provide a new, concise proof of the above mentioned problem of the stability of contact discontinuities. The basic idea we will use here is to combine the techniques employed by Li–Kong [5], especially both the decomposition of waves and the global behavior of waves on the contact discontinuity curves, with the method of using continuous Glimm’s functional. However, we must modify Glimm’s functional in order to take care of the presence of contact discontinuities. This makes our new analysis more complicated than those for the C 1 solutions of the Cauchy problem for linearly degenerate quasilinear hyperbolic systems in [11,14,15]. Our main results can be summarized as follows: Theorem 1.2. Suppose that system (1.1) is strictly hyperbolic and all characteristic fields are linearly degenerate. Suppose furthermore that u− (x) and u+ (x) are all C 1 vector functions on x ≤ 0 and on x ≥ 0 respectively with both bounded C 1 norm and u− (0) = u+ (0) = 0.

(1.21)

Suppose finally that − θ = |b u+ − b u− | = |u+ 0 (0) − u0 (0)| > 0

(1.22)

3794

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

is suitably small. Then for any constant M > 0, there exists a positive constant ε so small that if (1.19) holds together with +∞

Z

|u0+ (x)|dx, 0

Z

0

|u0− (x)|dx ≤ ε,

(1.23)

−∞

then the generalized Riemann problem (1.1) and (1.18) admits a unique global piecewise C 1 solution u = u(t , x) only containing n contact discontinuities x = xi (t ) (xi (0) = 0) (i = 1, . . . ,
n) defined for all x ∈ R and all t ≥ 0, which possesses a global structure similar to the one of the self-similar solution u = U xt of the corresponding Riemann problem (1.1) and (1.10). Precisely speaking,

 (0)  u (t , x), u = u(t , x) = u(i) (t , x),   (n) u (t , x),

x ≤ x1 (t ), xi (t ) ≤ x ≤ xi+1 (t )

(i = 1, . . . , n − 1),

(1.24)

x ≥ xn (t ),

where u(i) (t , x) (i = 0, 1, . . . , n) are all C 1 solutions to system (1.1) on each corresponding domains respectively and for i = 1, . . . , n, u(i−1) (t , x) and u(i) (t , x) are connected to each other by the ith contact discontinuity x = xi (t ). Remark 1.2. Our result indicates that the self-similar solution u = U xt of the Riemann problem (1.1) and (1.10) has a global nonlinear structure stability under a small BV perturbation of the Riemann initial data.




Remark 1.3. Suppose that (1.1) is a non-strictly hyperbolic system with characteristics with constant multiplicity, say, on the domain under consideration,

λ1 (u) ≡ · · · ≡ λp (u) < λp+1 (u) < · · · < λn (u) (1 ≤ p ≤ n).

(1.25)

Then the conclusion of Theorem 1.2 still holds (cf. [15]). Some of the results related to these topics are listed below. Chen et al. [16–19] investigated the asymptotic stability of Riemann waves for hyperbolic conservation laws. Hsiao and Tang [20] investigated the construction and qualitative behavior of the solution of the perturbated Riemann problem for the system of one-dimensional isentropic flow with damping. Xin et al. [21,22] proved the nonlinear stability of contact discontinuities in systems of conservation laws. Smoller et al. [23] investigated the instability of rarefaction shocks in systems of conservation laws. For the overcompressive shock waves, Liu [24] proved the nonlinear stability and instability. Bressan and LeFloch [25] investigated the structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws. Lions et al. [26] proved the existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Recently, L1 stability for systems of hyperbolic conservation laws was investigated by Bressan et al. [27] (cf. [9,28–30]). Liu and Xin [31] proved the nonlinear stability of discrete shocks for systems of conservation laws. Dafermos [32] studied the entropy and the stability of classical solutions of hyperbolic systems of conservation laws. For a relaxation system in several space dimensions, Luo and Xin [33] proved the nonlinear stability of shock fronts. Liu and Xin [34] investigated the nonlinear stability of rarefaction waves for compressible Navier–Stokes equations. Hsiao and Pan [35] investigated the nonlinear stability of rarefaction waves for a rate-type viscoelastic system. Moreover, the nonlinear stability of an undercompressive shock for complex Burgers equation was studied by Liu and Zumbrun [36]. For the viscous conservation laws, the theory of nonlinear stability of shock waves was established (see [37,38] and the references therein). This paper is organized as follows. For the sake of completeness, in Section 2, we briefly recall John’s formula on the decomposition of waves with some supplements and give a generalized Hörmander Lemma. In Section 3, we first review the definition of contact discontinuity, and then analyze some properties of waves on the contact discontinuity curves, which will play an important role in our proof. The main result, Theorem 1.2 is proved in Section 4. It is easy to see that Theorem 1.2 can be applied to all physical models discussed in [5,39,40], therefore we do not give the details in this paper. However, of particular interest is the system describing the motion of the relativistic string in Minkowski space R1+n , as an application of Theorem 1.2, the global existence and uniqueness of piecewise C 1 solution with contact discontinuities to the generalized Riemann problem for the system is presented in Section 5. Finally, we remark that the starting motivation of our study comes from two concrete applications. They are both the one-dimensional Born–Infeld system and the system of the motion of relativistic closed strings in the Minkowski space–time R1+n . Here we first give the former for which our work is of importance, the latter will be discussed in detail in Section 5. Example (Generalized Riemann Problem for the Born–Infeld System). The Born–Infeld model is a nonlinear version of Maxwell’s theory, it was introduced by Born and Infeld [41] in 1930s to cutoff (in a nonlinear fashion) the singularities created by point particles in classical Electrodynamics. Recently, the Born–Infeld system has attracted considerable attention because of its new applications in the string theory and high energy physics. We refer to [42,40,43] for mathematical analysis of the BI system and to [44] for its impact in modern high energy physics and string theory. The one-dimensional Born–Infeld

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

3795

system reads (cf. [40]):

   B3 + D2 P1 − D1 P2  = 0, ∂ D + ∂  t 2 x  h        −B2 + D3 P1 − D1 P3   ∂t D3 + ∂x = 0,   h     −D3 + B2 P1 − B1 P2 ∂t B2 + ∂x = 0,   h        D2 + B3 P1 − B1 P3   ∂t B3 + ∂x = 0,   h    p P (u) = D × B, h(u) = 1 + |B|2 + |D|2 + |D × B|2 ,

(1.26)

where u = (D2 , D3 , B2 , B3 )T are the unknown variables, B1 , D1 are real constants and B = (B1 , B2 , B3 )T ,

D = (D1 , D2 , D3 )T ,

P = (P1 , P2 , P3 )T .

From [39], we know that (1.26) is a linearly degenerate hyperbolic system with the following real eigenvalues:

λ1 (u) = λ2 (u) =

P1 − a h

,

λ3 (u) = λ4 (u) =

P1 + a h

,

(1.27)

where a=

q

1 + B21 + D21 > 0.

(1.28)

Hence, system (4.1) is non-strictly hyperbolic but with characteristics with constant multiplicity. They found that it enjoys many interesting properties like non-strictly hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, richness, existence of entropy–entropy flux pairs, etc. We are interested in the generalized Riemann problem for system (1.26), which is a Cauchy problem with a piecewise C 1 initial data of the form:

 t =0:u=

(D2l + D02l (x), D3l + D03l (x), B2l + B02l (x), B3l + B03l (x))T , (D2r + D02r (x), D3r + D03r (x), B2r + B02r (x), B3r + B03r (x))T ,

x ≤ 0, x ≥ 0,

(1.29)

where (D2l , D3l , B2l , B3l )T and (D2r , D3r , B2r , B3r )T are constant vectors, for i = {l, r }, (D02i (x), D03i (x), B02i (x), B03i (x))T ∈ C 1 with both bounded C 1 norm and small bounded variation, such that

kD02i (x)kC 1 , kD03i (x)kC 1 , kB02i (x)kC 1 , kB03i (x)kC 1 ≤ M ,

(1.30)

for some positive constant M (bounded but possibly large). By Theorem 1.2 and Remark 1.3, we get Theorem 1.3. Suppose that (D02l (x), D03l (x), B02l (x), B03l (x))T and (D02r (x), D03r (x), B02r (x), B03r (x))T are all C 1 vector functions on x ≤ 0 and on x ≥ 0 respectively with both bounded C 1 norm and small bounded variation. Suppose furthermore that

θ , |ul (0) − ur (0)| > 0 is suitably small.

(1.31)

Then for any constant M > 0, there exists a positive constant ε so small that if (1.30) holds together with +∞

Z

|D02r (x)|dx, 0

+∞

Z

|D03r (x)|dx,

+∞

Z

0

|B02r (x)|dx,

+∞

Z

0

|B03r (x)|dx ≤ ε

(1.32)

0

and

Z

0

|D02l (x)|dx, −∞

Z

0

|D03l (x)|dx, −∞

Z

0

|B02l (x)|dx, −∞

Z

0

|B03l (x)|dx ≤ ε,

(1.33)

−∞

then the generalized Riemann problem (1.26) and (1.29) admits a unique global piecewise C 1 solution u = u(t , x) only containing contact discontinuities defined for all x ∈ R and all t ≥ 0, which possesses a global structure similar to the one of the self-similar solution to the corresponding Riemann problem (1.26) and

 t =0:u=

(D2l + D02l (0), D3l + D03l (0), B2l + B02l (0), B3l + B03l (0))T , (D2r + D02r (0), D3r + D03r (0), B2r + B02r (0), B3r + B03r (0))T ,

x ≤ 0, x ≥ 0.

(1.34)

3796

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

2. John’s formula, generalized Hörmander lemma For the sake of completeness, in this section we briefly recall John’s formula on the decomposition of waves with some supplements, which will play an important role in our proof. Let

vi = li (u)u (i = 1, . . . , n)

(2.1)

wi = li (u)ux (i = 1, . . . , n),

(2.2)

and where li (u) = (li1 (u), . . . , lin (u)) denotes the ith left eigenvector. By (1.5), it is easy to see that u=

n X

v k r k ( u)

(2.3)

k=1

and ux =

n X

wk rk (u).

(2.4)

∂ ∂ + λi (u) ∂t ∂x

(2.5)

k=1

Let d

=

di t be the directional derivative along the ith characteristic. We have (cf. [45,2,5]) dv i di t

=

n X

βijk (u)vj wk (i = 1, . . . , n),

(2.6)

j,k=1

where

βijk (u) = (λk (u) − λi (u))li (u)∇ rj (u)rk (u).

(2.7)

Hence, we have

βiji (u) ≡ 0,

∀i, j.

(2.8)

On the other hand, we have (cf. [45,2,5]) dw i di t

=

n X

γijk (u)wj wk (i = 1, . . . , n),

(2.9)

j,k=1

where 1 {(λj (u) − λk (u))li (u)∇ rk (u)rj (u) − ∇λi (u)rj (u)δik + (j|k)}, 2 in which (j|k) denotes all the terms obtained by changing j and k in the previous terms. We have

γijk (u) =

γijj (u) ≡ 0,

∀j 6= i (i, j = 1, . . . , n).

(2.10)

(2.11)

Moreover, if the ith characteristic λi (u) is linearly degenerate in the sense of Lax, we have

γiii (u) ≡ 0.

(2.12)

Noting (2.4), by (2.9) we have (cf. [15]) n X ∂wi ∂(λi (u)wi ) def + = Γijk (u)wj wk = Gi (t , x), ∂t ∂x j,k=1

(2.13)

equivalently, d[wi (dx − λi (u)dt )] =

n X

Γijk (u)wj wk dt ∧ dx = Gi (t , x)dt ∧ dx,

(2.14)

j,k=1

where

Γijk (u) =

1 2

(λj (u) − λk (u))li (u)[∇ rk (u)rj (u) − ∇ rj (u)rk (u)].

(2.15)

Hence, we have

Γijj (u) ≡ 0,

∀i, j.

(2.16)

Lemma 2.1 (Generalized Hörmander Lemma). Suppose that u = u(t , x) is a piecewise C 1 solution to system (1.1), τ1 and τ2 are two C 1 arcs which are never tangent to the ith characteristic direction, and D is the domain bounded by τ1 , τ2 and two ith

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

3797

+ 1 characteristic curves L− i and Li . Suppose furthermore that the domain D contains mC curves of discontinuity of u, denoted by b Cj : x = xj (t ) (j = 1, . . . , m), which are never tangent to the ith characteristic direction. Then we have

Z τ1

|wi (dx − λi (u)dt )| ≤

Z τ2

|wi (dx − λi (u)dt )| +

m Z X j =1

|[wi ]dx − [wi λi (u)]dt |

b Cj

Z Z X n + Γijk (u)wj wk dtdx, D j,k=1

(2.17)

where Γijk (u) is given by (2.15) and [wi ] = wi+ − wi− denotes the jump of wi over the curve of discontinuity b Cj (j = 1, . . . , m), etc. The proof can be found in [5]. 3. Contact discontinuity In this section, we first review the definition of contact discontinuity, and then analyze some properties of waves on the contact discontinuity curves, which will play an important role in our proof. Definition 3.1. A piecewise C 1 vector function u = u(t , x) is called a piecewise C 1 solution containing a kth contact discontinuity x = xk (t )(xk (0) = 0) for system (1.1), if u = u(t , x) satisfies system (1.1) away from x = xk (t ) in the classical sense and satisfies on x = xk (t ) the Rankine–Hugoniot condition: f (u+ ) − f (u− ) = s(u+ − u− ),

(3.1)

and s = λk (u+ ) = λk (u− ), where u

±

(3.2)

= u (t , xk (t )) , u(t , xk (t ) ± 0) and s = ±

dxk (t ) . dt

Definition 3.1 can be found in [1] or [3]. The following lemma gives some properties of waves on the contact discontinuity curves. Lemma 3.1. Suppose that |u± | (u± = u(t , xk (t ) ± 0)) are suitably small. Then, on the kth contact discontinuity x = xk (t ) we have

vi+ = vi− + O(|v ± |2 ) (i = 1, . . . , k − 1, k + 1, . . . , n)

(3.3)

and

! wi = wi + O | u − u | · +

−

+

−

X

±

|wj |

(i = 1, . . . , k − 1, k + 1, . . . , n),

(3.4)

j6=k

where v = (v1 , . . . , vn )T is defined by (2.1) and v ± , v(t , xk (t ) ± 0), etc. The proof can be found in [5]. Corollary 3.1. On the kth contact discontinuity x = xk (t ), it holds that

! (wi λi (u))+ = (wi λi (u))− + O |u+ − u− | ·

X

|wj± |

(i = 1, . . . , k − 1, k + 1, . . . , n),

(3.5)

j6=k

provided that |u± | is small. Proof. Noting

(wi λi (u))+ − (wi λi (u))− = [wi+ − wi− ](λi (u))+ + wi− [(λi (u))+ − (λi (u))− ], from (3.4), we immediately get (3.5).

(3.6)



4. Proof of Theorem 1.2 For the sake of simplicity and without loss of generality, we may suppose that 0 < λ1 (0) < λ2 (0) < · · · < λn (0)

(4.1)

3798

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

and

|b u± | ≤ θ .

(4.2)

By the existence and uniqueness of local classical discontinuous solutions of quasilinear hyperbolic systems of conservation laws (see [3]), when θ > 0 is suitably small, the generalized Riemann problem (1.1) and (1.18) admits a unique piecewise C 1 solution u = u(t , x) only containing n contact discontinuities x = xi (t ) (i = 1, . . . , n) on the strip [0, h] × R, where h > 0 is a small number; moreover, this solution has a local structure similar to the one of the self-similar solution to the corresponding Riemann problem. In order to prove Theorem 1.2, it suffices to establish a uniform a priori estimate for the piecewise C 0 norm of u and ux on any given domain of existence of the piecewise C 1 solution u = u(t , x). By (4.1), there exist sufficiently small positive constants δ and δ0 such that

λi+1 (u) − λi (v) ≥ δ0 ,

∀|u|, |v| ≤ δ (i = 1, . . . , n − 1).

(4.3)

For the time being it is supposed that on the domain of existence of the piecewise C solution u = u(t , x) to the generalized Riemann problem (1.1) and (1.18), we have 1

|u(t , x)| ≤ δ.

(4.4)

At the end of the proof of Lemma 4.5, we will explain that this hypothesis is reasonable. For any fixed T > 0, let U∞ (T ) = sup sup |u(t , x)|,

(4.5)

0≤t ≤T x∈R

V∞ (T ) = sup sup |v(t , x)|,

(4.6)

0≤t ≤T x∈R

W∞ (T ) = sup sup |w(t , x)|,

(4.7)

0≤t ≤T x∈R

e1 (T ) = max max sup W i=1,...,n j6=i

Z

|wi (t , x)|dt ,

(4.8)

e Cj

e Cj

where | · | stands for the Euclidean norm in Rn , v = (v1 , . . . , vn )T and w = (w1 , . . . , wn )T in which vi and wi are defined by (2.1) and (2.2) respectively, while e Cj stands for any given jth characteristic on the domain [0, T ] × R. In (4.4)–(4.7), on any contact discontinuity curve x = xk (t ) the values of u(t , x), v(t , x) and w(t , x) are taken to be u± (t , x) = u(t , xk (t ) ± 0), v ± (t , x) = v(t , xk (t ) ± 0) and w± (t , x) = w(t , xk (t ) ± 0). Clearly, V∞ (T ) is equivalent to U∞ (T ). First we recall some basic L1 estimates. They are essentially due to [46,47,14]. Lemma 4.1. Let φ = φ(t , x) ∈ C 1 satisfy

φt + (λ(t , x)φ)x = F (t , x),

0 ≤ t ≤ T , x ∈ R,

φ(0, x) = g (x),

where λ ∈ C . Then 1

+∞

Z

|φ(t , x)|dx ≤ −∞

Z

+∞

|g (x)|dx +

T

Z

−∞

+∞

Z

|F (t , x)|dxdt ,

∀t ≤ T ,

(4.9)

−∞

0

provided that the right-hand side of the inequality is bounded. Lemma 4.2. Let φ = φ(t , x) and ψ = ψ(t , x) be C 1 functions satisfying

φt + (λ(t , x)φ)x = F1 (t , x),

0 ≤ t ≤ T , x ∈ R,

φ(0, x) = g1 (x),

and

ψt + (µ(t , x)ψ)x = F2 (t , x),

0 ≤ t ≤ T , x ∈ R,

ψ(0, x) = g2 (x),

respectively, where λ, µ ∈ C such that there exists a positive constant δ0 independent of T verifying 1

µ(t , x) − λ(t , x) ≥ δ0 ,

0 ≤ t ≤ T , x ∈ R.

Then T

Z 0

Z

+∞

|φ(t , x)| |ψ(t , x)|dxdt ≤ C

+∞

Z

−∞

|g1 (x)|dx + −∞

Z

|g2 (x)|dx + −∞

+∞

Z

|F1 (t , x)|dxdt



−∞

0

+∞

×

T

Z Z

0

T

Z

+∞

 |F2 (t , x)|dxdt ,

−∞

provided that the two factors on the right-hand side of the inequality is bounded.

(4.10)

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

3799

Fig. 1. The domain ABCD in (t , x) plane.

In the present situation, similar to the above basic L1 estimates (4.9)–(4.10), we have Lemma 4.3. Under the assumptions of Theorem 1.2, on any given domain of existence [0, T ] × R of the piecewise C 1 solution u = u(t , x) to the generalized Riemann problem (1.1) and (1.18), there exists a positive constant k1 independent of ε , T and M such that

Z

+∞

|wi (t , x)|dx ≤ k1

(Z

−∞

0

|u0− (x)|dx

+∞

Z

|u0+ (x)|dx

+

−∞

e1 (T ) + + V∞ ( T ) W

T

Z

0

)

+∞

Z

|Gi (t , x)|dxdt , −∞

0

∀t ≤ T ,

(4.11)

provided that the right-hand side of the inequality is bounded.

R +∞

Proof. To estimate −∞ |wi (t , x)|dx, we need only to estimate

Z

a

|wi (t , x)|dx

(4.12)

−a

for any given a > 0 and then let a → +∞. For i = 1, . . . , n, for any given t with 0 ≤ t ≤ T , passing through point A(t , a) (a > xn (t )) (resp. B(t , −a)), we draw the ith backward characteristic which intersects the x-axis at a point D(0, xD ) (resp. C (0, xC )), see Fig. 1. Then, applying (2.17) on the domain ABCD, we have

Z

|wi (t , x)|dx ≤ BA

xD

Z

|wi (0, x)|dx +

xC

n Z X

|([wi ]x0k (t ) − [wi λi (u)])dt | +

ZZ

b Ck

k=1

|Gi |dxdt ,

(4.13)

ABCD

where b Ck : x = xk (t ) stands for the kth contact discontinuity passing through the origin, which is contained in the region ABCD. Thus, noting (3.2), we get

Z

a

|wi (t , x)|dx ≤ −a

Z n X

+∞

Z

|wi (0, x)|dx + −∞

k=1,k6=i

|([wi ]xk (t ) − [wi λi (u)])dt | + 0

T

Z

b Ck

+∞

Z

0

|Gi |dxdt .

(4.14)

−∞

Using (3.4)–(3.5) and (4.4), it is easy to see that

Z

a

|wi (t , x)|dx ≤ −a

Z

+∞

e1 (T ) + |wi (0, x)|dx + c1 V∞ (T )W

−∞

(Z

T

Z

|u0− (x)|dx

≤ c2 −∞

+∞

|Gi |dxdt 0

0

Z

−∞

+∞

Z

|u0+ (x)|dx

+ 0

e1 (T ) + + V∞ ( T ) W

T

Z 0

Z

+∞

) |Gi |dxdt ,

(4.15)

−∞

where here and henceforth, ci (i = 1, 2, . . .) will denote positive constants independent of ε , T and M. Letting a → +∞, we immediately get the assertion in (4.11). The proof of Lemma 4.3 is finished.  Lemma 4.4. Under the assumptions of Theorem 1.2, on any given domain of existence [0, T ] × R of the piecewise C 1 solution u = u(t , x) to the generalized Riemann problem (1.1) and (1.18), there exists a positive constant k2 independent of ε , T and M such that

3800

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808 T Z

Z

+∞

|wi (t , x)| |wj (t , x)|dxdt

−∞

0

0

Z

|u0− (x)|dx +

Z

|u0− (x)|dx +

Z

≤ k2 −∞

×

e1 (T ) + |u0+ (x)|dx + V∞ (T )W

Z

e1 (T ) + |u0+ (x)|dx + V∞ (T )W

Z

−∞

T

+∞

−∞

T

Z

+∞

 |Gi (t , x)|dxdt ! |Gj (t , x)|dxdt ,

−∞

0

0

+∞

Z

0

0

0

Z

+∞

∀i 6= j (i, j = 1, . . . , n),

(4.16)

provided that the right-hand side of the inequality is bounded. Proof. To estimate T

Z

Z

+∞

|wi (t , x)| |wj (t , x)|dxdt ,

(4.17)

−∞

0

it is enough to estimate T

Z 0

Z

L

|wi (t , x)| |wj (t , x)|dxdt

(4.18)

−L

for any given L > 0 and then let L → +∞. For i, j ∈ {1, . . . , n} and i 6= j, without loss of generality, we suppose that i < j. Let x = xi (t , L) (0 ≤ t ≤ T ) be the ith forward characteristic passing through point (0, L) (L > xn (T )). Then, we draw the ith backward characteristic x = si (t ) (0 ≤ t ≤ T ) passing through point (T , a) (a > xi (T , L)). In the meantime, passing through the point (T , −L), we draw the jth characteristic x = sj (t ) (0 ≤ t ≤ T ) which intersects the x-axis at a point. We introduce the ‘‘continuous Glimm’s functional’’ (cf. [11,48,49,14]) Q (t ) =

ZZ sj (t )
|wj (t , x)| |wi (t , y)|dxdy.

(4.19)

Because of the piecewise C 1 solution u = u(t , x) containing only n contact discontinuities x = xk (t )(xk (0) = 0) (k = e , {(x, y)|sj (t ) < x < y < si (t )} by the straight lines y = xk (t ) (k = 1, . . . , n) 1, . . . , n), we divide the bounded domain Ω e reveal that into some parts. Then, the straightforward calculations on each part of the domain Ω dQ (t ) dt

= s0i (t )|wi (t , si (t ))| +

n X

Z

si (t ) sj (t )

xk (t ){|wi (t , xk (t ) − 0)| − |wi (t , xk (t ) + 0)|} 0

ZZ sj (t )
= s0i (t )|wi (t , si (t ))| +

n X

Z

∂ (|wj (t , x)|)|wi (t , y)|dxdy + ∂t si (t ) sj (t )

sj (t )

sj (t )

x0k (t ){|wi (t , xk (t ) − 0)| − |wi (t , xk (t ) + 0)|}

|wj (t , x)|dx

sj (t )
Z

si (t )

sj (t )

xk ( t )

Z

sj (t )

∂ (λj (u)|wj (t , x)|)|wi (t , y)|dxdy − sj (t )
|wi (t , x)|dx

ZZ

|wj (t , x)|dx − s0j (t )|wj (t , sj (t ))|

k=1

si (t )

xk ( t )

Z

k=1

+

Z

|wj (t , x)|dx − s0j (t )|wj (t , sj (t ))|

|wj (t , x)|

|wi (t , x)|dx

|wj (t , x)|dx

ZZ

−

sj (t )
Z

si (t )

=− sj (t )

sj (t )
sj (t )
(λj (u(t , x)) − λi (u(t , x)))|wi (t , x)| |wj (t , x)|dx

+ (s0i (t ) − λi (u(t , si (t ))))|wi (t , si (t ))|

Z

+ (λj (u(t , sj (t ))) − sj (t ))|wj (t , sj (t ))|

Z

0

si (t )

sj (t ) si (t ) sj (t )

|wj (t , x)|dx |wi (t , x)|dx

∂ (|wi (t , y)|)dxdy ∂t

|wj (t , x)|

∂ (λi (u)|wi (t , y)|)dxdy ∂y

|wj (t , x)|sgn(wi )Gi (t , y)dxdy

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

+ (x0i (t ) − λi (u(t , xi (t ) − 0)))|wi (t , xi (t ) − 0)|

Z

+ (λi (u(t , xi (t ) + 0)) − x0i (t ))|wi (t , xi (t ) + 0)|

Z

n X

+

xi (t )

|wj (t , x)|dx

sj (t ) xi (t )

|wj (t , x)|dx

sj (t )

x0k (t ){|wi (t , xk (t ) − 0)| − |wi (t , xk (t ) + 0)|}

n X

xk ( t )

Z

k=1,k6=i

+

3801

|wj (t , x)|dx

sj (t )

{λi (u(t , xk (t ) + 0))|wi (t , xk (t ) + 0)| − λi (u(t , xk (t ) − 0))|wi (t , xk (t ) − 0)|}

k=1,k6=i

ZZ + sj (t )
ZZ

sgn(wj )Gj (t , x)|wi (t , y)|dxdy +

sj (t )
Z

xk (t ) sj (t )

|wj (t , x)|dx

|wj (t , x)|sgn(wi )Gi (t , y)dxdy.

(4.20)

Noting (3.2) and (4.1) and using (4.3), we get from (4.20) that dQ (t )

si (t )

Z ≤ −δ0

dt

sj (t )

|wi (t , x)| |wj (t , x)|dx +

X

x0k (t ){|wi (t , xk (t ) − 0)

k6=i

− wi (t , xk (t ) + 0)|}

xk (t )

Z

sj (t )

X {|λi (u(t , xk (t ) + 0))wi (t , xk (t ) + 0)

|wj (t , x)|dx +

k6=i xk (t )

Z

− λi (u(t , xk (t ) − 0))wi (t , xk (t ) − 0)|} |wj (t , x)|dx sj (t ) Z si (t ) Z si (t ) Z si (t ) Z si (t ) + |Gj (t , x)|dx |wi (t , x)|dx + |Gi (t , x)|dx |wj (t , x)|dx sj (t ) sj (t ) sj (t ) sj (t ) Z si (t ) X x0k (t ){|wi (t , xk (t ) − 0) ≤ −δ0 |wi (t , x)| |wj (t , x)|dx + sj (t )

k6=i

− wi (t , xk (t ) + 0)|}

+∞

Z

|wj (t , x)|dx +

X {|λi (u(t , xk (t ) + 0))wi (t , xk (t ) + 0)

−∞

Z

k6=i +∞

− λi (u(t , xk (t ) − 0))wi (t , xk (t ) − 0)|} |wj (t , x)|dx −∞ Z +∞ Z +∞ Z +∞ Z + |Gj (t , x)|dx |wi (t , x)|dx + |Gi (t , x)|dx −∞

−∞

−∞

+∞

|wj (t , x)|dx.

(4.21)

−∞

It then follows from Lemma 4.3 that dQ (t )

+ δ0

dt

si (t )

Z

sj (t )

|wi (t , x)| |wj (t , x)|dx

+∞

Z

|Gj (t , x)|dx

≤ k1 −∞

|u0− (x)|dx

+∞

Z

|u0+ (x)|dx

+

−∞

X

+ k1

0

Z

e1 (T ) + + V∞ (T )W

0

T

Z 0

x0k (t ){|wi (t , xk (t ) − 0) − wi (t , xk (t ) + 0)|} +

Z

!

+∞

|Gi (t , x)|dxdt

−∞

X {|λi (u(t , xk (t ) + 0))wi (t , xk (t ) + 0)

k6=i

k6=i

− λi (u(t , xk (t ) − 0))wi (t , xk (t ) − 0)|} +

Z

!

+∞

|Gi (t , x)|dx −∞

Z

0

|u0− (x)|dx

×

+∞

Z

|u0+ (x)|dx

+

−∞

e1 (T ) + + V∞ (T )W

0

T

Z 0

Z

+∞

! |Gj (t , x)|dxdt .

−∞

Therefore

δ0

T

Z 0

Z

si (t ) sj (t )

|wi (t , x)| |wj (t , x)|dxdt

≤ Q (0) + k1

T

Z 0

Z

+∞

|Gj (t , x)|dxdt −∞

Z

0

|u0− (x)|dx + −∞

+∞

Z

e1 (T ) |u0+ (x)|dx + V∞ (T )W 0

(4.22)

3802

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808 T

Z

!

+∞

Z

|Gi (t , x)|dxdt + k1

+ −∞

0

|u0− (x)|dx

×

|u0+ (x)|dx

+

−∞

|[wi ]|λk (u )dt +

+∞

Z

XZ

±

b Ck

k6=i

0

Z

XZ

e1 (T ) + + V∞ (T )W

T

Z

!

+∞

Z

0

|Gi (t , x)|dxdt −∞

!

+∞

Z

|Gj (t , x)|dxdt .

(4.23)

−∞

0

0

|[wi λi (u)]|dt +

b Ck

k6=i

T

Z

Using (3.4)–(3.5) and noting (4.4), we obtain

δ0

T

Z

si (t )

Z

sj (t )

0

|wi (t , x)| |wj (t , x)|dxdt

≤ Q (0) + c3

Z

0

|u0− (x)|dx + −∞

|u0− (x)|dx +

×

e1 (T ) + |u0+ (x)|dx + V∞ (T )W

0

+∞

Z

−∞

T

Z

0

0

Z

+∞

Z

e1 (T ) + |u0+ (x)|dx + V∞ (T )W

T

Z

+∞

|Gi (t , x)|dxdt



−∞

 |Gj (t , x)|dxdt .

(4.24)

−∞

0

0

+∞

Z

Z

Noting +∞

Z

Q (0) ≤

|wi (0, x)|dx

+∞

Z

−∞

|wj (0, x)|dx,

(4.25)

−∞

we get

δ0

T

Z

si (t )

Z

sj (t )

0

|wi (t , x)| |wj (t , x)|dxdt

0

Z

|u0− (x)|dx +

Z

|u0− (x)|dx +

Z

≤ c4 −∞

×

e1 (T ) + |u0+ (x)|dx + V∞ (T )W

Z

e1 (T ) + |u0+ (x)|dx + V∞ (T )W

Z

0

0

Z

+∞

−∞

T

Z

0

|Gi (t , x)|dxdt



−∞

0

+∞

+∞

T

+∞

Z

 |Gj (t , x)|dxdt .

(4.26)

−∞

0

It then follows T

Z

Z

si (t )

sj (t )

0

≤

|wi (t , x)| |wj (t , x)|dxdt

Z

c4

δ0

0

|u0− (x)|dx

e1 (T ) + + V∞ (T )W

|u0− (x)|dx + −∞

Z

e1 (T ) + |u0+ (x)|dx + V∞ (T )W

T

Z

0

+∞

|Gi (t , x)|dxdt



−∞

0

+∞

Z

T

Z

0

0

Z

|u0+ (x)|dx

+

−∞

×

+∞

Z

+∞

Z

 |Gj (t , x)|dxdt .

(4.27)

−∞

0

Therefore T

Z 0

≤

Z

L

|wi (t , x)| |wj (t , x)|dxdt

−L

c4

Z

δ0

0

|u0− (x)|dx + −∞

Z ×

−∞

e1 (T ) + |u0+ (x)|dx + V∞ (T )W |u0+ (x)|dx

+ 0

Z

e1 (T ) + + V∞ (T )W

Z 0

T

+∞

|Gi (t , x)|dxdt



−∞

0

+∞

Z

T

Z

0

0

|u0− (x)|dx

+∞

Z

Z

+∞

|Gj (t , x)|dxdt

 (4.28)

−∞

and the desired conclusion follows by taking L → +∞. The proof of Lemma 4.4 is finished.



Lemma 4.5. Under the assumptions of Theorem 1.2, for small θ > 0 there exists a constant ε > 0 so small that on any given domain of existence [0, T ] × R of the piecewise C 1 solution u = u(t , x) to the generalized Riemann problem (1.1) and (1.18), there exist positive constants k3 , k4 and k5 independent of θ , ε , T and M, such that the following uniform a priori estimates hold:

e1 (T ) ≤ k3 ε, W

(4.29)

U∞ (T ), V∞ (T ) ≤ k4 θ

(4.30)

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

3803

Fig. 2. The ith contact discontinuity passing through O(0, 0).

and W∞ (T ) ≤ k 5 M .

(4.31)

Proof. We introduce QW (T ) =

n XZ X

T

|wi (t , x)| |wj (t , x)|dxdt .

(4.32)

−∞

0

j=1 i6=j

+∞

Z

By (2.13), it follows from Lemma 4.4 that QW (T ) ≤ c5

Z

0

|u0− (x)|dx + −∞

+∞

Z

e1 (T ) + |u0+ (x)|dx + V∞ (T )W 0

T

Z 0

Z

+∞

|G(t , x)|dxdt

2

,

(4.33)

−∞

where G = (G1 , G2 , . . . , Gn ), here and henceforth ci (i = 5, 6, . . .) will denote positive constants independent of θ , ε , T and M. Noting (2.16), we have T

Z

+∞

Z

|G(t , x)|dxdt ≤ c6 QW (T ).

(4.34)

−∞

0

Substituting (4.34) into (4.33) and noting (1.23), we obtain


2

e1 (T ) + QW (T ) QW (T ) ≤ c7 ε + V∞ (T )W

.

(4.35)

e1 (T ). We next estimate W Let e Cj : x = xj (t ) (0 ≤ t1 ≤ t ≤ t2 ≤ T )

(4.36)

be any given jth characteristic on the domain [0, T ] × R. Then, passing through the point P1 (t1 , xj (t1 )) (resp. P2 (t2 , xj (t2 ))) we draw the ith characteristic which intersects the x-axis at a point A1 (0, y1 ) (resp. A2 (0, y2 )). Without loss of generality, we assume that the ith contact discontinuity x = xi (t ) passing through O(0, 0) intersects e Cj with point P0 (t0 , xj (t0 )) (see Fig. 2). It is easy to see that

Z

|wi (t , x)|dt =

e Cj

Z

t0

|wi (t , xj (t ))|dt +

t1

We now estimate

R t2 t0

Z

t2

|wi (t , xj (t ))|dt .

(4.37)

t0

|wi (t , xj (t ))|dt.

Applying (2.17) on the domain P0 OA2 P2 bounded by the ith contact discontinuity x = xi (t ), e Cj , the ith characteristic passing through A2 and the x-axis, and noting (2.16), it is easy to see that

Z

t2

|wi (t , xj (t ))| |λj (u(t , xj (t ))) − λi (u(t , xj (t )))|dt

t0 y2

Z ≤ 0

|wi (0, x)|dx +

XZ k∈S1

b Ck

|([wi ]x0k (t ) − [wi λi (u)])dt | +

ZZ

X

P0 OA2 P2 j6=k

|Γijk (u)wj wk |dtdx,

(4.38)

3804

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

where S1 stands for the set of all indices k such that the kth contact discontinuity b Ck : x = xk (t ) is partly contained in the domain P0 OA2 P2 . Using (1.23), (3.4), (3.5) and (4.4), and noting the fact that i 6∈ S1 , we obtain t2

Z

 e1 (T ) + QW (T ) . |wi (t , xj (t ))|dt ≤ c8 ε + V∞ (T )W

(4.39)

t0

Similarly, we have t0

Z

 e1 (T ) + QW (T ) . |wi (t , xj (t ))|dt ≤ c9 ε + V∞ (T )W

(4.40)

t1

Thus, we get

e1 (T ) ≤ c10 ε + V∞ (T )W e1 (T ) + QW (T ) . W 



(4.41)

We next estimate U∞ (T ) and V∞ (T ). Passing through any fixed point (t , x) ∈ [0, T ]× R, we draw the ith backward characteristic Ci which intersects the x-axis at a point (0, y). Integrating (2.6) along this characteristic Ci and noting (2.8) yields

vi (t , x) = vi (0, y) +

Z X [vi ]k + k∈S2

n X

βijk (u)vj wk dt ,

(4.42)

Ci j,k=1,k6=i

where S2 denotes the set of all indices k such that this characteristic Ci intersects the kth contact discontinuity x = xk (t ) at a point (tk , xk (tk )), and [vi ]k = vi (tk , xk (tk ) + 0) − vi (tk , xk (tk ) − 0). Noting (1.21) and using (1.23), we have

|u+ (x)| ≤

+∞

Z

|u0+ (x)|dx ≤ ε,

∀x ∈ R+

(4.43)

∀x ∈ R− .

(4.44)

0

and

|u− (x)| ≤

Z

0

|u0− (x)|dx ≤ ε, −∞

Therefore, noting the fact that i 6∈ S2 , and using (2.1), (3.3), (4.2) and (4.4), we get from (4.42)–(4.44) that

e1 (T )]}. V∞ (T ) ≤ c11 {θ + ε + V∞ (T )[V∞ (T ) + W

(4.45)

We now prove (4.29)–(4.30) and QW (T ) ≤ k6 ε 2 ,

(4.46)

where k6 is a positive constant independent of θ , ε and T . Recalling (4.2), (4.43) and (4.44), evidently we have U∞ (0), V∞ (0) ≤ c12 θ

(4.47)

e1 (0) = 0, QW (0) = W

(4.48)

and

provided that ε  θ . Thus, by continuity there exist positive constants k3 , k4 and k6 independent of θ , ε and T such that (4.29)–(4.30) and (4.46) hold at least for 0 ≤ T ≤ τ0 , where τ0 is a small positive number. Hence, in order to prove (4.29)–(4.30) and (4.46) it suffices to show that we can choose k3 , k4 and k6 in such a way that for any fixed T0 (0 < T0 ≤ T ) such that

e1 (T0 ) ≤ 2k3 ε, W

(4.49)

V∞ (T0 ) ≤ 2k4 θ ,

(4.50)

QW (T0 ) ≤ 2k6 ε ,

(4.51)

2

we have

e1 (T0 ) ≤ k3 ε, W

(4.52)

V∞ (T0 ) ≤ k4 θ ,

(4.53)

QW (T0 ) ≤ k6 ε .

(4.54)

2

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

3805

To this end, substituting (4.49)–(4.51) into the right-hand side of (4.35), (4.41) and (4.45) (in which we take T = T0 ), it is easy to see that, when θ > 0 is suitably small, we have QW (T0 ) ≤ 4c7 ε 2 ,

(4.55)

e1 (T0 ) ≤ 2c10 ε, W

(4.56)

V∞ (T0 ) ≤ 3c11 θ ,

(4.57)

provided that ε  θ . Hence, if k3 ≥ 2c10 , k4 ≥ 3c11 and k6 ≥ 4c7 , then we get (4.52)–(4.54), provided that θ is suitably small. This proves (4.29)–(4.30) and (4.46). We finally estimate W∞ (T ). For any fixed point (t , x) ∈ [0, T ] × R, we draw the ith backward characteristic Ci passing through the point (t , x), which intersects the x-axis at a point (0, y). Integrating (2.9) along this characteristic Ci and noting (2.11)–(2.12) yields

wi (t , x) = wi (0, y) +

Z X [wi ]k + k∈S3

n X

γijk (u)wj wk dt ,

(4.58)

Ci j,k=1,j6=k

where S3 denotes the set of all indices k such that this characteristic Ci intersects the kth contact discontinuity x = xk (t ) at a point (tk , xk (tk )), and [wi ]k = wi (tk , xk (tk ) + 0) − wi (tk , xk (tk ) − 0). Using (3.4) and (4.4) and noting the fact that i 6∈ S3 , we have

e1 (T )}. W∞ (T ) ≤ W∞ (0) + c13 {V∞ (T )W∞ (T ) + W∞ (T )W

(4.59)

Hence, noting (1.19), (4.29) and (4.30), it is easy to see that W∞ (T ) ≤ c14 {M + θ W∞ (T ) + ε W∞ (T )},

(4.60)

which implies (4.31). Finally, we observe that when θ > 0 is suitably small, by (4.30) we have U∞ (T ) ≤ k4 θ ≤

1 2

δ.

This implies the validity of hypothesis (4.4). The proof of Lemma 4.5 is finished.

(4.61) 

Proof of Theorem 1.2. Under the assumptions of Theorem 1.2, from (4.30) and (4.31), we know that for small θ > 0 there exists ε > 0 suitably small such that on any given domain of existence [0, T ] × R of the piecewise C 1 solution u = u(t , x) to the generalized Riemann problem (1.1) and (1.18), the piecewise C 1 norm of the solution possesses a uniform a priori estimate independent of T . This leads to the conclusion of Theorem 1.2 immediately. The proof of Theorem 1.2 is finished.  5. Application Now let us consider the generalized Riemann problem for the system of the motion of relativistic strings in the Minkowski space R1+n . Recall Kong et al.’s work [50] at first. We denote by X = (t , x1 , . . . , xn ) points in the (1 + n)-dimensional Minkowski space R1+n . Then the scalar product of two vectors X and Y = (e t , y1 , . . . , yn ) in R1+n is defined by X ·Y =

n X

xi yi − te t,

(5.1)

i=1

in particular, X2 =

n X

x2i − t 2 .

(5.2)

i=1

The Lorentzian metric of R1+n can be written as ds2 =

n X

dx2i − dt 2 .

(5.3)

i=1

To describe the motion of a relativistic string in the Minkowski space R1+n , we consider the local equation of an extremal time-like surface S in R1+n taking the following parameter form in a suitable coordinate system (cf. [50]): xi = xi (t , θ ) (i = 1, . . . , n).

(5.4)

3806

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

Then, in the surface coordinates t and θ , the Lorentzian metric (5.3) is expressed as ds2 = (dt , dθ )M (dt , dθ )T ,

(5.5)

where,

 hxt , xθ i , |xθ |2

|xt |2 − 1 M = hxt , xθ i 

(5.6)

in which x = (x1 , . . . , xn )T and

hxt , xθ i =

n X

xi,t xi,θ ,

|xt |2 = hxt , xt i and |xθ |2 = hxθ , xθ i.

(5.7)

i =1

Since the surface S is C 2 and time-like, i.e., det M < 0,

(5.8)

equivalently,

hxt , xθ i2 − (|xt |2 − 1)|xθ |2 > 0,

(5.9)

it follows that the area element of the surface S is dA =

p

hxt , xθ i2 − (|xt |2 − 1)|xθ |2 dtdθ .

(5.10)

The surface S is called to be extremal surface, if x = x(t , θ ) is the critical point of the area functional I =

ZZ p

hxt , xθ i2 − (|xt |2 − 1)|xθ |2 dtdθ .

(5.11)

The corresponding Euler–Lagrange equation is (cf. [50,51])

|xθ |2 xt − hxt , xθ ixθ p

hxt , xθ i2 − (|xt |2 − 1)|xθ |2

! − t

hxt , xθ ixt − (|xt |2 − 1)xθ p hxt , xθ i2 − (|xt |2 − 1)|xθ |2

! = 0.

(5.12)

θ

By computation, it follows from (5.12) that

|xθ |2 xtt − 2hxt , xθ ixt θ + (|xt |2 − 1)xθθ = 0.

(5.13)

Remark 5.1. Taking θ = x1 and n = 2, we observe that Eq. (5.13) is just the classical Born–Infeld equation. Therefore, in this sense, Eq. (5.13) is called the generalized Born–Infeld equation. Let u = xt ,

v = xθ ,

(5.14)

where u = (u1 , . . . , un ) and v = (v1 , . . . , vn ) , then (5.12) can be equivalently rewritten as T

 v − uθ = 0,   t ! |v|2 u − hu, viv  −  p hu, vi2 − (|u|2 − 1)|v|2 t

T

hu, viu − (|u|2 − 1)v p

!

hu, vi2 − (|u|2 − 1)|v|2

= 0.

(5.15)

θ

We are interested in the generalized Riemann problem for system (5.15), which is a Cauchy problem with a piecewise C 1 initial data of the form: t = 0 : (u, v) =

 (e u0 + u− (θ ),e v0 + v− (θ )), (e u0 + u+ (θ ), e v0 + v+ (θ )),

θ ≤ 0, θ ≥ 0,

(5.16)

where

(u+ (0), v+ (0)) 6= (u− (0), v− (0)), e u0 = (e u01 , . . . ,e u0n )T and e v0 = (e v10 , . . . ,e vn0 )T are constant vectors with |e v0 | =

(5.17)

q

(e v10 )2 + · · · + (e vn0 )2 > 0, (u± (θ ), v± (θ )) ∈

C 1 with both bounded C 1 norm and small bounded variation, such that 0 ku± (θ )kC 0 , kv± (θ )kC 0 , ku0± (θ )kC 0 , kv± (θ )kC 0 ≤ M ,

(5.18)

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

3807

for some positive constant M (bounded but possibly large). Let

  U =

u

v

.

(5.19)

Then, we can rewrite system (5.15) as Ut + A(U )Uθ = 0,

(5.20)

where

 A(U ) =

−

2hu, vi

In×n

|v|2 −In×n

 |u|2 − 1 I n × n . |v|2

(5.21)

0

  e u

0 It is easy to see that in a neighborhood of U0 = e v0 , (5.15) is a hyperbolic system with the following real eigenvalues:

λ1 (U ) ≡ · · · ≡ λn (U ) = λ− < 0 < λn+1 (U ) ≡ · · · ≡ λ2n (U ) = λ+ ,

(5.22)

where

λ± =

−hu, vi ±

p

hu, vi2 − (|u|2 − 1)|v|2 . |v|2

(5.23)

The corresponding left and right eigenvectors are li (U ) = (ei , λ+ ei )

(i = 1, . . . , n),

li (U ) = (ei−n , λ− ei−n )

(i = n + 1, . . . , 2n)

(5.24)

and ri (U ) = (−λ− ei , ei )T

(i = 1, . . . , n),

ri (U ) = (−λ+ ei−n , ei−n )T

(i = n + 1, . . . , 2n)

(5.25)

respectively, where (i)

ei = (0, . . . , 0, 1 , 0, . . . , 0)

(i = 1, . . . , n).

(5.26)

When n = 1, (5.15) is a strictly hyperbolic system; while, when n ≥ 2, (5.15) is a non-strictly hyperbolic system with characteristics with constant multiplicity. It is easy to see that all characteristic fields are linearly degenerate in the sense of Lax, i.e.,

∇λi (U )ri (U ) ≡ 0 (i = 1, . . . , 2n).

(5.27)

By Theorem 1.2 we get Theorem 5.1. Suppose that u− (θ ), v− (θ ), u+ (θ ) and v+ (θ ) are all C 1 vector functions on θ ≤ 0 and on θ ≥ 0 respectively with both bounded C 1 norm and small bounded variation. Suppose furthermore that

η , |(u+ (0), v+ (0)) − (u− (0), v− (0))| > 0 is suitably small.

(5.28)

Then for any constant M > 0, there exists a positive constant ε so small that if (5.18) holds together with +∞

Z

|u0+ (θ )|dθ ,

+∞

Z

0

0 |v+ (θ )|dθ , 0

Z

0

|u0− (θ )|dθ , −∞

Z

0

0 |v− (θ )|dx ≤ ε,

(5.29)

−∞

then the generalized Riemann problem (5.15) and (5.16) admits a unique global piecewise C 1 solution U = U (t , θ ) only containing contact discontinuities defined for all θ ∈ R and all t ≥ 0, which possesses a global structure similar to the one of the self-similar solution to the corresponding Riemann problem (5.15) and t = 0 : (u, v) =



(e u0 + u− (0),e v0 + v− (0)), (e u0 + u+ (0),e v0 + v+ (0)),

θ ≤ 0, θ ≥ 0.

(5.30)

Acknowledgements The author would like to express his gratitude to the referees for their valuable comments and careful reading. The author would like to thank Prof. Ta-tsien Li very much for his valuable lectures and guidance in Fuzhou in December 2003. The author also would like to thank Prof. Jiaxing Hong very much for his guidance and constant encouragement.

3808

Z.-Q. Shao / Nonlinear Analysis: Real World Applications 11 (2010) 3791–3808

References [1] P.D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957) 537–556. [2] D.-X. Kong, Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: Shocks and contact discontinuities, J. Differential Equations 188 (2003) 242–271. [3] T.T. Li, W.C. Yu, Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Series V, Duke University, Durham, 1985. [4] A. Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc. 41 (275) (1983) 1–95. [5] T.T. Li, D.-X. Kong, Global classical discontinuous solutions to a class of generalized Riemann problem for general quasilinear hyperbolic systems of conservations laws, Comm. Partial Differential Equations 24 (1999) 801–820. [6] D.-X. Kong, Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: Rarefaction waves, J. Differential Equations 219 (2005) 421–450. [7] Z.-Q. Shao, Global structure stability of Riemann solutions for general hyperbolic systems of conservation laws in the presence of a boundary, Nonlinear Anal. 69 (2008) 2651–2676. [8] Z.-Q. Shao, Global structure instability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary, J. Math. Anal. Appl. 330 (2007) 511–540. [9] A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford University Press, 2000. [10] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 697–715. [11] A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421. [12] S. Schochet, Sufficient condition for local existence via Glimm’s scheme for large BV data, J. Differential Equations 89 (1991) 317–354. [13] A. Corli, M. Sable-Tougeron, Stability of contact discontinuities under perturbations of bounded variation, Rend. Semin. Mat. Univ. Padova 97 (1997) 35–60. [14] Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Ann. Math. Ser. B 25 (2004) 37–56. [15] W.-R. Dai, D.-X. Kong, Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields, J. Differential Equations 235 (2007) 127–165. [16] G.-Q. Chen, H. Frid, Asymptotic stability of Riemann waves for conservation laws, Z. Angew. Math. Phys. 48 (1997) 30–44. [17] G.-Q. Chen, H. Frid, Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations, Trans. Amer. Math. Soc. 353 (2001) 1103–1117. [18] G.-Q. Chen, H. Frid, Y.C. Li, Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics, Comm. Math. Phys. 228 (2002) 201–217. [19] G.-Q. Chen, Y.C. Li, Stability of Riemann solutions with large oscillation for the relativistic Euler equations, J. Differential Equations 202 (2004) 332–353. [20] L. Hsiao, S.Q. Tang, Construction and qualitative behavior of the solution of the perturbated Riemann problem for the system of one-dimensional isentropic flow with damping, J. Differential Equations 123 (1995) 480–503. [21] Z.P. Xin, On nonlinear stability of contact discontinuities, in: Hyperbolic Problems: Theory, Numerics, Applications, Stony Brook, NY, 1994, World Scientific Publishing, River Edge, NJ, 1996, pp. 249–257. [22] F.M. Huang, Z.P. Xin, T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math. 219 (2008) 1246–1297. [23] J.A. Smoller, J.B. Temple, Z.P. Xin, Instability of rarefaction shocks in systems of conservation laws, Arch. Ration. Mech. Anal. 112 (1990) 63–81. [24] T.P. Liu, Nonlinear stability and instability of overcompressive shock waves, in: J.E. Dunn, R. Posdick, M. Slemrod (Eds.), Shock Induced Transitions and Phase Structures in General Media, in: IMA Volumes in Mathematical Applications, vol. 52, Springer, New York, 1993, pp. 159–167. [25] A. Bressan, P.G. LeFloch, Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws, Indiana Univ. Math. J. 48 (1999) 43–84. [26] P.L. Lions, B. Perthame, P.E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 49 (1996) 599–638. [27] A. Bressan, T.P. Liu, T. Yang, L1 stability estimates for n × n conservation laws, Arch. Ration. Mech. Anal. 149 (1999) 1–22. [28] A. Bressan, G. Crasta, B. Piccoli, Well-posedness of the Cauchy problem for n × n systems of conservation laws, Mem. Amer. Math. Soc. 146 (694) (2000) 1–134. [29] T.P. Liu, T. Yang, L1 stability of conservation laws with coinciding Hugoniot and characteristic curves, Indiana Univ. Math. J. 48 (1999) 237–247. [30] T.P. Liu, T. Yang, L1 stability for 2 × 2 systems of hyperbolic conservation laws, J. Amer. Math. Soc. 12 (1999) 729–774. [31] J.G. Liu, Z.P. Xin, Nonlinear stability of discrete shocks for systems of conservation laws, Arch. Ration. Mech. Anal. 125 (1993) 217–256. [32] C.M. Dafermos, Entropy and the stability of classical solutions of hyperbolic systems of conservation laws, in: T. Ruggeri (Ed.), Recent Mathematical Methods in Nonlinear Wave Propagation, Montecatini Terme, 1994, in: Lecture Notes in Mathematics, vol. 1640, Springer, Berlin, 1996, pp. 48–69. [33] T. Luo, Z.P. Xin, Nonlinear stability of shock fronts for a relaxation systemin several space dimensions, J. Differential Equations 139 (1997) 365–408. [34] T.P. Liu, Z.P. Xin, Nonlinear stability of rarefaction waves for compressible Navier–Stokes equations, Comm. Math. Phys. 118 (1988) 451–465. [35] L. Hsiao, R.H. Pan, Nonlinear stability of rarefaction waves for a rate-type viscoelastic system, Chinese Ann. Math. Ser. B 20 (1999) 223–232. [36] T.P. Liu, K. Zumbrun, Nonlinear stability of an undercompressive shock for complex Burgers equation, Comm. Math. Phys. 168 (1995) 163–186. [37] T.P. Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 56 (328) (1985) 1–108. [38] Z.P. Xin, Theory of viscous conservation laws, in: L. Hsiao, Z. Xin (Eds.), Some Current Topics on Nonlinear Conservation Laws, in: AMS/IP Studies in Advanced Mathematics, vol. 15, Amer. Math. Soc., Providence, RI, 2000, pp. 141–193. [39] T.T. Li, Y.J. Peng, J. Ruiz, Entropy solutions for linearly degenerate hyperbolic systems of rich type, J. Math. Pures Appl. 91 (2009) 553–568. [40] Y. Brenier, Hydrodynamic structure of the augmented Born–Infeld equations, Arch. Ration. Mech. Anal. 172 (2004) 65–91. [41] M. Born, L. Infeld, Foundations of the new field theory, Proc. R. Soc. Lond. Ser. A 144 (1934) 425–451. [42] G. Bolliat, C.M. Dafermos, P.D. Lax, T.P. Liu, Recent Mathematical Methods in Nonlinear Wave Propagation, in: Lecture Notes in Mathematics, vol. 1640, Springer-Verlag, Berlin, 1996. [43] D. Serre, Hyperbolicity of the nonlinear models of Maxwell’s equations, Arch. Ration. Mech. Anal. 172 (2004) 309–331. [44] G.W. Gibbons, Aspects of Born–Infeld theory and string/M-theory, in: Proceedings of the 4th Mexican School on Gravitation and Mathematical Physics, Membrane, 2000. Or http://arxiv.org/abs/hep-th/0106059. [45] F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 (1974) 377–405. [46] M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemann problem, Indiana Univ. Math. J. 34 (1985) 533–589. [47] M. Schatzman, The Geometry of Continuous Glimm Functionals, in: Lectures in Applied Mathematics, vol. 23, American Mathematical Society, Providence, RI, 1986, pp. 417–439. [48] A. Bressan, A locally contractive metric for systems of conservation laws, Ann. Sc. Norm. Super. Pisa IV 22 (1995) 109–135. [49] Z.-Q. Shao, The mixed initial-boundary value problem for quasilinear hyperbolic systems with linearly degenerate characteristics, Nonlinear Anal. 71 (2009) 1350–1368. [50] D.-X. Kong, Q. Zhang, Q. Zhou, The dynamics of relativistic strings moving in the Minkowski space R1+n , Comm. Math. Phys. 269 (2007) 153–174. [51] Z.-Q. Shao, Global weakly discontinuous solutions to the mixed initial-boundary value problem for quasilinear hyperbolic systems, Math. Models Methods Appl. Sci. 19 (2009) 1099–1138.