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Riemann problem and wave interactions for the one-dimensional relativistic string equation in Minkowski space
J. Math. Anal. Appl. 486 (2020) 123932
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Riemann problem and wave interactions for the one-dimensional relativistic string equation in Minkowski space ✩ Jianli Liu ∗ , Ruixi Liu Department of Mathematics, Shanghai University, Shanghai 200444, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 3 October 2019 Available online 6 February 2020 Submitted by T. Yang
In this paper, we consider the Riemann problem for the one-dimensional relativistic string equation in Minkowski space. It is shown that the delta shock wave will be constructed in the Riemann solutions in some certain situations. Furthermore, the interactions of elementary waves including contact discontinuity and delta shock wave are obtained. © 2020 Elsevier Inc. All rights reserved.
Keywords: Riemann problem Relativistic string equation Delta shock wave Wave interactions
1. Introduction In this paper, we will investigate the Riemann problem of the one-dimensional hyperbolic system of conservation laws as follows ⎧ ⎪ ⎨ ut − vx = 0, (1.1) v u ⎪ )t − ( √ )x = 0, ⎩ (√ 1 + u2 − v 2 1 + u2 − v 2 which can be obtained from the equation to one-dimensional relativistic string in Minkowski space (
φt 1+
φ2x
−
φ2t
φx )t − ( )x = 0, 1 + φ2x − φ2t
where u = φx , v = φt denote the functions of (x, t) ∈ R × R+ . Relativistic string equation is very important in Lorentz geometry and particle physics. Kong, Zhang and Zhou [15] studied the dynamics of a relativistic string in Minkowski space R1+n . For the higher dimensional case, Kong and Zhang [14] studied the global ✩
This work was partially supported by NSF of China 11771274.
* Corresponding author. E-mail addresses: [email protected] (J. Liu), [email protected] (R. Liu). https://doi.org/10.1016/j.jmaa.2020.123932 0022-247X/© 2020 Elsevier Inc. All rights reserved.
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J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
motion of relativistic strings in Minkowski space R1+n . Hoppe [13] gave the classical solutions of the equation of motion of a relativistic membrane. The spherical symmetric solutions for the relativistic membranes moving in Schwarzschild spacetime were obtained in [12]. Wang and Wei [29] studied a global smooth solution to the Cauchy problem with a class of large initial date for the relativistic membrane equation embedded in R1+(1+n) with n = 2, 3. The related results can also be found in [16,18,19]. Riemann problem is a kind of initial problem with the special piecewise initial data. Generally, the solution of the Riemann problem involves shock, rarefaction and contact discontinuities. There is one major milestone result by P.D. Lax for small BV solutions of general hyperbolic conservation laws [17], see also [7]. The blowup BV solution for p-system can be found in [1] and reference therein. Chang and Hsiao [2] made a completely understood the solution of Riemann problem for full Euler equation (non-isentropic). Smoller [27] studied Riemann problem for isentropic p-systems and the general hyperbolic systems of conservation laws with the sufficiently close initial states. Sheng and Zhang [26] studied the one-dimensional and twodimensional Riemann problems for the transportation equations in gas dynamics and proved uniqueness and stability for viscous perturbations. Zhang and Lai [34] studied the Riemann problem and interaction of elementary waves for the relativistic p-system with general pressure law p = p(ρ). Zhang and Sheng [35] studied the Riemann problem and waves interactions for the chromatography equations with conservative form. There are lots of references related to the results about the delta shock waves. The author in [31] studied the Riemann problems for a class of coupled hyperbolic systems of conservation laws and gave delta-shock wave solutions and vacuum solutions. The authors in [32] studied new developments of delta shock waves and its applications in systems of conservation laws. The authors in [33] studied the Riemann problems for a class of non-strictly and weakly hyperbolic systems of conservation laws. The Riemann problem with delta initial data and the vanishing pressure limit problem were considered in [8] and [9] respectively. Wang and Zhang [30] studied the Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations, and they obtained the global existence and the stability of generalized solutions. Recently, Pang [21] studied delta shock wave in the compressible Euler equations for a Chaplygin gas and interaction of waves in [22]. Shen and Sun [25] studied the interactions of δ-shock waves and the vacuum states between the two contact discontinuities for the transport equations. Sun [28] studied the interactions of the delta shock waves with the shock waves and the rarefaction waves for the simplified chromatography equations. More results about delta shock waves, we can see [3–6,10,11,23] and the references cited therein. In this paper, we will consider Riemann problem and interaction of elementary waves of the relativistic string equation in the hyperbolic region {(u, v)| 1 + u2 − v 2 > 0}. The existence of solutions of Riemann problems is obtained by the method of phase plane analysis. We will construct the Riemann solution of contact discontinuity and delta shock wave, and also give the interactions of those elementary waves. In particular, for one case, the interactions of contact discontinuities produce a delta shock wave, whereas for another case, after the interactions of a delta shock wave and a contact discontinuity the delta shock wave disappears, which are very special phenomena in wave interactions. This paper is organized as follows. In Section 2, we present the properties of system (1.1). The Riemann solution for system (1.1) is given in Section 3. In Section 4, we discuss the interactions of the contact discontinuities, interactions of a delta shock wave and a contact discontinuity, and interactions of the delta shock waves. 2. Properties of system (1.1) In this section, we will give some properties of the system (1.1), let U (t, x) = (u, v)T , then system (1.1) can be written as follows A(U )Ut + B(U )Ux = 0,
(2.1)
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
3
where A(U ) =
1
0
−uv w
1+u2 w
, B(U ) =
0
−1
v 2 −1 w
−uv w
3
and w = (1 + u2 − v 2 ) 2 . It is easy to get the two eigenvalues λ1 =
−uv −
√
√ 1 + u2 − v 2 −uv + 1 + u2 − v 2 , λ2 = , 1 + u2 1 + u2
(2.2)
and the corresponding right eigenvectors T T − → − r1 = (1, −λ1 ) , → r2 = (1, −λ2 ) .
(2.3)
→ → ∇λ1 · − r1 = 0, ∇λ2 · − r2 = 0,
(2.4)
Furthermore, we can also get
i.e. the system is linearly degenerate in the sense of Lax [17]. In the region
Ω = (u, v)|1 + u2 − v 2 > 0 ,
(2.5)
we can easily get the system is strictly hyperbolic. Therefore, the elementary waves are contact discontinuities. 3. The Riemann problem In this section, we will consider the Riemann problem of the system (1.1) in the region Ω with initial data U− = (u− , v− ), x < 0, U (0, x) = U0 (x) = (3.1) U+ = (u+ , v+ ), x > 0. When the solutions are contact discontinuities, we have already known that the Rankine-Hugoniot condition is satisfied on the discontinuous curve (see [2,26]). Therefore, we have ⎧ ⎪ ⎨σ(u − u− ) + (v − v− ) = 0, v v− u u− σ( √ − ) + (√ − ) = 0, ⎪ 2 − v2 ⎩ 2 2 2 1 + u2 − v 2 1 + u 1 + u− − v− 1 + u2− − v− where σ =
dx dt .
(3.2)
Then, we can obtain σ=
−uv ∓
√
1 + u2 − v 2 = λ1,2 . 1 + u2
(3.3)
Here we will give the properties of wave curves (Fig. 1). Noting the first equation of (3.2) and (3.3), we have √ uv ± 1 + u2 − v 2 dv . (3.4) = −λ1,2 = du 1 + u2
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J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932 v
6
1 + u2 − v 2 = 0
v=u
v=1 O
-
u
v = −1
v = −u
Fig. 1. The curve 1 + u2 − v 2 = 0 in the (u, v) plane.
Therefore, the backward contact discontinuity and forward contact discontinuity in the (u, v) phase plane can be calculated as (see [35]) ⎧ 2 ⎪ −u v − 1 + u2− − v− ⎪ − − ⎪ ⎪ σ = , ⎪ ⎨ 1 1 + u2− J1 (u− , v− ) : ⎪ 2 ⎪
u v + 1 + u2− − v− ⎪ − − ⎪ ⎪ ⎩v − v− = (u − u− ), 1 + u2− ⎧ 2 ⎪ −u v + 1 + u2− − v− ⎪ − − ⎪ ⎪ σ = , ⎪ 2 ⎨ 1 + u2− J2 (u− , v− ) : ⎪ 2 ⎪
u v − 1 + u2− − v− ⎪ − − ⎪ ⎪ ⎩v − v− = (u − u− ). 1 + u2−
(3.5)
(3.6)
2
d v By the easy computation, we have du 2 = 0. Then, the wave curves in the (u, v) plane are straight line. Furthermore, J1 , J2 are tangent to the curve 1 + u2 − v 2 = 0 in the (u, v) plane. In the following, we shall construct the solution to Riemann problem of the system (1.1) with initial data (3.1) by contact discontinuities or delta shock wave (see [26,31,32]). Here we will only give the detail about construction of solutions of four cases of U− = (u− , v− ) which lies in the first quadrant of the (u, v) plane (| v− |= 1) or the positive-half axis of u and v. The other cases can be obtained using the similar procedures. Here we will give some notations used in this paper. Note that J1 and J2 are straight lines. The tangent line of the curve Γ is important to divide the region. In the tangent point, the system will be degenerate. Denote k1i
and k2i are the slopes of the two lines J1i , J2i through (ui , vi ) tangent to the curve Γ : 1 + u2 − v 2 = 0 in the (u, v) plane. λi1 , λi2 mean the eigenvalues of the two lines J1i , J2i , i = +, −, m.
Case 1. u− > 0, v− > 1. Subcase 1.1. u− < v− . We can get
J1 : v − v− = k1− (u − u− ), k1− > 0, J2 : v − v− = k2− (u − u− ), k2− > 0,
(3.7)
and the existing region of U+ is Ω1 = {(u, v)|u > 0, v > 1, 1 + u2 − v 2 > 0}.
(3.8)
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Fig. 2. Characteristic curves of J1 and J2 in the region Ω1 .
In the region Ω1 , we note J1 and J2 are intersecting at the point U− = (u− , v− ), the lines of J1 and J2
are tangent to the curve Γ : 1 + u2 − v 2 = 0 at the points A1 and B1 . Therefore, the region Ω1 is divided into six parts as shown in Fig. 2. We will give the construction of Riemann solutions part by part. Subcase 1.1.1. (u+ , v+ ) ∈I ∪ II ∪ III ∪ IV ∪ V: J1 + J2 . We assume that the state (um , vm ) is connected with (u− , v− ) on the left by J1 -wave and connected with (u+ , v+ ) on the right by J2 -wave. Then, we have k1− =
k2m
u− v− +
2 1 + u2− − v−
1+
u2−
=
um vm +
2 1 + u2m − vm = k1m , 2 1 + um
(3.9)
2 u v − 1 + u2+ − v+ 2 2 + + um vm − 1 + um − vm = = = k2+ . 1 + u2m 1 + u2+
(3.10)
Thus, by the calculation as [24], the intermediate state (um , vm ) of the solution is denoted by (um , vm ) =
k1− [v+ − v− + k2+ (u− − u+ )] v+ − v− + k1− u− − k2+ u+ , v + − k1− − k2+ k1− − k2+
Subcase 1.1.2. (u+ , v+ ) ∈ VI: δ-shock wave. Denote M = rewritten the system (1.1) as follows
1 √ , 1+u2 −v 2
u =
⎧ ⎪ 1 ⎪ ⎪ + v 2 − 1 − vx = 0, ⎪ ⎪ M2 ⎨ t ⎪ ⎪ 1 ⎪ ⎪ + v2 − 1 M = 0. ⎪(vM )t − ⎩ M2
1 M2
.
(3.11)
+ v 2 − 1. Then, we can
(3.12)
x
+ In this case, there will be λ− 1 > λ2 . Then the characteristic lines from initial data will overlap in the domain Ω0 show in Fig. 3. Therefore, the singularity must form in Ω0 . It is easy to know that singularity is impossible to be a jump with the finite amplitude. In other words, there is no solution which is the piecewise smooth and bounded. Then, we will give the definition of delta shock wave as follows (see [26,32,33]).
Definition 3.1. A two-dimensional weighted delta function w(s)δL supported on a smooth curve L parametrized as t = t(s), x = x(s)(a ≤ s ≤ b) is defined by
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J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
+ Fig. 3. λ− 1 > λ2 , δ – shock wave.
b
w(s)δL , φ =
w(s), φ(t(s), x(s))ds,
(3.13)
a
for all φ ∈ C0∞ (R2 ). Noting 1 + u2 − v 2 = 0, then M = ∞. We will construct the delta shock wave of the system (1.1) and (3.1) as follows ⎧ ⎪ (v , M ), x < σt, ⎪ ⎨ − − (3.14) (v, M )(t, x) = (vδ , w(t)δ(x − σt)), x = σt, ⎪ ⎪ ⎩ (v+ , M+ ), x > σt, where w(t), σ denote the weight and propagation speed of the δ-wave, respectively. Then, the generalized Rankine-Hugoniot relations for the delta wave are ⎧ dx ⎪ = σ, ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ 2 ⎪0 = σ + v − 1 + [v], ⎨ M2 ⎪ ⎪ ⎪ dw(t)v 1 δ ⎪ ⎪ + v2 − 1 M , = σ[vM ] + ⎪ ⎪ dt M2 ⎪ ⎪ ⎪ ⎪ ⎩ t = 0 : x(0) = 0, w(0) = 0,
(3.15)
with ⎧ 1 ⎪ ⎪ , ⎪ ⎪ M− ⎪ ⎨ 1 = 0, ⎪ M ⎪ ⎪ 1 ⎪ ⎪ , ⎩ M+
x < σt, x = σt,
(3.16)
x > σt.
By (3.15), we can get σ=
1 2 M+
v− − v+ . 1 2 −1− 2 −1 + v+ + v− M2
(3.17)
−
Note that there are two variables vδ and w(t) in the third equation of (3.15), which is under-determined. − Here we choose the special case vδ = v+ +v . Using the similar procedure as [30], we can get 2
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
dw(t) w0 = = dt
2[(M+ + M− )(v+ v− −
1 2 M+
2 −1 + v+
1 2 M−
2 − 1 − 1) + + v−
1 2 −1− 2 − 1) (v+ + v− )( M12 + v+ + v− M2 +
7
1 M+
+
1 M− ]
.
(3.18)
−
Then, we can get w(t) = w0 t, x(t) = σt.
(3.19)
Furthermore, we can also check the above constructed delta wave satisfying the Lax entropy condition − λ+ 2 < σ < λ1 .
(3.20)
Remark. By the construction of delta wave, the equation of (vδ , w(t)) is under-determined. This case is different from the many constructions of delta shock wave in Euler system. Here we choose the case of − vδ = v+ +v as the general case in Euler system. However, there is only one special construction case and 2 no uniqueness. It is worthy to consider the reason in the future. Subcase 1.2. u− ≥ v− . Similarly, we can get J1 , J2 as (3.7) and the existing region of U+ is Ω2 = {(u, v)|u > 0, v > 1, 1 + u2 − v 2 > 0}.
(3.21)
The region Ω2 is divided into five parts: I, II, III, IV, V. Now, for any given (u+ , v+ ) ∈I ∪ II ∪ III ∪ IV ∪ V, we can give the construction of the solution by J1 + J2 , the intermediate state (um , vm ) of the solution can be given by (3.11). Case 2. u− > 0, 0 < v− < 1. Subcase 2.1. u− < v− . Similarly, we can get
J1 : v − v− = k1− (u − u− ), k1− > 0, J2 : v − v− = k2− (u − u− ), k2− < 0,
(3.22)
and the existing region of U+ is Ω3 = {(u, v)| − 1 < v < 1, 1 + u2 − v 2 > 0}.
(3.23)
As shown in Fig. 4, in region Ω3 , J1 and J2 are intersecting at the point (u− , v− ), the extension lines of
J1 and J2 are tangent to the curve Γ : 1 + u2 − v 2 = 0 at the points A3 and B3 . Therefore, the region Ω3 is divided into four parts: I, II, III, IV. Now, for any given (u+, v+ ) ∈I ∪ II ∪ III ∪ IV, we can give the construction of the solution by two contact discontinuities J1 + J2 and the intermediate state (um , vm ) of the solution can be given by (3.11). Subcase 2.2. u− ≥ v− . Similarly, we can get J1 , J2 as (3.22) and the existing region of U+ is Ω4 = {(u, v)| − 1 < v < 1, 1 + u2 − v 2 > 0}.
(3.24)
The region Ω4 is divided into four parts: I, II, III, IV. Now, for any given (u+ , v+ ) ∈I ∪ II ∪ III ∪ IV, we can give the construction of the solution by J1 + J2 , the intermediate state (um , vm ) of the solution can be given by (3.11). Case 3. u− > 0, v− = 0.
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
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Fig. 4. Characteristic curves of J1 and J2 in the region Ω3 .
Fig. 5. Characteristic curves of J1 and J2 in the region Ω5 .
We can get J1 : v = k1− (u − u− ),
k1− > 0,
(3.25)
J2 : v = k2− (u − u− ), k2− < 0, and the existing region of U+ is Ω5 = {(u, v)| − 1 < v < 1, 1 + u2 − v 2 > 0}.
(3.26)
As shown in Fig. 5, in the region Ω5 , J1 and J2 are intersecting at the point U− = (u− , v− ), the extension
lines of J1 and J2 are tangent to the curve Γ : 1 + u2 − v 2 = 0 at the points A5 and B5 . Then, the region Ω5 is also divided into four parts: I, II, III, IV. Now, for any given (u+ , v+ ) ∈I ∪ II ∪ III ∪ IV, we can give the construction of the solution by two contact discontinuities J1 + J2 , the intermediate state (um , vm ) of the solution can be given by (um , vm ) =
v+ + k1− u− − k2+ u+ k1− [v+ + k2+ (u− − u+ )] , v + − k1− − k2+ k1− − k2+
.
(3.27)
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
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Case 4. u− = 0, 0 < v− < 1. We can also obtain J1 , J2 and the existing region of U+ is Ω6 = {(u, v)| − 1 < v < 1, 1 + u2 − v 2 > 0}.
(3.28)
The region Ω6 is divided into four parts: I, II, III, IV. Now, for any given (u+ , v+ ) ∈I ∪ II ∪ III ∪ IV, we can give the construction of the solution by J1 + J2 , the intermediate state (um , vm ) of the solution is denoted by (um , vm ) =
v+ − v− − k2+ u+ k− (v+ − v− − k2+ u+ ) , v− + 1 − + k1 − k2 k1− − k2+
.
(3.29)
Theorem 3.1. There exist global piecewise solutions for the Riemann problems (1.1) and (3.1) when U− = (u− , v− ) lies in the region Ω (| v− |= 1). 4. Interaction of elementary waves In this section, we will give the interactions of elementary waves, including contact discontinuity and delta shock wave (see [20–22,25,28]). Then, we consider the system (3.1) with three pieces constant initial data, ⎧ ⎪ ⎨(u− , v− ), x < −ε, (4.1) (u, v)(0, x) = (um , vm ), ε < x < ε, ⎪ ⎩(u , v ), x > ε. + + Let ui , vi ≥ 0, ε > 0, i = −, m, +. We will consider the following four cases according to the different wave combinations from (0, −ε) and (0, ε): → ← − − ← − → − (1) J −m J m+ , (2) δ−m J m+ , (3) δ−m J m+ , (4) δ−m δm+ . − → ← − Case 1. J −m J m+ . → − In this case, we consider the interaction of forward contact discontinuity J −m starting from the point ← − (0, −ε) and backward contact discontinuity J m+ starting from the point (0, ε). The occurrence of this case depends on that the initial data (4.1) meet the conditions
2 vm
−u− v− +
2 1 + u2− − v− −um vm + 1 + − = = = λ− 2, 1 + u2m 1 + u2− 2 −u+ v+ − 1 + u2+ − v+ 2 −um vm − 1 + u2m − vm m λ1 = = = λ+ 1. 1 + u2m 1 + u2+
λm 2
u2m
(4.2)
(4.3)
→ − ← − → − m The propagation speed of J −m is λm 2 and that of J m+ is λ1 . Thus, it is easy to see that J −m overtakes ← − m −1 J m+ in a finite time, whose intersection point is (t1 , x1 ), which implies that t1 = 2ε(λm , x1 = 2 − λ1 ) m λ1 t1 + ε. Then, at time t = t1 , we get the Riemann problem (1.1) with initial data (u− , v− ), x < x1 , (u, v)(t, x) = (u+ , v+ ), x > x1 .
(4.4)
+ According to the relative sizes of λ− 1 and λ2 , we should divide our discussion into the following two subcases. − + − Subcase 1.1. λ1 < λ2 : According to λ1 < λ+ 2 , the solution of the Riemann problem can be expressed as follows
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J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
+ Fig. 7. λ− 1 > λ2 .
+ Fig. 6. λ− 1 < λ2 .
⎧ ⎪ ⎨(u− , v− ), (u, v)(t, x) = (u∗ , v∗ ), ⎪ ⎩(u , v ), + +
x < λ− 1 (t − t1 ) + x1 , − λ1 (t − t1 ) + x1 < x < λ+ 2 (t − t1 ) + x1 , x > λ+ (t − t ) + x . 1 1 2
(4.5)
Using the similar procedures of getting (um , vm ) of system (3.11), we can get the state (u∗ , v∗ ) as shown in ← − → − Fig. 6. Thus, a new backward contact discontinuity J and a new forward contact discontinuity J will be → − ← − generated after the interaction of J and J . + Subcase 1.2. λ− 1 > λ2 : In this case, the solution of the Riemann problem can be given as the system (3.14), where x(t), w(t) is obtained by making a substitution t = t − t1 , x(t) = x(t) − x1 in the system (3.19), as → − ← − shown in Fig. 7. Thus, a new delta shock wave will be generated after the interaction of J and J . → − Theorem 4.1. When the initial data (4.1) satisfy (4.2) and (4.3), a forward contact discontinuity J starting ← − → − from the point (0, −ε) will collide with a backward discontinuity J starting from the point (0, ε). J overtakes ← − J in a finite time and after the interaction new contact discontinuities or a new delta shock wave will appear. ← − Case 2. δ−m J m+ . In this case, we consider the interaction of delta shock δ−m starting from the point (0, −ε) and backward ← − contact discontinuity J m+ starting from the point (0, ε). The occurrence of this case depends on that the + m m initial data (4.1) meet the conditions λ− 1 > λ2 , λ1 = λ1 . The propagation speed of δ−m is σ−m and that ← − ← − m of J m+ is λ1 . Thus, it is easy to see that δ−m overtakes J m+ in a finite time, whose intersection point is −1 (t1 , x1 ), which implies that t1 = 2ε(σ−m − λm , x 1 = λm 1 ) 1 t1 + ε. Then, at time t = t1 , we get the Riemann problem (1.1) with delta initial data (4.6) ⎧ ⎪ ⎨(v− , M− ), (v, M )(t, x) = (vδ , w1 δ(x − x1 )), ⎪ ⎩(v , M ), + +
x < x1 , x = x1 , x > x1 ,
(4.6)
+ where w(t1 ) = w1 , M = √1+u12 −v2 . According to the relative sizes of λ− 1 and λ2 , we should divide our discussion into the following two subcases. + − + Subcase 2.1. λ− 1 < λ2 : According to λ1 < λ2 , the solution of the Riemann problem can be expressed as ← − the system (4.5), as shown in Fig. 8. Thus, a new backward contact discontinuity J and a new forward → − ← − contact discontinuity J will be generated after the interaction of δ-shock and J . + − + Subcase 2.2. λ− 1 > λ2 : According to λ1 > λ2 , the solution of the Riemann problem can be expressed as the system (3.14), where x(t), w(t) is obtained by making a substitution t = t − t1 , x(t) = x(t) − x1 in the system (3.19), as shown in Fig. 9. Thus, a new delta shock wave will be generated after the interaction of ← − δ-shock and J .
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
+ Fig. 8. λ− 1 < λ2 .
+ Fig. 9. λ− 1 > λ2 .
+ Fig. 10. λ− 1 < λ2 .
+ Fig. 11. λ− 1 > λ2 .
11
+ m m Theorem 4.2. When the initial data (4.1) satisfy λ− 1 > λ2 and λ1 = λ1 , a delta shock wave starting from ← − ← − (0, −ε) will collide with a backward discontinuity J starting from (0, ε). δ-shock overtakes J in a finite time and after the interaction new contact discontinuities or a new delta shock wave will appear.
→ − Case 3. δ−m J m+ . In this case, we consider the interaction of delta shock δ−m starting from (0, −ε) and forward contact → − discontinuity J m+ starting from (0, ε). The occurrence of this case depends on that the initial data (4.1) → − + m m m meet the conditions λ− 1 > λ2 , λ2 = λ2 . The propagation speed of δ−m is σ−m and that of J m+ is λ2 . → − Thus, it is easy to see that δ−m overtakes J m+ in a finite time, whose intersection point is (t1 , x1 ), which −1 implies that t1 = 2ε(σ−m − λm , x 1 = λm 2 ) 2 t1 + ε. Then, at time t = t1 , we get the Riemann problem + m (1.1) with delta initial data (4.6), and notice that λ− 1 > λ2 = λ2 , thus, the solution of the Riemann problem can be expressed as the system (3.14), where x(t), w(t) is obtained by making a substitution t = t − t1 , x(t) = x(t) − x1 in the system (3.19), as shown in Fig. 10. Thus, a new delta shock wave will be → − generated after the interaction of δ-shock and J . Case 4. δ−m δm+ . In this case, we consider the interaction of delta shock δ−m starting from (0, −ε) and delta shock δm+ starting from (0, ε). The occurrence of this case depends on that the initial data (4.1) meet the conditions + m m λ− 1 > λ2 , λ1 > λ2 . The propagation speed of δ−m is σ−m and that of δm+ is σm+ . Thus, it is easy to see that δ−m overtakes δm+ in a finite time, whose intersection point is (t1 , x1 ), which implies that t1 = 2ε(σ−m − σm+ )−1 , x1 = σm+ t1 + ε. Then, at time t = t1 , we get the Riemann problem (1.1) with delta + m initial data (4.6), and notice that λ− 1 > λ2 > λ2 , thus, the solution of the Riemann problem can be expressed as the system (3.14), where x(t), w(t) is obtained by making a substitution t = t − t1 , x(t) = x(t) − x1 in the system (3.19), as shown in Fig. 11. Thus, a new delta shock wave will be generated after the interaction of δ-shocks.
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+ m m Theorem 4.3. When the initial data (4.1) satisfy λ− 1 > λ2 and λ2 = λ2 , a delta shock wave starting from → − the point (0, −ε) will collide with a forward contact discontinuity J starting from the piont (0, ε), δ-shock → − overtakes J in a finite time and after the interaction a new delta shock wave will appear. + m m Theorem 4.4. When the initial data (4.1) satisfy λ− 1 > λ2 and λ1 > λ2 , a delta shock wave starting from the point (0, −ε) will collide with another delta shock wave starting from the point (0, ε). After the interaction, there is a new delta shock wave.
Acknowledgments The authors would like to thank Prof. Wancheng Sheng and Prof. Geng Lai of Shanghai University for their helpful discussions and suggestions. References [1] A. Bressan, G. Chen, Q.T. Zhang, On finite time BV blow-up for the p-system, Commun. Partial Differ. Equ. 43 (8) (2018) 1242–1280. [2] T. Chang, L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, Longman Sci. & Tech., 1989. [3] H. Cheng, Delta shock waves for a linearly degenerate hyperbolic system of conservation laws of Keyfitz-Kranzer type, Adv. Math. Phys. (2013) 958120. [4] H. Cheng, H. Yang, Riemann problem for the relativistic Chaplygin Euler equations, J. Math. Anal. Appl. 381 (1) (2011) 17–26. [5] H. Cheng, H. Yang, Delta shock waves in chromatography equations, J. Math. Anal. Appl. 380 (2) (2011) 475–485. [6] H. Cheng, H. Yang, Riemann problem for the isentropic relativistic Chaplygin Euler equations, Z. Angew. Math. Phys. 63 (3) (2012) 429–440. [7] C.M. Dafermos, Structure of solutions of the Riemann problem for hyperbolic systems of conservation laws, Arch. Ration. Mech. Anal. 53 (1974) 203–217. [8] L. Guo, T. Li, L. Pan, et al., The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations with a source term, Nonlinear Anal., Real World Appl. 41 (2018) 588–606. [9] L. Guo, T. Li, G. Yin, The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term, Commun. Pure Appl. Anal. 16 (1) (2017) 295–310. [10] L. Guo, T. Li, G. Yin, The limit behavior of the Riemann solutions to the generalized Chaplygin gas equations with a source term, J. Math. Anal. Appl. 455 (1) (2017) 127–140. [11] L. Guo, Y. Zhang, G. Yin, Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions, Math. Methods Appl. Sci. 38 (11) (2015) 2132–2148. [12] C.L. He, D.X. Kong, Spherical symmetric solutions for the motion of relativistic membranes in the Schwarzschild spacetime, J. Math. Phys. 50 (8) (2009) 083516. [13] J. Hoppe, Some classical solutions of relativistic membrane equations in 4 space-time dimensions, Phys. Lett. B 329 (1) (1994) 10–14. [14] D.X. Kong, Q. Zhang, Solution formula and time-periodicity for the motion of relativistic strings in the Minkowski space, Physica D 238 (9–10) (2009) 902–922. [15] D.X. Kong, Q. Zhang, Q. Zhou, The dynamics of relativistic strings moving in the Minkowski space R1+n , J. Math. Phys. 269 (1) (2007) 153–174. [16] N. Lai, J. Liu, Global weak and smooth solutions of the equations for timelike extremal surface in Minkowski space, J. Math. Anal. Appl. 428 (2) (2015) 1135–1146. [17] P.D. Lax, Hyperbolic systems of conservation laws. II, Commun. Pure Appl. Math. 10 (4) (1957) 537–566. [18] J. Liu, Y. Zhou, Initial-boundary value problem of the timelike extremal surface in Minkowski space, J. Math. Phys. 49 (4) (2008) 043507. [19] J. Liu, Y. Zhou, The initial-boundary value problem on a strip for the equation of time-like extremal surfaces, Discrete Contin. Dyn. Syst. 23 (1/2) (2009) 381–397. [20] Y. Liu, W. Sun, Riemann problem and wave interactions in magnetogasdynamics, J. Math. Anal. Appl. 397 (2) (2013) 454–466. [21] Y. Pang, Delta shock wave in the compressible Euler equations for a Chaplygin gas, J. Math. Anal. Appl. 448 (1) (2017) 245–261. [22] Y. Pang, Y. Ding, Y. Wen, Interaction of waves in one-dimensional compressible Euler equations for a Chaplygin gas, Math. Pract. Theory 47 (11) (2017) 261–271. [23] A. Qu, Z. Wang, Stability of the Riemann solutions for a Chaplygin gas, J. Math. Anal. Appl. 409 (1) (2014) 347–361. [24] C. Shen, W. Sheng, M. Sun, The asymptotic limits of Riemann solutions to the scaled Leroux system, Commun. Pure Appl. Anal. 17 (2) (2017) 391–411. [25] C. Shen, M. Sun, Interactions of delta shock waves for the transport equations with split delta functions, J. Math. Anal. Appl. 351 (2) (2009) 747–755.
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[26] W. Sheng, T. Zhang, The Riemann problem for the transportation equations in gas dynamics, Mem. Am. Math. Soc. 654 (1999). [27] J.A. Smoller, Shock waves and reaction-diffusion equations, Math. Gaz. 258 (2) (1994) 177–194. [28] M. Sun, Interactions of delta shock waves for the chromatography equations, Appl. Math. Lett. 26 (6) (2013) 631–637. [29] J. Wang, C. Wei, Global existence of smooth solution to relativistic membrane equation with large data, arXiv:1708. 03839v1, 2017. [30] Z. Wang, Q. Zhang, The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations, Acta Math. Sci. 32B (3) (2012) 825–841. [31] H. Yang, Riemann problems for a class of coupled hyperbolic systems of conservation laws, J. Differ. Equ. 159 (2) (1999) 447–484. [32] H. Yang, Y. Zhang, New developments of delta shock waves and its applications in systems of conservation on laws, J. Differ. Equ. 252 (11) (2012) 5951–5993. [33] H. Yang, Y. Zhang, Delta shock waves with Dirac delta function in both components for systems of conservation laws, J. Differ. Equ. 257 (12) (2014) 4369–4402. [34] J. Zhang, G. Lai, Riemann problem and interaction of elementary waves for the relativistic p-system, J. Appl. Comput. Math. 32 (04) (2018) 813–830. [35] T. Zhang, W. Sheng, Global solutions of the perturbed Riemann problem for the chromatography equations, Acta Math. Sci. 39B (1) (2019) 57–82.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Riemann problem and wave interactions for the one-dimensional relativistic string equation in Minkowski space ✩ Jianli Liu ∗ , Ruixi Liu Department of Mathematics, Shanghai University, Shanghai 200444, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 3 October 2019 Available online 6 February 2020 Submitted by T. Yang
In this paper, we consider the Riemann problem for the one-dimensional relativistic string equation in Minkowski space. It is shown that the delta shock wave will be constructed in the Riemann solutions in some certain situations. Furthermore, the interactions of elementary waves including contact discontinuity and delta shock wave are obtained. © 2020 Elsevier Inc. All rights reserved.
Keywords: Riemann problem Relativistic string equation Delta shock wave Wave interactions
1. Introduction In this paper, we will investigate the Riemann problem of the one-dimensional hyperbolic system of conservation laws as follows ⎧ ⎪ ⎨ ut − vx = 0, (1.1) v u ⎪ )t − ( √ )x = 0, ⎩ (√ 1 + u2 − v 2 1 + u2 − v 2 which can be obtained from the equation to one-dimensional relativistic string in Minkowski space (
φt 1+
φ2x
−
φ2t
φx )t − ( )x = 0, 1 + φ2x − φ2t
where u = φx , v = φt denote the functions of (x, t) ∈ R × R+ . Relativistic string equation is very important in Lorentz geometry and particle physics. Kong, Zhang and Zhou [15] studied the dynamics of a relativistic string in Minkowski space R1+n . For the higher dimensional case, Kong and Zhang [14] studied the global ✩
This work was partially supported by NSF of China 11771274.
* Corresponding author. E-mail addresses: [email protected] (J. Liu), [email protected] (R. Liu). https://doi.org/10.1016/j.jmaa.2020.123932 0022-247X/© 2020 Elsevier Inc. All rights reserved.
2
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
motion of relativistic strings in Minkowski space R1+n . Hoppe [13] gave the classical solutions of the equation of motion of a relativistic membrane. The spherical symmetric solutions for the relativistic membranes moving in Schwarzschild spacetime were obtained in [12]. Wang and Wei [29] studied a global smooth solution to the Cauchy problem with a class of large initial date for the relativistic membrane equation embedded in R1+(1+n) with n = 2, 3. The related results can also be found in [16,18,19]. Riemann problem is a kind of initial problem with the special piecewise initial data. Generally, the solution of the Riemann problem involves shock, rarefaction and contact discontinuities. There is one major milestone result by P.D. Lax for small BV solutions of general hyperbolic conservation laws [17], see also [7]. The blowup BV solution for p-system can be found in [1] and reference therein. Chang and Hsiao [2] made a completely understood the solution of Riemann problem for full Euler equation (non-isentropic). Smoller [27] studied Riemann problem for isentropic p-systems and the general hyperbolic systems of conservation laws with the sufficiently close initial states. Sheng and Zhang [26] studied the one-dimensional and twodimensional Riemann problems for the transportation equations in gas dynamics and proved uniqueness and stability for viscous perturbations. Zhang and Lai [34] studied the Riemann problem and interaction of elementary waves for the relativistic p-system with general pressure law p = p(ρ). Zhang and Sheng [35] studied the Riemann problem and waves interactions for the chromatography equations with conservative form. There are lots of references related to the results about the delta shock waves. The author in [31] studied the Riemann problems for a class of coupled hyperbolic systems of conservation laws and gave delta-shock wave solutions and vacuum solutions. The authors in [32] studied new developments of delta shock waves and its applications in systems of conservation laws. The authors in [33] studied the Riemann problems for a class of non-strictly and weakly hyperbolic systems of conservation laws. The Riemann problem with delta initial data and the vanishing pressure limit problem were considered in [8] and [9] respectively. Wang and Zhang [30] studied the Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations, and they obtained the global existence and the stability of generalized solutions. Recently, Pang [21] studied delta shock wave in the compressible Euler equations for a Chaplygin gas and interaction of waves in [22]. Shen and Sun [25] studied the interactions of δ-shock waves and the vacuum states between the two contact discontinuities for the transport equations. Sun [28] studied the interactions of the delta shock waves with the shock waves and the rarefaction waves for the simplified chromatography equations. More results about delta shock waves, we can see [3–6,10,11,23] and the references cited therein. In this paper, we will consider Riemann problem and interaction of elementary waves of the relativistic string equation in the hyperbolic region {(u, v)| 1 + u2 − v 2 > 0}. The existence of solutions of Riemann problems is obtained by the method of phase plane analysis. We will construct the Riemann solution of contact discontinuity and delta shock wave, and also give the interactions of those elementary waves. In particular, for one case, the interactions of contact discontinuities produce a delta shock wave, whereas for another case, after the interactions of a delta shock wave and a contact discontinuity the delta shock wave disappears, which are very special phenomena in wave interactions. This paper is organized as follows. In Section 2, we present the properties of system (1.1). The Riemann solution for system (1.1) is given in Section 3. In Section 4, we discuss the interactions of the contact discontinuities, interactions of a delta shock wave and a contact discontinuity, and interactions of the delta shock waves. 2. Properties of system (1.1) In this section, we will give some properties of the system (1.1), let U (t, x) = (u, v)T , then system (1.1) can be written as follows A(U )Ut + B(U )Ux = 0,
(2.1)
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
3
where A(U ) =
1
0
−uv w
1+u2 w
, B(U ) =
0
−1
v 2 −1 w
−uv w
3
and w = (1 + u2 − v 2 ) 2 . It is easy to get the two eigenvalues λ1 =
−uv −
√
√ 1 + u2 − v 2 −uv + 1 + u2 − v 2 , λ2 = , 1 + u2 1 + u2
(2.2)
and the corresponding right eigenvectors T T − → − r1 = (1, −λ1 ) , → r2 = (1, −λ2 ) .
(2.3)
→ → ∇λ1 · − r1 = 0, ∇λ2 · − r2 = 0,
(2.4)
Furthermore, we can also get
i.e. the system is linearly degenerate in the sense of Lax [17]. In the region
Ω = (u, v)|1 + u2 − v 2 > 0 ,
(2.5)
we can easily get the system is strictly hyperbolic. Therefore, the elementary waves are contact discontinuities. 3. The Riemann problem In this section, we will consider the Riemann problem of the system (1.1) in the region Ω with initial data U− = (u− , v− ), x < 0, U (0, x) = U0 (x) = (3.1) U+ = (u+ , v+ ), x > 0. When the solutions are contact discontinuities, we have already known that the Rankine-Hugoniot condition is satisfied on the discontinuous curve (see [2,26]). Therefore, we have ⎧ ⎪ ⎨σ(u − u− ) + (v − v− ) = 0, v v− u u− σ( √ − ) + (√ − ) = 0, ⎪ 2 − v2 ⎩ 2 2 2 1 + u2 − v 2 1 + u 1 + u− − v− 1 + u2− − v− where σ =
dx dt .
(3.2)
Then, we can obtain σ=
−uv ∓
√
1 + u2 − v 2 = λ1,2 . 1 + u2
(3.3)
Here we will give the properties of wave curves (Fig. 1). Noting the first equation of (3.2) and (3.3), we have √ uv ± 1 + u2 − v 2 dv . (3.4) = −λ1,2 = du 1 + u2
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J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932 v
6
1 + u2 − v 2 = 0
v=u
v=1 O
-
u
v = −1
v = −u
Fig. 1. The curve 1 + u2 − v 2 = 0 in the (u, v) plane.
Therefore, the backward contact discontinuity and forward contact discontinuity in the (u, v) phase plane can be calculated as (see [35]) ⎧ 2 ⎪ −u v − 1 + u2− − v− ⎪ − − ⎪ ⎪ σ = , ⎪ ⎨ 1 1 + u2− J1 (u− , v− ) : ⎪ 2 ⎪
u v + 1 + u2− − v− ⎪ − − ⎪ ⎪ ⎩v − v− = (u − u− ), 1 + u2− ⎧ 2 ⎪ −u v + 1 + u2− − v− ⎪ − − ⎪ ⎪ σ = , ⎪ 2 ⎨ 1 + u2− J2 (u− , v− ) : ⎪ 2 ⎪
u v − 1 + u2− − v− ⎪ − − ⎪ ⎪ ⎩v − v− = (u − u− ). 1 + u2−
(3.5)
(3.6)
2
d v By the easy computation, we have du 2 = 0. Then, the wave curves in the (u, v) plane are straight line. Furthermore, J1 , J2 are tangent to the curve 1 + u2 − v 2 = 0 in the (u, v) plane. In the following, we shall construct the solution to Riemann problem of the system (1.1) with initial data (3.1) by contact discontinuities or delta shock wave (see [26,31,32]). Here we will only give the detail about construction of solutions of four cases of U− = (u− , v− ) which lies in the first quadrant of the (u, v) plane (| v− |= 1) or the positive-half axis of u and v. The other cases can be obtained using the similar procedures. Here we will give some notations used in this paper. Note that J1 and J2 are straight lines. The tangent line of the curve Γ is important to divide the region. In the tangent point, the system will be degenerate. Denote k1i
and k2i are the slopes of the two lines J1i , J2i through (ui , vi ) tangent to the curve Γ : 1 + u2 − v 2 = 0 in the (u, v) plane. λi1 , λi2 mean the eigenvalues of the two lines J1i , J2i , i = +, −, m.
Case 1. u− > 0, v− > 1. Subcase 1.1. u− < v− . We can get
J1 : v − v− = k1− (u − u− ), k1− > 0, J2 : v − v− = k2− (u − u− ), k2− > 0,
(3.7)
and the existing region of U+ is Ω1 = {(u, v)|u > 0, v > 1, 1 + u2 − v 2 > 0}.
(3.8)
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
5
Fig. 2. Characteristic curves of J1 and J2 in the region Ω1 .
In the region Ω1 , we note J1 and J2 are intersecting at the point U− = (u− , v− ), the lines of J1 and J2
are tangent to the curve Γ : 1 + u2 − v 2 = 0 at the points A1 and B1 . Therefore, the region Ω1 is divided into six parts as shown in Fig. 2. We will give the construction of Riemann solutions part by part. Subcase 1.1.1. (u+ , v+ ) ∈I ∪ II ∪ III ∪ IV ∪ V: J1 + J2 . We assume that the state (um , vm ) is connected with (u− , v− ) on the left by J1 -wave and connected with (u+ , v+ ) on the right by J2 -wave. Then, we have k1− =
k2m
u− v− +
2 1 + u2− − v−
1+
u2−
=
um vm +
2 1 + u2m − vm = k1m , 2 1 + um
(3.9)
2 u v − 1 + u2+ − v+ 2 2 + + um vm − 1 + um − vm = = = k2+ . 1 + u2m 1 + u2+
(3.10)
Thus, by the calculation as [24], the intermediate state (um , vm ) of the solution is denoted by (um , vm ) =
k1− [v+ − v− + k2+ (u− − u+ )] v+ − v− + k1− u− − k2+ u+ , v + − k1− − k2+ k1− − k2+
Subcase 1.1.2. (u+ , v+ ) ∈ VI: δ-shock wave. Denote M = rewritten the system (1.1) as follows
1 √ , 1+u2 −v 2
u =
⎧ ⎪ 1 ⎪ ⎪ + v 2 − 1 − vx = 0, ⎪ ⎪ M2 ⎨ t ⎪ ⎪ 1 ⎪ ⎪ + v2 − 1 M = 0. ⎪(vM )t − ⎩ M2
1 M2
.
(3.11)
+ v 2 − 1. Then, we can
(3.12)
x
+ In this case, there will be λ− 1 > λ2 . Then the characteristic lines from initial data will overlap in the domain Ω0 show in Fig. 3. Therefore, the singularity must form in Ω0 . It is easy to know that singularity is impossible to be a jump with the finite amplitude. In other words, there is no solution which is the piecewise smooth and bounded. Then, we will give the definition of delta shock wave as follows (see [26,32,33]).
Definition 3.1. A two-dimensional weighted delta function w(s)δL supported on a smooth curve L parametrized as t = t(s), x = x(s)(a ≤ s ≤ b) is defined by
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+ Fig. 3. λ− 1 > λ2 , δ – shock wave.
b
w(s)δL , φ =
w(s), φ(t(s), x(s))ds,
(3.13)
a
for all φ ∈ C0∞ (R2 ). Noting 1 + u2 − v 2 = 0, then M = ∞. We will construct the delta shock wave of the system (1.1) and (3.1) as follows ⎧ ⎪ (v , M ), x < σt, ⎪ ⎨ − − (3.14) (v, M )(t, x) = (vδ , w(t)δ(x − σt)), x = σt, ⎪ ⎪ ⎩ (v+ , M+ ), x > σt, where w(t), σ denote the weight and propagation speed of the δ-wave, respectively. Then, the generalized Rankine-Hugoniot relations for the delta wave are ⎧ dx ⎪ = σ, ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ 2 ⎪0 = σ + v − 1 + [v], ⎨ M2 ⎪ ⎪ ⎪ dw(t)v 1 δ ⎪ ⎪ + v2 − 1 M , = σ[vM ] + ⎪ ⎪ dt M2 ⎪ ⎪ ⎪ ⎪ ⎩ t = 0 : x(0) = 0, w(0) = 0,
(3.15)
with ⎧ 1 ⎪ ⎪ , ⎪ ⎪ M− ⎪ ⎨ 1 = 0, ⎪ M ⎪ ⎪ 1 ⎪ ⎪ , ⎩ M+
x < σt, x = σt,
(3.16)
x > σt.
By (3.15), we can get σ=
1 2 M+
v− − v+ . 1 2 −1− 2 −1 + v+ + v− M2
(3.17)
−
Note that there are two variables vδ and w(t) in the third equation of (3.15), which is under-determined. − Here we choose the special case vδ = v+ +v . Using the similar procedure as [30], we can get 2
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
dw(t) w0 = = dt
2[(M+ + M− )(v+ v− −
1 2 M+
2 −1 + v+
1 2 M−
2 − 1 − 1) + + v−
1 2 −1− 2 − 1) (v+ + v− )( M12 + v+ + v− M2 +
7
1 M+
+
1 M− ]
.
(3.18)
−
Then, we can get w(t) = w0 t, x(t) = σt.
(3.19)
Furthermore, we can also check the above constructed delta wave satisfying the Lax entropy condition − λ+ 2 < σ < λ1 .
(3.20)
Remark. By the construction of delta wave, the equation of (vδ , w(t)) is under-determined. This case is different from the many constructions of delta shock wave in Euler system. Here we choose the case of − vδ = v+ +v as the general case in Euler system. However, there is only one special construction case and 2 no uniqueness. It is worthy to consider the reason in the future. Subcase 1.2. u− ≥ v− . Similarly, we can get J1 , J2 as (3.7) and the existing region of U+ is Ω2 = {(u, v)|u > 0, v > 1, 1 + u2 − v 2 > 0}.
(3.21)
The region Ω2 is divided into five parts: I, II, III, IV, V. Now, for any given (u+ , v+ ) ∈I ∪ II ∪ III ∪ IV ∪ V, we can give the construction of the solution by J1 + J2 , the intermediate state (um , vm ) of the solution can be given by (3.11). Case 2. u− > 0, 0 < v− < 1. Subcase 2.1. u− < v− . Similarly, we can get
J1 : v − v− = k1− (u − u− ), k1− > 0, J2 : v − v− = k2− (u − u− ), k2− < 0,
(3.22)
and the existing region of U+ is Ω3 = {(u, v)| − 1 < v < 1, 1 + u2 − v 2 > 0}.
(3.23)
As shown in Fig. 4, in region Ω3 , J1 and J2 are intersecting at the point (u− , v− ), the extension lines of
J1 and J2 are tangent to the curve Γ : 1 + u2 − v 2 = 0 at the points A3 and B3 . Therefore, the region Ω3 is divided into four parts: I, II, III, IV. Now, for any given (u+, v+ ) ∈I ∪ II ∪ III ∪ IV, we can give the construction of the solution by two contact discontinuities J1 + J2 and the intermediate state (um , vm ) of the solution can be given by (3.11). Subcase 2.2. u− ≥ v− . Similarly, we can get J1 , J2 as (3.22) and the existing region of U+ is Ω4 = {(u, v)| − 1 < v < 1, 1 + u2 − v 2 > 0}.
(3.24)
The region Ω4 is divided into four parts: I, II, III, IV. Now, for any given (u+ , v+ ) ∈I ∪ II ∪ III ∪ IV, we can give the construction of the solution by J1 + J2 , the intermediate state (um , vm ) of the solution can be given by (3.11). Case 3. u− > 0, v− = 0.
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
8
Fig. 4. Characteristic curves of J1 and J2 in the region Ω3 .
Fig. 5. Characteristic curves of J1 and J2 in the region Ω5 .
We can get J1 : v = k1− (u − u− ),
k1− > 0,
(3.25)
J2 : v = k2− (u − u− ), k2− < 0, and the existing region of U+ is Ω5 = {(u, v)| − 1 < v < 1, 1 + u2 − v 2 > 0}.
(3.26)
As shown in Fig. 5, in the region Ω5 , J1 and J2 are intersecting at the point U− = (u− , v− ), the extension
lines of J1 and J2 are tangent to the curve Γ : 1 + u2 − v 2 = 0 at the points A5 and B5 . Then, the region Ω5 is also divided into four parts: I, II, III, IV. Now, for any given (u+ , v+ ) ∈I ∪ II ∪ III ∪ IV, we can give the construction of the solution by two contact discontinuities J1 + J2 , the intermediate state (um , vm ) of the solution can be given by (um , vm ) =
v+ + k1− u− − k2+ u+ k1− [v+ + k2+ (u− − u+ )] , v + − k1− − k2+ k1− − k2+
.
(3.27)
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
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Case 4. u− = 0, 0 < v− < 1. We can also obtain J1 , J2 and the existing region of U+ is Ω6 = {(u, v)| − 1 < v < 1, 1 + u2 − v 2 > 0}.
(3.28)
The region Ω6 is divided into four parts: I, II, III, IV. Now, for any given (u+ , v+ ) ∈I ∪ II ∪ III ∪ IV, we can give the construction of the solution by J1 + J2 , the intermediate state (um , vm ) of the solution is denoted by (um , vm ) =
v+ − v− − k2+ u+ k− (v+ − v− − k2+ u+ ) , v− + 1 − + k1 − k2 k1− − k2+
.
(3.29)
Theorem 3.1. There exist global piecewise solutions for the Riemann problems (1.1) and (3.1) when U− = (u− , v− ) lies in the region Ω (| v− |= 1). 4. Interaction of elementary waves In this section, we will give the interactions of elementary waves, including contact discontinuity and delta shock wave (see [20–22,25,28]). Then, we consider the system (3.1) with three pieces constant initial data, ⎧ ⎪ ⎨(u− , v− ), x < −ε, (4.1) (u, v)(0, x) = (um , vm ), ε < x < ε, ⎪ ⎩(u , v ), x > ε. + + Let ui , vi ≥ 0, ε > 0, i = −, m, +. We will consider the following four cases according to the different wave combinations from (0, −ε) and (0, ε): → ← − − ← − → − (1) J −m J m+ , (2) δ−m J m+ , (3) δ−m J m+ , (4) δ−m δm+ . − → ← − Case 1. J −m J m+ . → − In this case, we consider the interaction of forward contact discontinuity J −m starting from the point ← − (0, −ε) and backward contact discontinuity J m+ starting from the point (0, ε). The occurrence of this case depends on that the initial data (4.1) meet the conditions
2 vm
−u− v− +
2 1 + u2− − v− −um vm + 1 + − = = = λ− 2, 1 + u2m 1 + u2− 2 −u+ v+ − 1 + u2+ − v+ 2 −um vm − 1 + u2m − vm m λ1 = = = λ+ 1. 1 + u2m 1 + u2+
λm 2
u2m
(4.2)
(4.3)
→ − ← − → − m The propagation speed of J −m is λm 2 and that of J m+ is λ1 . Thus, it is easy to see that J −m overtakes ← − m −1 J m+ in a finite time, whose intersection point is (t1 , x1 ), which implies that t1 = 2ε(λm , x1 = 2 − λ1 ) m λ1 t1 + ε. Then, at time t = t1 , we get the Riemann problem (1.1) with initial data (u− , v− ), x < x1 , (u, v)(t, x) = (u+ , v+ ), x > x1 .
(4.4)
+ According to the relative sizes of λ− 1 and λ2 , we should divide our discussion into the following two subcases. − + − Subcase 1.1. λ1 < λ2 : According to λ1 < λ+ 2 , the solution of the Riemann problem can be expressed as follows
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+ Fig. 7. λ− 1 > λ2 .
+ Fig. 6. λ− 1 < λ2 .
⎧ ⎪ ⎨(u− , v− ), (u, v)(t, x) = (u∗ , v∗ ), ⎪ ⎩(u , v ), + +
x < λ− 1 (t − t1 ) + x1 , − λ1 (t − t1 ) + x1 < x < λ+ 2 (t − t1 ) + x1 , x > λ+ (t − t ) + x . 1 1 2
(4.5)
Using the similar procedures of getting (um , vm ) of system (3.11), we can get the state (u∗ , v∗ ) as shown in ← − → − Fig. 6. Thus, a new backward contact discontinuity J and a new forward contact discontinuity J will be → − ← − generated after the interaction of J and J . + Subcase 1.2. λ− 1 > λ2 : In this case, the solution of the Riemann problem can be given as the system (3.14), where x(t), w(t) is obtained by making a substitution t = t − t1 , x(t) = x(t) − x1 in the system (3.19), as → − ← − shown in Fig. 7. Thus, a new delta shock wave will be generated after the interaction of J and J . → − Theorem 4.1. When the initial data (4.1) satisfy (4.2) and (4.3), a forward contact discontinuity J starting ← − → − from the point (0, −ε) will collide with a backward discontinuity J starting from the point (0, ε). J overtakes ← − J in a finite time and after the interaction new contact discontinuities or a new delta shock wave will appear. ← − Case 2. δ−m J m+ . In this case, we consider the interaction of delta shock δ−m starting from the point (0, −ε) and backward ← − contact discontinuity J m+ starting from the point (0, ε). The occurrence of this case depends on that the + m m initial data (4.1) meet the conditions λ− 1 > λ2 , λ1 = λ1 . The propagation speed of δ−m is σ−m and that ← − ← − m of J m+ is λ1 . Thus, it is easy to see that δ−m overtakes J m+ in a finite time, whose intersection point is −1 (t1 , x1 ), which implies that t1 = 2ε(σ−m − λm , x 1 = λm 1 ) 1 t1 + ε. Then, at time t = t1 , we get the Riemann problem (1.1) with delta initial data (4.6) ⎧ ⎪ ⎨(v− , M− ), (v, M )(t, x) = (vδ , w1 δ(x − x1 )), ⎪ ⎩(v , M ), + +
x < x1 , x = x1 , x > x1 ,
(4.6)
+ where w(t1 ) = w1 , M = √1+u12 −v2 . According to the relative sizes of λ− 1 and λ2 , we should divide our discussion into the following two subcases. + − + Subcase 2.1. λ− 1 < λ2 : According to λ1 < λ2 , the solution of the Riemann problem can be expressed as ← − the system (4.5), as shown in Fig. 8. Thus, a new backward contact discontinuity J and a new forward → − ← − contact discontinuity J will be generated after the interaction of δ-shock and J . + − + Subcase 2.2. λ− 1 > λ2 : According to λ1 > λ2 , the solution of the Riemann problem can be expressed as the system (3.14), where x(t), w(t) is obtained by making a substitution t = t − t1 , x(t) = x(t) − x1 in the system (3.19), as shown in Fig. 9. Thus, a new delta shock wave will be generated after the interaction of ← − δ-shock and J .
J. Liu, R. Liu / J. Math. Anal. Appl. 486 (2020) 123932
+ Fig. 8. λ− 1 < λ2 .
+ Fig. 9. λ− 1 > λ2 .
+ Fig. 10. λ− 1 < λ2 .
+ Fig. 11. λ− 1 > λ2 .
11
+ m m Theorem 4.2. When the initial data (4.1) satisfy λ− 1 > λ2 and λ1 = λ1 , a delta shock wave starting from ← − ← − (0, −ε) will collide with a backward discontinuity J starting from (0, ε). δ-shock overtakes J in a finite time and after the interaction new contact discontinuities or a new delta shock wave will appear.
→ − Case 3. δ−m J m+ . In this case, we consider the interaction of delta shock δ−m starting from (0, −ε) and forward contact → − discontinuity J m+ starting from (0, ε). The occurrence of this case depends on that the initial data (4.1) → − + m m m meet the conditions λ− 1 > λ2 , λ2 = λ2 . The propagation speed of δ−m is σ−m and that of J m+ is λ2 . → − Thus, it is easy to see that δ−m overtakes J m+ in a finite time, whose intersection point is (t1 , x1 ), which −1 implies that t1 = 2ε(σ−m − λm , x 1 = λm 2 ) 2 t1 + ε. Then, at time t = t1 , we get the Riemann problem + m (1.1) with delta initial data (4.6), and notice that λ− 1 > λ2 = λ2 , thus, the solution of the Riemann problem can be expressed as the system (3.14), where x(t), w(t) is obtained by making a substitution t = t − t1 , x(t) = x(t) − x1 in the system (3.19), as shown in Fig. 10. Thus, a new delta shock wave will be → − generated after the interaction of δ-shock and J . Case 4. δ−m δm+ . In this case, we consider the interaction of delta shock δ−m starting from (0, −ε) and delta shock δm+ starting from (0, ε). The occurrence of this case depends on that the initial data (4.1) meet the conditions + m m λ− 1 > λ2 , λ1 > λ2 . The propagation speed of δ−m is σ−m and that of δm+ is σm+ . Thus, it is easy to see that δ−m overtakes δm+ in a finite time, whose intersection point is (t1 , x1 ), which implies that t1 = 2ε(σ−m − σm+ )−1 , x1 = σm+ t1 + ε. Then, at time t = t1 , we get the Riemann problem (1.1) with delta + m initial data (4.6), and notice that λ− 1 > λ2 > λ2 , thus, the solution of the Riemann problem can be expressed as the system (3.14), where x(t), w(t) is obtained by making a substitution t = t − t1 , x(t) = x(t) − x1 in the system (3.19), as shown in Fig. 11. Thus, a new delta shock wave will be generated after the interaction of δ-shocks.
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+ m m Theorem 4.3. When the initial data (4.1) satisfy λ− 1 > λ2 and λ2 = λ2 , a delta shock wave starting from → − the point (0, −ε) will collide with a forward contact discontinuity J starting from the piont (0, ε), δ-shock → − overtakes J in a finite time and after the interaction a new delta shock wave will appear. + m m Theorem 4.4. When the initial data (4.1) satisfy λ− 1 > λ2 and λ1 > λ2 , a delta shock wave starting from the point (0, −ε) will collide with another delta shock wave starting from the point (0, ε). After the interaction, there is a new delta shock wave.
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