Open Access

Two-Dimension Riemann Initial-Boundary Value Problem of Scalar Conservation Laws with Curved Boundary

Boundary Value Problems20112011:138396

DOI: 10.1155/2011/138396

Received: 16 December 2010

Accepted: 1 February 2011

Published: 24 February 2011

Abstract

This paper is concerned with the structure of the weak entropy solutions to two-dimension Riemann initial-boundary value problem with curved boundary. Firstly, according to the definition of weak entropy solution in the sense of Bardos-Leroux-Nedelec (1979), the necessary and sufficient condition of the weak entropy solutions with piecewise smooth is given. The boundary entropy condition and its equivalent formula are proposed. Based on Riemann initial value problem, weak entropy solutions of Riemann initial-boundary value problem are constructed, the behaviors of solutions are clarified, and we focus on verifying that the solutions satisfy the boundary entropy condition. For different Riemann initial-boundary value data, there are a total of five different behaviors of weak entropy solutions. Finally, a worked-out specific example is given.

1. Introduction

Multidimensional conservation laws are a famous hard problem that plays an important role in mechanics and physics [13]. For Cauchy problem of multi-dimensional scalar conservation laws, Conway and Smoller [4] and Kruzkov [1] have proved that weak solution uniquely exists if it also satisfies entropy condition, and it is called weak entropy solutions. In order to further understand qualitative behavior of solutions, it is also important to investigate multi-dimensional Riemann problems. For two-dimensional case, Lindquist [5], Wagner [6], Zhang and Zheng [7] Guckenheimer [8], Zheng [9] among others, have discussed some relating Riemann problems. In a previous discussion, initial value contains several constant states with discontinuity lines so that self-similar transformations can be applied to reduce two-dimensional problem to one-dimensional case. The situation that initial value contains two constant states divided by a curve can not be solved by selfsimilar transformations, and Yang [10] proposed a new approach for construction of shock wave and rarefaction wave solutions; especially, rarefaction wave was got by constructing implicit function instead of the usual selfsimilar method. This approach can be expanded to general https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq1_HTML.gif -dimension. In addition, multi-dimensional scalar conservation laws with boundary are more common in practical problems. Bardos et al. [2] have proved the existence and uniqueness of the weak entropy solution of initial-boundary problems of multi-dimensional scalar conservation laws. The main difficulty for nonlinear conservation laws with boundary is to have a good formation of the boundary condition. Namely, for a fixed initial value, we really can not impose such a condition at the boundary, and the boundary condition is necessarily linked to the entropy condition. Moreover the behavior of solutions for one-dimensional problem with boundary was discussed in [1118]. However, for multi-dimensional problem with boundary, the behaviors of solutions are still hard to study.

In this paper, two-dimensional case as an example of Yang's multi-dimensional Riemann problem [10] is expanded to the case with boundary. Considering two-dimensional Riemann problem for scalar conservation laws with curved boundary,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq3_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq4_HTML.gif are both constants, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq7_HTML.gif is a smooth manifold and divides https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq8_HTML.gif into two infinite parts, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq9_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq10_HTML.gif and denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq11_HTML.gif .

In Section 2, weak entropy solution of Riemann initial-boundary value problem (1.1) is defined, and the boundary entropy condition is discussed. In Section 3, weak entropy solutions of the corresponding Riemann initial value problem are expressed. In Section 4, using the weak entropy solutions of the corresponding Riemann initial value problem, we construct the weak entropy solutions of Riemann initial-boundary value problem, and prove that they satisfy the boundary entropy condition. The weak entropy solutions include a total of five different shock and rarefaction wave solutions based on different Riemann data. Finally, in Section 5, we give a worked-out specific example.

2. Preliminaries

According to the definition of the weak entropy solution and the boundary entropy condition to the general initial-boundary problems of multi-dimensional scalar conservation laws which was proposed by Bardos et al. [2] and Pan and Lin [13], we can obtain the following definition and three lemmas for Riemann initial-boundary value problem (1.1).

Definition 2.1.

A locally bounded and bounded variation function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq12_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq13_HTML.gif is called a weak entropy solution of Riemann initial-boundary value problem (1.1) if, for any real constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq14_HTML.gif and for any nonnegative function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq15_HTML.gif , it satisfies the following inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ2_HTML.gif
(2.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq16_HTML.gif is the outward normal vector of curve https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq17_HTML.gif .

Lemma 2.2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq18_HTML.gif is a weak entropy solution of (1.1), then it satisfies the following boundary: entropy condition
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ3_HTML.gif
(2.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq19_HTML.gif .

It can be easily proved that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq20_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq21_HTML.gif , so (2.2) can be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ4_HTML.gif
(2.3)
thus one can get https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq22_HTML.gif or
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ5_HTML.gif
(2.4)
and one notices that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq23_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq24_HTML.gif , then boundary entropy condition (2.2) is equivalent to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ6_HTML.gif
(2.5)

The proof for one-dimension case of Lemma 2.2 can be found in Pan and Lin's work [13], and the proof for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq25_HTML.gif -dimension case is totally similar to one-dimension case; actually the idea of the proof first appears in Bardos et al.'s work [2], so the proof details for Lemma 2.2 are omitted here.

Lemma 2.3.

A piecewise smooth function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq26_HTML.gif with smooth discontinuous surface is a weak entropy solution to the Riemann initial-boundary value problem (1.1) in the sense of (2.1) if and only if the following conditions are satisfied.
  1. (i)
    Rankine-Hugoniot condition: At any point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq27_HTML.gif on discontinuity surface https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq28_HTML.gif of solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq29_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq30_HTML.gif is a unit normal vector to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq31_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq32_HTML.gif if
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ7_HTML.gif
    (2.6)
     
then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ8_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq34_HTML.gif .

For any constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq35_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq36_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ9_HTML.gif
(2.8)
or equivalently
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ10_HTML.gif
(2.9)
  1. (ii)
    Boundary entropy condition:
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ11_HTML.gif
    (2.10)
     
  1. (iii)
    Initial value condition:
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ12_HTML.gif
    (2.11)
     

For piecewise smooth solution with smooth discontinuous surface, Rankine-Hugoniot condition (2.7), entropy conditions (2.8), (2.9) and initial value condition (2.11) are obviously satisfied, see also the previous famous works in [4, 79]. As in Lemma 2.2, boundary entropy condition (2.10) also holds. The proof of the converse in not difficult and is omitted here.

According to Bardos et al.'s work [2], we have the following Lemma.

Lemma 2.4.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq37_HTML.gif is piecewise smooth weak entropy solution of (1.1) which satisfies the conditions of Lemma 2.3, then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq38_HTML.gif is unique.

According to the uniqueness of weak entropy solution, as long as the piecewise smooth function satisfying Lemma 2.3 is constructed, the weak entropy solution of Riemann initial-boundary value problem can be obtained.

3. Solution of Riemann Initial Value Problem

First, we study the Riemann initial value problem corresponding to the Riemann initial-boundary value problem (1.1) as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ13_HTML.gif
(3.1)
Condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq39_HTML.gif For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq40_HTML.gif , 
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ14_HTML.gif
(3.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq41_HTML.gif is a certain interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq42_HTML.gif can be a finite number or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq43_HTML.gif .

Condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq44_HTML.gif combines flux functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq45_HTML.gif and curved boundary manifold https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq46_HTML.gif , providing necessary condition for the convex property of the new flux function which will be constructed in formula (4.5). The convex property clarifies whether the characteristics intersect or not, whether the weak solution satisfied internal entropy conditions (2.8) and (2.9) and boundary entropy condition (2.10), In addition, Condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq47_HTML.gif is easily satisfied, for example, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq48_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq49_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq50_HTML.gif , so Condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq51_HTML.gif holds. Here https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq52_HTML.gif is a cubic curve on the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq53_HTML.gif plane, and it is strictly bending.

Yang's work [10] showed that depending on whether the characteristics intersect or not, the weak entropy solution of (3.1) has two forms as follows.

Lemma 3.1 (see [10]).

Suppose ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq54_HTML.gif ) holds. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq55_HTML.gif , then weak entropy solution of (3.1) is shock wave solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq56_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ15_HTML.gif
(3.3)
and discontinuity surface https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq57_HTML.gif is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ16_HTML.gif
(3.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq58_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq59_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq60_HTML.gif .

Lemma 3.2 (see [10]).

Suppose that ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq61_HTML.gif ) holds. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq62_HTML.gif , then weak entropy solution of (3.1) is rarefaction wave solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq63_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ17_HTML.gif
(3.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq64_HTML.gif is the implicit function which satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ18_HTML.gif
(3.6)

Theorem 3.3 (see [10]).

Suppose that ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq65_HTML.gif ) holds. Given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq66_HTML.gif , then

(i)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq67_HTML.gif , weak entropy solution of(3.1)is S and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq69_HTML.gif has a form as (3.3);

(ii)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq70_HTML.gif , weak entropy solution of (3.1) is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq72_HTML.gif has a form as (3.5);

(iii) weak entropy solutions formed as (3.3) and (3.5) uniquely exist.

The weak entropy solutions constructed here are piecewise smooth and satisfy conditions (i) and (iii) of Lemma 2.3.

4. Solution of Riemann Initial-Boundary Value Problem

Now we restrict the weak entropy solutions of the Riemann initial value problem (3.1) constructed in Section 3 in region https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq73_HTML.gif , and they still satisfy conditions (i) and (iii) of Lemma 2.3. If they also satisfy the boundary entropy condition (ii) of Lemma 2.3, then they are the weak entropy solutions of Riemann initial-boundary value problem (1.1).

Based on different Riemann data of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq75_HTML.gif , the weak entropy solutions of the Riemann initial value problem (3.1) have the following five different behaviors when restricted in region https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq76_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq77_HTML.gif , the solution of (3.1) is shock wave S and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ19_HTML.gif
(4.1)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq79_HTML.gif is formed by moving https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq80_HTML.gif along the direction of the vector https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq81_HTML.gif , and the outward normal vector https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq82_HTML.gif of curve https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq83_HTML.gif is equal to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq84_HTML.gif . According to the angle between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq85_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq86_HTML.gif , the solution restricted in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq87_HTML.gif has two behaviors as follows.

Case 1.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq88_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq89_HTML.gif .

See also Figure 1(a); it shows that the angle between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq91_HTML.gif is an acute angle, the shock wave surface https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq92_HTML.gif is outside region https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq93_HTML.gif , and the solution is constant state formed as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ20_HTML.gif
(4.2)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Fig1_HTML.jpg
Figure 1

Case 1. The constant solutionThe phase plane https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq94_HTML.gif

Case 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq95_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq96_HTML.gif .

See also Figure 2; it shows that the angle between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq97_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq98_HTML.gif is an obtuse angle, the shock wave surface https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq99_HTML.gif is inside region https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq100_HTML.gif , and the solution is shock wave formed as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ21_HTML.gif
(4.3)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Fig2_HTML.jpg
Figure 2

The shock wave solution of Case 2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq101_HTML.gif , the solution of (3.1) is rarefaction wave https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq102_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ22_HTML.gif
(4.4)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq103_HTML.gif is formed by moving https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq104_HTML.gif along the direction of the vector https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq105_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq106_HTML.gif is formed by moving https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq107_HTML.gif along the direction of the vector https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq108_HTML.gif , and the outward normal vector https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq109_HTML.gif of curve https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq110_HTML.gif is equal to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq111_HTML.gif .

We construct a new flux function
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ23_HTML.gif
(4.5)
according to condition ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq112_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq113_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq114_HTML.gif is convex, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq115_HTML.gif is monotonically increasing function, so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq116_HTML.gif . And also
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ24_HTML.gif
(4.6)

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq117_HTML.gif . According to the angles between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq118_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq119_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq120_HTML.gif , the solution restricted in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq121_HTML.gif has three behaviors as follows.

Case 3.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq122_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq123_HTML.gif .

See also Figure 3; it shows that the angles between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq124_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq126_HTML.gif are obtuse angles, the rarefaction wave surfaces https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq127_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq128_HTML.gif are both inside region https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq129_HTML.gif , and the solution is rarefaction wave formed as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ25_HTML.gif
(4.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq130_HTML.gif is the implicit function which satisfies (3.6).
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Fig3_HTML.jpg
Figure 3

The rarefaction wave solution of Case 3.

Case 4.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq131_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq132_HTML.gif .

See also Figure 4(a); it shows that the angle between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq134_HTML.gif is an obtuse angle, the angle between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq135_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq136_HTML.gif is an acute angles, the rarefaction wave surface https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq137_HTML.gif is inside region https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq138_HTML.gif , the rarefaction wave surface https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq139_HTML.gif is outside region https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq140_HTML.gif , and the solution is rarefaction wave formed as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ26_HTML.gif
(4.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq141_HTML.gif is the implicit function which satisfies (3.6).
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Fig4_HTML.jpg
Figure 4

Case 4. The rarefaction wave solutionThe phase plane https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq142_HTML.gif

Case 5.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq144_HTML.gif .

See also Figure 5(a); it shows that the angles between https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq145_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq146_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq147_HTML.gif are acute angles, the rarefaction wave surfaces https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq148_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq149_HTML.gif are both outside region https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq150_HTML.gif , and the solution is constant state formed as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ27_HTML.gif
(4.9)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Fig5_HTML.jpg
Figure 5

Case 5. The constant solutionThe phase plane https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq151_HTML.gif

Next, we verify the above five solutions all satisfying the boundary entropy condition (ii) of Lemma 2.3. By noticing the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq152_HTML.gif (4.5) and its convex property, the boundary entropy condition (ii) of Lemma 2.3 can be equivalent to the following formula
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ28_HTML.gif
(4.10)

and thus we verify the above five solutions all satisfying the boundary entropy condition (4.10).

Case 1.

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq153_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq154_HTML.gif , the shock wave solution is formed as (4.2). In this case, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq155_HTML.gif since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ29_HTML.gif
(4.11)
and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq156_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq157_HTML.gif is the extreme point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq158_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq159_HTML.gif , according to the convex property of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq160_HTML.gif , we have that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ30_HTML.gif
(4.12)

and so the boundary entropy condition (4.10) is verified.

Case 2.

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq161_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq162_HTML.gif , the shock wave solution is formed as (4.3). In this case, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq163_HTML.gif , so the boundary entropy condition (4.10) is naturally verified.

Case 3.

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq164_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq165_HTML.gif , the rarefaction wave solution is formed as (4.7). In this case, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq166_HTML.gif , and so the boundary entropy condition (4.10) is naturally verified.

Case 4.

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq167_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq168_HTML.gif , the rarefaction wave solution is formed as (4.8). In this case, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq169_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq170_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq171_HTML.gif (see also Figure 4(b)), namely, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq172_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq173_HTML.gif , according to the convex property of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq174_HTML.gif and Lagrange mean value theorem, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq175_HTML.gif , satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ31_HTML.gif
(4.13)

and so the boundary entropy condition (4.10) is verified.

Case 5.

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq176_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq177_HTML.gif , the rarefaction wave solution is formed as (4.9). In this case, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq178_HTML.gif since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq179_HTML.gif (see also Figure 5(b)) For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq180_HTML.gif , according to the convex property of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq181_HTML.gif and Lagrange mean value theorem, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq182_HTML.gif , satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ32_HTML.gif
(4.14)

and so the boundary entropy condition (4.10) is verified.

In summary, we have the following theorem.

Theorem 4.1.

Suppose that ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq183_HTML.gif ) holds. Given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq184_HTML.gif , then

(i)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq185_HTML.gif and   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq186_HTML.gif , the solution of (1.1) is constant state and has form as (4.2),

(ii)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq187_HTML.gif and   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq188_HTML.gif , the solution of (1.1) is shock wave T , and has form as (4.3),

(iii)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq190_HTML.gif and   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq191_HTML.gif , the solution of (1.1) is rarefaction wave https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq192_HTML.gif and has a form as (4.7),

(iv)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq193_HTML.gif and   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq194_HTML.gif , the solution of (1.1) is rarefaction wave https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq195_HTML.gif and has a form as (4.8);

(v)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq196_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq197_HTML.gif , the solution of (1.1) is constant state and has a form as (4.9).

In addition the solutions formed as (4.2), (4.3), (4.7), (4.8), and (4.9) uniquely exist.

Corollary 4.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq198_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq199_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq200_HTML.gif can be finite or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq201_HTML.gif , and when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq202_HTML.gif ,

(i)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq203_HTML.gif and   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq204_HTML.gif , the solution of (1.1) is rarefaction wave https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq205_HTML.gif and has a form as (4.7),

(ii)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq206_HTML.gif and   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq207_HTML.gif , the solution of (1.1) is rarefaction wave https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq208_HTML.gif and has a form as (4.8),

(iii)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq209_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq210_HTML.gif , the solution of (1.1) is constant state and has a form as (4.9),

(iv)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq211_HTML.gif and   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq212_HTML.gif , the solution of (1.1) is constant state and has a form as (4.2),

(v)if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq213_HTML.gif and   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq214_HTML.gif , the solution of (1.1) is shock wave https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq215_HTML.gif and has a form as (4.3).

Corollary 4.3.

The approach here for two-dimensional Riemann initial-boundary problem can be expanded to the case of general https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq216_HTML.gif -dimension.

5. An Example

Solve the following Riemann initial-boundary problem:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ33_HTML.gif
(5.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq217_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq218_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq219_HTML.gif , and it denotes https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq220_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq221_HTML.gif , we easily get https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq222_HTML.gif , and condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq223_HTML.gif holds.

According to the different data of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq224_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq225_HTML.gif , the behavior of the solution to Riemann initial-boundary problem (5.1) has a total of five situations; they can be described by the following five cases: (i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq226_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq227_HTML.gif ; (ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq228_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq229_HTML.gif ; (iii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq230_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq231_HTML.gif ; (iv) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq232_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq233_HTML.gif ; (v) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq234_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq235_HTML.gif .

For case (i), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq236_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ34_HTML.gif
(5.2)
and thus the solution is constant state formed as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ35_HTML.gif
(5.3)
For case (ii), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq237_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ36_HTML.gif
(5.4)
and thus the solution is shock wave solution formed as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ37_HTML.gif
(5.5)
For case (iii), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq238_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ38_HTML.gif
(5.6)
namely, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq239_HTML.gif , thus the solution is rarefaction wave formed as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ39_HTML.gif
(5.7)
Here, we only need to solve https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq240_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ40_HTML.gif
(5.8)
To solve the following equation of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq241_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ41_HTML.gif
(5.9)
using Cardano formula, we can get the unique solution as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ42_HTML.gif
(5.10)

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq242_HTML.gif is the solution of implicit function, we still need to verify https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq243_HTML.gif satisfying the following three conditions: (a) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq244_HTML.gif ; (b) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq245_HTML.gif ; (c) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq246_HTML.gif . In fact, according to the next proposition, the above three conditions can be easily verified, and the detail the omitted here.

Proposition 5.1.

For any real number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq247_HTML.gif , the following formula holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ43_HTML.gif
(5.11)

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ44_HTML.gif
(5.12)
then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq248_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ45_HTML.gif
(5.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq249_HTML.gif must be one root of (5.13). In fact, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq250_HTML.gif . Equation (5.13) at most has one real root; but https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq251_HTML.gif is its real root, thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq252_HTML.gif , and the proposition holds.

For case (iv), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq253_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ46_HTML.gif
(5.14)
namely, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq254_HTML.gif , and thus the solution is rarefaction wave formed as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ47_HTML.gif
(5.15)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq255_HTML.gif has the same form as (5.10).

For case (v), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq256_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ48_HTML.gif
(5.16)
namely, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq257_HTML.gif , and thus the solution is constant state formed as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ49_HTML.gif
(5.17)

Declarations

Acknowledgment

This work is supported by the National Natural Science Foundation of China (10771087, 61078040), the Natural Science Foundation of Guangdong Province (7005948).

Authors’ Affiliations

(1)
Department of Mathematics, Shanghai University
(2)
Key Laboratory of Optoelectronic Information and Sensing Technologies of Guangdong Higher Educational Institutes, Jinan University

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Copyright

© H. Chen and T. Pan. 2011

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