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 http://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,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq3_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq4_HTML.gif are both constants, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq7_HTML.gif is a smooth manifold and divides http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq8_HTML.gif into two infinite parts, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq9_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq10_HTML.gif and denote http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq12_HTML.gif on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq14_HTML.gif and for any nonnegative function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq15_HTML.gif , it satisfies the following inequality:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ2_HTML.gif
(2.1)

where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq17_HTML.gif .

Lemma 2.2.

If http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ3_HTML.gif
(2.2)

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

It can be easily proved that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq20_HTML.gif , http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ4_HTML.gif
(2.3)
thus one can get http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq22_HTML.gif or
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ5_HTML.gif
(2.4)
and one notices that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq23_HTML.gif , http://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
http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq27_HTML.gif on discontinuity surface http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq28_HTML.gif of solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq30_HTML.gif is a unit normal vector to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq31_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq32_HTML.gif if
    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ7_HTML.gif
    (2.6)
     
then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ8_HTML.gif
(2.7)

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

For any constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq36_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ9_HTML.gif
(2.8)
or equivalently
http://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:
    http://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:
    http://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 http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ13_HTML.gif
(3.1)
Condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq39_HTML.gif For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq40_HTML.gif , 
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ14_HTML.gif
(3.2)

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

Condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq44_HTML.gif combines flux functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq45_HTML.gif and curved boundary manifold http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq47_HTML.gif is easily satisfied, for example, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq48_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq49_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq50_HTML.gif , so Condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq51_HTML.gif holds. Here http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq52_HTML.gif is a cubic curve on the http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq54_HTML.gif ) holds. If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq56_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ15_HTML.gif
(3.3)
and discontinuity surface http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq57_HTML.gif is
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ16_HTML.gif
(3.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq58_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq59_HTML.gif , http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq61_HTML.gif ) holds. If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq63_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ17_HTML.gif
(3.5)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq64_HTML.gif is the implicit function which satisfies
http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq65_HTML.gif ) holds. Given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq66_HTML.gif , then

(i)if http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq71_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq74_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq76_HTML.gif .

If http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ19_HTML.gif
(4.1)

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq79_HTML.gif is formed by moving http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq80_HTML.gif along the direction of the vector http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq81_HTML.gif , and the outward normal vector http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq82_HTML.gif of curve http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq83_HTML.gif is equal to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq84_HTML.gif . According to the angle between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq85_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq86_HTML.gif , the solution restricted in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq88_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq90_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq92_HTML.gif is outside region http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ20_HTML.gif
(4.2)
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq94_HTML.gif

Case 2.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq95_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq97_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq99_HTML.gif is inside region http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ21_HTML.gif
(4.3)
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq102_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ22_HTML.gif
(4.4)

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

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

Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq117_HTML.gif . According to the angles between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq118_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq119_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq120_HTML.gif , the solution restricted in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq122_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq124_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq125_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq127_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq128_HTML.gif are both inside region http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ25_HTML.gif
(4.7)
where http://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).
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq131_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq133_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq135_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq137_HTML.gif is inside region http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq138_HTML.gif , the rarefaction wave surface http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq139_HTML.gif is outside region http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ26_HTML.gif
(4.8)
where http://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).
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq142_HTML.gif

Case 5.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq143_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq145_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq146_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq148_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq149_HTML.gif are both outside region http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ27_HTML.gif
(4.9)
http://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 http://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 http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq153_HTML.gif , http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq155_HTML.gif since
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ29_HTML.gif
(4.11)
and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq156_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq157_HTML.gif is the extreme point of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq158_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq159_HTML.gif , according to the convex property of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq160_HTML.gif , we have that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq161_HTML.gif , http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq164_HTML.gif , http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq167_HTML.gif , http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq169_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq170_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq171_HTML.gif (see also Figure 4(b)), namely, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq172_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq173_HTML.gif , according to the convex property of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq175_HTML.gif , satisfying
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq176_HTML.gif , http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq178_HTML.gif since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq179_HTML.gif (see also Figure 5(b)) For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq180_HTML.gif , according to the convex property of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq182_HTML.gif , satisfying
http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq183_HTML.gif ) holds. Given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq184_HTML.gif , then

(i)if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq185_HTML.gif and   http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq187_HTML.gif and   http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq190_HTML.gif and   http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq193_HTML.gif and   http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq196_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq198_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq199_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq200_HTML.gif can be finite or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq201_HTML.gif , and when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq202_HTML.gif ,

(i)if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq203_HTML.gif and   http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq206_HTML.gif and   http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq209_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq211_HTML.gif and   http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq213_HTML.gif and   http://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 http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ33_HTML.gif
(5.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq217_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq218_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq219_HTML.gif , and it denotes http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq220_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq221_HTML.gif , we easily get http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq222_HTML.gif , and condition http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq224_HTML.gif and http://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) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq226_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq227_HTML.gif ; (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq228_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq229_HTML.gif ; (iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq230_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq231_HTML.gif ; (iv) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq232_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq233_HTML.gif ; (v) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq234_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq235_HTML.gif .

For case (i), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq236_HTML.gif and
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ35_HTML.gif
(5.3)
For case (ii), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq237_HTML.gif and
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ37_HTML.gif
(5.5)
For case (iii), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq238_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ38_HTML.gif
(5.6)
namely, http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq240_HTML.gif , where
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq241_HTML.gif :
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ42_HTML.gif
(5.10)

Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq243_HTML.gif satisfying the following three conditions: (a) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq244_HTML.gif ; (b) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq245_HTML.gif ; (c) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq247_HTML.gif , the following formula holds:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ43_HTML.gif
(5.11)

Proof.

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

where http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq251_HTML.gif is its real root, thus http://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), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq253_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ46_HTML.gif
(5.14)
namely, http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ47_HTML.gif
(5.15)

where http://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), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_IEq256_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F138396/MediaObjects/13661_2010_Article_25_Equ48_HTML.gif
(5.16)
namely, http://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
http://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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