Global Solution of a Nonlinear Conservation Law with Weak Discontinuous Flux in the Half Space

This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space { } | 0 R x x + = > ( ) ( ) ( ) 0, , 0 , 0 ,0 , 0, , 0 t x m u f u x R t u x a u x u x a u t u t +

The main purpose of our present manuscript is devoted to studying the structure of the global weak entropy solution for the above initial-boundary value problem under the condition of ( ) ( ) . By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial-boundary value problem, and investigate the interaction of elementary waves with the boundary and the boundary behavior of the weak entropy solution.

Introduction
Consider the initial-boundary value problem of a nonlinear conservation law in the half space with the initial condition ( ) ( ) and the boundary condition where 0 a > , u ± and m u are constant, and the flux f is a given continuous function of u, which satisfies the following conditions: (A 1 ) Its derivative function f ′ is piecewise C 1 -smooth with one discontinuous point d u , and ( ) exists, where f − ′ and f + ′ represent the left and right derivatives of f, respectively; For such an initial-boundary value problem ( 2)-( 4), under the conditions of (A 1 ) and (A 2 ), the global weak entropy solution was constructed in [1] for the case of ( ) ( ) . We want to study the structure of the global weak entropy solution of the problem ( 2)-( 4) for the case of ( ) ( ) the conditions of (A 1 ) and (A 2 ).As first step, we investigate the Riemann type of initial-boundary value problem, i.e., the problem ( 2)-( 4) with m u u u in our present manuscript.The more general problem ( 2)-( 4) with m u u ± ≠ will be investigated in our forthcoming paper.The main difficulty in studying the initial-boundary value problem of hyperbolic conservation laws is that the appearance of boundary results in obstacle in analysis.The difficulty lies in two respects: on one hand, the initial-boundary value problem of hyperbolic conservation laws is generally ill-posed; on the other hand, the nonlinear elementary waves will perhaps collide and interact with the boundary at finite time, so that the boundary layer may appear, which requires to give a reasonable boundary entropy condition to ensure the well-posedness of the global weak solution satisfying the relevant physical meaning.Bardos-Leroux-Nedelec [2] first established the existence and uniqueness of global weak entropy solution in the BV-setting for the initial-boundary value problem of scalar conservation laws with several space variables by vanishing viscosity method and by Kruzkov's method, respectively, and they gave a boundary entropy condition which requires only that the boundary data and the boundary value of solution satisfy an inequality.The other results of existence and uniqueness have been done for the initial-boundary value problem of scalar conservation laws after [2].The interested readers are referred to [3]- [9].Because of the influence of boundary, the geometric structure of the solution of initial-boundary value problem for scalar conservation laws is much more difficult than that of corresponding Cauchy problem.In recent years, for the initial-boundary value problem of one-dimensional nonlinear hyperbolic conservation laws (2)-( 4) with C 2 -smooth flux, some results have been obtained in this regard.The authors in papers [10] [11] [12] constructed the global weak entropy solutions to the initial-boundary problems on a bounded interval for some special initial-boundary data with three pieces of constant corresponding to the practical problem of continuous sedimentation of an ideal suspension.Liu-Pan [13] [14] [15] gave a construction method to the global weak entropy solution of the initial-boundary value problem with piecewise smooth initial dada and constant boundary data for scalar nonlinear hyperbolic conservation laws, and clarified the structure and boundary behavior of the weak entropy solution.
The present paper is organized as follows.In Section 2, we introduce the definition of weak entropy solution and the boundary entropy condition for the initial-boundary value problem ( 2)-( 4) and give a lemma to be used to construct the piecewise smooth solution of ( 2)-(4).In Section 3, based on the analysis method in [13], we use the lemma on piecewise smooth solution given in Section 2 to construct the global weak entropy solution of the initial-boundary value problem ( 2)-( 4) with m u u u under the conditions of (A 1 ) and (A 2 ) for the case of ( ) ( ) , and state the geometric structure and the behavior of boundary for the weak entropy solution.

Definition of Weak Entropy Solution and Related Lemma
where ( ) , u x t is a weak entropy solution of ( 2)-( 4), then it satisfies the following boundary entropy condition: where  .For the initial-boundary value problem ( 2)-( 4) with general initial-boundary data of bounded variation, its global weak entropy solution in the sense of (5) exists and is unique (see [2] [3] [6] [11]).In order to clarify the structure of the global weak entropy solution for the initial-boundary value problem ( 2)-( 4) under the assumptions (A 1 ) and (A 2 ), we need the following lemma 2.
Lemma 2 Suppose that the conditions (A 1 ) and (A 2 ) are valid.A piecewise smooth function ( ) , u x t with piecewise smooth discontinuity curves is a weak entropy solution of ( 2)-( 4) in the sense of ( 5) if and only if the following conditions are satisfied: 1) ( ) , u x t satisfies the Equation (2) on its smooth domains; 2) If ( ) , and when and the Oleinik's entropy condition hold, where , and u is any number between u − and u + .
By using the analogous technique in references [3] [16], Lemma 2 is easy to be proved by Definition 1 and Lemma 1, we omit it here.

Solution Structures
In this section, for the initial-boundary value problem ( 2)-( 4) with m u u u , we shall construct the global weak entropy solution under the conditions of (A) 1 and (A) 2 and ( ) ( ) by employing Lemma 2 and the structure of weak entropy solution of the corresponding initial value problem, and investigate the interaction of elementary waves with the boundary 0 x = and the boundary behaviors of the global weak entropy solution.The methods which will be used to construct the weak entropy solutions of the initial value problem and the initial-boundary value problem here are the characteristic method (see also [17]) and the truncation method developed in [13], respectively.
We only discuss the case of ( ) ( ) . The other cases can be dealt with similarly.
For the convenience of our construction work, we first introduce some notations.We denote , ; ,  7), ( 8) and the contact condition , respectively.The left or right or double-contact discontinuity waves are collectively referred to as the contact-discontinuity waves.It is well known that the solution of the shock wave where t c > .
If ( ) , v x t is an increasing (or decreasing) function with respect to x, which connects u − and u + from the leftmost to the rightmost, then ( ) , v x t is called an expansion wave (or compression wave) connecting u − and u + , we denote it by ( ) 4) is degenerated into a corres- ponding problem with 2 f C ∈ (see [13]).Throughout this section, we always suppose that ( )( ) We divide our problem into two cases: (I)

Case (I): m u u u
+ −

= ≠
According to the discussion framework in [13], we first investigate the solution structure of the following Riemann problem ( ) and then by which and Lemma 2, we construct the global weak entropy solution for the initial-boundary problem ( 2)-( 4).
We divide this case into two sub-cases: 1) u u + ≤ , similar to the discussion in [17], the weak entropy solution ( ) , v x t of Riemann problem (12) includes only a shock wave ( ) ( ) starting at point (0, 0) (see Figure 2), and this shock wave solution can be expressed as follows: , , Let ( ) ( ) , 0 , , , as , 0 0 , , as , 0 We can easily verify that this ( ) , u x t satisfies all conditions in Lemma 2, therefore it is the global weak entropy solution of the initial-boundary problem ( 2)-( 4).( ) , u x t includes only a constant state u + as ( ) 2).
is an unknown function of x R + ∈ and 0 t > , u ± , m u are three given constants satisfying m u the flux function f is a given continuous function with a weak discontinuous point d u .

Following the papers [ 2 ]
[3], we give the definition and the boundary entropy condition of weak entropy solution for the initial-boundary value problem (2)-(4).Definition 1 Let ( ) , u x t be a bounded and local bounded variation function on

u
denote a rarefaction wave connecting l u and r u from the left to the right, centered at point ( ) from the left to the right, emanating from point ( ) b c in x t− plane is respectively expressed as:

Figure 1 .
Figure 1.The location of 1* u in the

Figure 2 .
Figure 2. The shock wave ( ) , S u u − + of the problem (12) for the case of

Figure 3 .
Figure 3.The compression wave ( ) , C u u − + of the problem (12) for the case of By the assumptions on the flux function f, there exist two numbers 2

Figure 4 .
Figure 4.The location of 2* u and 3* u in the

Figure 5 .
Figure 5.The expansion wave (), E u u − +of the problem(12) for the case of

,
u x t is the global weak entropy solution of the problem (2)-(4).( ) not interact with the boundary 0 x = and can be written as follows:

Figure 6 .
Figure 6.The expansion wave (), E u u − +of the problem(12) for the case of

Figure 7 .
Figure 7.The expansion wave (), E u u − +of the problem(12) for the case of