1. Introduction

AJCM

American Journal of Computational Mathematics

2161-1203

Scientific Research Publishing

10.4236/ajcm.2018.84026

AJCM-89633

Articles

Physics&Mathematics

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

Xiaoqian

¹ Jing

Zhang

Department of Optoelectronic Engineering, Jinan University, Guangzhou, China

Department of Mathematics, Jinan University, Guangzhou, China

03 12 2018

08 04 326 342 12, December 2018 26, December 2018 29, December 2018

2014

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R ⁺= { x | x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R⁺ and t>0 , u_± , u_m are three given constants satisfying u_m=u₊≠u_- or u_m = u_- ≠ u₊ , and the flux function f is a given continuous function with a weak discontinuous point u_d. 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 f '_-(u_d) > f '₊(u_d). 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.

Nonlinear Conservation Laws with Weak Discontinuous Flux Initial-Boundary Value Problem Shock Wave Rarefaction Wave Contact Discontinuity Interaction Structure of Global Weak Entropy Solution

1. Introduction

Consider the initial-boundary value problem of a nonlinear conservation law in the half space R + = { x | x > 0 }

u t + f ( u ) x = 0 , x ∈ R + , t > 0 (2)

with the initial condition

u ( x , 0 ) = u 0 ( x ) : = { u m , 0 < x < a u + , x > a (3)

and the boundary condition

u ( 0 , t ) = u b ( t ) : = u − , t > 0 (4)

where a > 0 , u ± and u m are constant, and the flux f is a given continuous function of u, which satisfies the following conditions:

(A₁) Its derivative function f ′ is piecewise C¹-smooth with one discontinuous point u d , and f ′ ± ( u d ) exists, where f ′ − and f ′ + represent the left and right derivatives of f, respectively;

(A₂) f ″ ( u ) > 0 for u ≠ u d .

For such an initial-boundary value problem (2)-(4), under the conditions of (A₁) and (A₂), the global weak entropy solution was constructed in [1] for the case of f ′ − ( u d ) < f ′ + ( u d ) . We want to study the structure of the global weak entropy solution of the problem (2)-(4) for the case of f ′ − ( u d ) > f ′ + ( u d ) under the conditions of (A₁) and (A₂). As first step, we investigate the Riemann type of initial-boundary value problem, i.e., the problem (2)-(4) with u m = u + ≠ u − or u m = u − ≠ u + in our present manuscript. The more general problem (2)-(4) with u m ≠ 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 C²-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 u m = u + ≠ u − or u m = u − ≠ u + under the conditions of (A₁) and (A₂) for the case of f ′ − ( u d ) > f ′ + ( u d ) , and state the geometric structure and the behavior of boundary for the weak entropy solution.

2. Definition of Weak Entropy Solution and Related Lemma

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 [ 0 , ∞ ) × [ 0 , ∞ ) . If for each k ∈ ( − ∞ , ∞ ) and for any nonnegative test function ϕ ∈ C 0 ∞ ( [ 0 , ∞ ) × [ 0 , ∞ ) ) , it satisfies the following inequality

∫ 0 ∞ ∫ 0 ∞ ( | u − k | ϕ t + sgn ( u − k ) ( f ( u ) − f ( k ) ) ϕ x ) d x d t + ∫ 0 ∞ | u 0 ( x ) − k | ϕ ( x , 0 ) d x + ∫ 0 ∞ sgn ( u b ( t ) − k ) ( f ( u ( 0 , t ) ) − f ( k ) ) ϕ ( 0 , t ) d t ≥ 0 , (5)

where

sgn ( x ) = { 1 x ≥ 0 − 1 x < 0

then u ( x , t ) is called a weak entropy solution of the initial-boundary problem (2)-(4).

Lemma 1 If u ( x , t ) is a weak entropy solution of (2)-(4), then it satisfies the following boundary entropy condition:

u ( 0 + , t ) = u b ( t )

or f ( u ( 0 + , t ) ) − f ( k ) u ( 0 + , t ) − k ≤ 0 , k ∈ I ( u ( 0 + , t ) , u b ( t ) ) , k ≠ u ( 0 + , t ) , a . e . t ≥ 0 , (6)

where k ∈ I ( u ( 0 + , t ) , u b ( t ) ) = [ min { u ( 0 + , t ) , u b ( t ) } , max { u ( 0 + , t ) , u b ( t ) } ] .

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 (A₁) and (A₂), we need the following lemma 2.

Lemma 2 Suppose that the conditions (A₁) and (A₂) 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 x = x ( t ) is a weak discontinuity curves of u ( x , t ) , then when

u ( x ( t ) , t ) is not the discontinuous point of f ′ , d x d t = f ′ ( u ( x ( t ) , t ) ) , and when u ( x ( t ) , t ) is the discontinuous point of f ′ ,

d x d t = f ′ − ( u ( x ( t ) , t ) ) or d x d t = f ′ + ( u ( x ( t ) , t ) ) ;

If x = x ( t ) is a strong discontinuity curves of u ( x , t ) , then the Rankine-Hugoniot’s discontinuity condition

d x d t = f ( u − ) − f ( u + ) u − − u + , (7)

and the Oleinik’s entropy condition

f ( u − ) − f ( u + ) u − − u + ≤ f ( u − ) − f ( u ) u − − u (8)

hold, where u ± = u ( x ( t ) ± 0 , t ) , and u is any number between u − and u + .

3) The boundary entropy condition (6) is valid.

4) u ( x , 0 ) = u 0 ( x ) a . e . x ≥ 0.

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.

3. Solution Structures

In this section, for the initial-boundary value problem (2)-(4) with u m = u + ≠ u − or u m = u − ≠ u + , we shall construct the global weak entropy solution under the conditions of (A)₁ and (A)₂ and f ′ − ( u d ) > f ′ + ( u d ) 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 x = 0 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 f ′ − ( u d ) > 0 > f ′ + ( u d ) . The other cases can be dealt with similarly.

For the convenience of our construction work, we first introduce some notations. We denote

ω ( u 1 , u 2 ) = f ( u 1 ) − f ( u 2 ) u 1 − u 2 (9)

Let R ( u l , u r ; ( b , c ) ) denote a rarefaction wave connecting u l and u r from the left to the right, centered at point ( b , c ) in the x − t plane, and S ( u l , u r ; ( b , c ) ) denote a shock wave x = x ( t ) connecting u l and u r from the left to the right, starting at point ( b , c ) in the x − t plane, where x = x ( t ) satisfies (7), (8) and the Lax’s shock wave condition f ′ − ( u − ) > x ′ ( t ) > f ′ + ( u + ) . We denote a left or right or double-contact discontinuity wave x = x ( t ) connecting u l and u r from the left to the right, emanating from point ( b , c ) in the x − t plane by S l ( u l , u r ; ( b , c ) ) or S r ( u l , u r ; ( b , c ) ) or S l r ( u l , u r ; ( b , c ) ) , respectively, where x = x ( t ) satisfies (7), (8) and the contact condition f ′ − ( u − ) = x ′ ( t ) > f ′ + ( u + ) or f ′ − ( u − ) > x ′ ( t ) = f ′ + ( u + ) or f ′ − ( u − ) = x ′ ( t ) = f ′ + ( u + ) , 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 S ( u l , u r ; ( b , c ) ) (or the contact-discontinuity waves S l ( u l , u r ; ( b , c ) ) or S r ( u l , u r ; ( b , c ) ) or S l r ( u l , u r ; ( b , c ) ) and the solution of the central rarefaction wave R ( u l , u r ; ( b , c ) ) in x − t plane is respectively expressed as:

u ( x , t ) = { u l , x b + ω ( u l , u r ) ( t − c ) (10)

and

u ( x , t ) = { u l , x b + f ′ − ( u r ) ( t − c ) (11)

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 E ( u − , u + ) (or C ( u − , u + ) ).

When u ± ≤ u d or u ± ≥ u d , the problem (2)-(4) is degenerated into a corresponding problem with f ∈ C 2 (see [13] ). Throughout this section, we always suppose that ( u − − u d ) ( u + − u d ) < 0 . We divide our problem into two cases: (I) u m = u + ≠ u − ; (II) u m = u − ≠ u + .

3.1. Case (I): u m = u + ≠ u −

According to the discussion framework in [13] , we first investigate the solution structure of the following Riemann problem

{ v t + f ( v ) x = 0 , − ∞ < x < ∞ , t > 0 v ( x , 0 ) = { u − , x < 0 u + , x > 0 (12)

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 d < u − ; 2) u − < u d < u + .

3.1.1. u + < u d < u −

Let u 1 * denote the abscissa of the intersection point of the secant passing through point ( u − , f ( u − ) ) and ( u d , f ( u d ) ) with the image of f in u − f ( u ) plane (see Figure 1).

If u + ≤ u 1 * , similar to the discussion in [17] , the weak entropy solution v ( x , t ) of Riemann problem (12) includes only a shock wave S ( u − , u + ; ( 0 , 0 ) ) starting at point (0, 0) (see Figure 2), and this shock wave solution can be expressed as follows:

v ( x , t ) = { u − , x < ω ( u − , u + ) t u + , x > ω ( u − , u + ) t (13)

Let u ( x , t ) = v ( x , t ) | x , t > 0 , then

u ( 0 + , ) t = { u + , as ω ( u − , u + ) ≤ 0 u − , as ω ( u − , u + ) > 0 (14)

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 ω ( u − , u + ) ≤ 0 or a shock wave S ( u − , u + ; ( 0 , 0 ) ) as ω ( u − , u + ) > 0 (see Figure 2).

If u 1 * < u + < u d , by ω ( u − , u d ) < ω ( u d , u + ) , (7), (8) and the Lax’s shock wave condition, we have that a compression wave C ( u − , u + ) , which includes two shock waves S ( u − , u d ; ( 0 , 0 ) ) and S ( u d , u + ; ( 0 , 0 ) ) , i.e., C ( u − , u + ) = S ( u − , u d ; ( 0 , 0 ) ) ∪ S ( u d , u + ; ( 0 , 0 ) ) , appears in the weak entropy solution v ( x , t ) of the Riemann problem (12) (see Figure 3), this compression wave solution v ( x , t ) can be written as:

v ( x , t ) = { u − , x < ω ( u − , u d ) t u d , ω ( u − , u d ) t < x < ω ( u d , u + ) t u + , x > ω ( u d , u + ) t . (15)

Set u ( x , t ) = v ( x , t ) | x , t > 0 , then it holds

u ( 0 + , t ) = { u − , as ω ( u − , u d ) > 0 u d , as ω ( u − , u d ) ≤ 0 and ω ( u d , u + ) > 0 u + , as ω ( u d , u + ) ≤ 0 (16)

By Lemma 2, this u ( x , t ) is the global weak entropy solution of the initial-boundary problem (2)-(4). u ( x , t ) includes only a compression wave C ( u − , u + ) = S ( u − , u d ; ( 0 , 0 ) ) ∪ S ( u d , u + ; ( 0 , 0 ) ) as ω ( u − , u d ) > 0 or a shock wave S ( u d , u + ; ( 0 , 0 ) ) as ω ( u − , u d ) ≤ 0 and ω ( u d , u + ) > 0 or a constant state u + as ω ( u d , u + ) ≤ 0 (see Figure 3).

3.1.2. u − < u d < u +

By the assumptions on the flux function f, there exist two numbers u 2 * , u 3 * such that

u 2 * < u d < u 3 * and ω ( u 2 * , u 3 * ) = f ′ ( u 2 ) = f ′ ( u 3 ) (see Figure 4). By using of the similar analysis in [17] , the weak entropy solution v ( x , t ) of Riemann problem (12) includes only an expansion wave E ( u − , u + ) .

When u − < u 2 * , E ( u − , u + ) can be expressed as follows:

E ( u − , u + ) = { R ( u − , u 2 * ; ( 0 , 0 ) ) ∪ S l r ( u 2 * , u 3 * ; ( 0 , 0 ) ) , if u + = u 3 * R ( u − , u 2 * ; ( 0 , 0 ) ) ∪ S l r ( u 2 * , u 3 * ; ( 0 , 0 ) ) ∪ R ( u 3 * , u + ; ( 0 , 0 ) ) , if u + > u 3 * R ( u − , u 4 * ; ( 0 , 0 ) ) ∪ S l ( u 4 * , u + ; ( 0 , 0 ) ) , if u d < u + < u 3 * , (17)

where u 4 * ∈ ( u 2 * , u d ) satisfies ω ( u 4 * , u + ) = f ′ ( u 4 * ) . Thus this expansion wave solution can be written as:

v ( x , t ) = { u − , x ≤ f ′ ( u − ) t ( f ′ ) − 1 ( x t ) , f ′ ( u − ) t < x < f ′ ( u 2 * ) t u + , x > f ′ ( u 2 * ) t (18)

for u + = u 3 * (see Figure 5), and for u + > u 3 * ,

v ( x , t ) = { u − , x ≤ f ′ ( u − ) t ( f ′ ) − 1 ( x t ) , f ′ ( u − ) t < x < f ′ ( u 2 * ) t or f ′ ( u 3 * ) t < x ≤ f ′ ( u + ) t u + , x > f ′ ( u + ) t (19)

(see Figure 6), and for u d < u + < u 3 * ,

v ( x , t ) = { u − , x ≤ f ′ ( u − ) t ( f ′ ) − 1 ( x t ) , f ′ ( u − ) t < x < f ′ ( u 4 * ) t u + , x > f ′ ( u 4 * ) t (20)

(see Figure 7).

If we take u ( x , t ) = v ( x , t ) | x , t > 0 , then for the case of u + = u 3 * ,

u ( 0 + , t ) = { u + , if ω ( u 2 * , u 3 * ) ≤ 0 u − , if ω ( u 2 * , u 3 * ) > 0 and f ′ ( u − ) ≥ 0 u 5 * , if ω ( u 2 * , u 3 * ) > 0 and f ′ ( u − ) < 0 (21)

where u 5 * ∈ ( u − , u 2 * ) satisfies f ′ ( u 5 * ) = 0 ; and for the case of u + > u 3 * ,

u ( 0 + , t ) = { u + , if f ′ ( u + ) ≤ 0 u 6 * , if f ′ ( u + ) > 0 and ω ( u 2 * , u 3 * ) ≤ 0 u − , if f ′ ( u + ) > 0 and ω ( u 2 * , u 3 * ) > 0 and f ′ ( u − ) ≥ 0 u 5 * , if f ′ ( u + ) > 0 and ω ( u 2 * , u 3 * ) > 0 and f ′ ( u − ) < 0 (22)

where u 6 * ∈ [ u 3 * , u + ) satisfies f ′ ( u 6 * ) = 0 ; and for the case of u d < u + < u 3 * ,

u ( 0 + , t ) = { u + , if ω ( u 4 * , u + ) ≤ 0 u − , if ω ( u 4 * , u + ) > 0 and f ′ ( u − ) ≥ 0 u 5 * , if ω ( u 4 * , u + ) > 0 and f ′ ( u − ) < 0 (23)

Therefore, from Lemma 2, we can also easily verify that u ( x , t ) is the global weak entropy solution of the problem (2)-(4). u ( x , t ) includes only an expansion wave E ¯ = E ( u − , u + ) | x , t > 0 , which does not interact with the boundary x = 0 and can be written as follows:

E ¯ = { u + , if ω ( u 2 * , u 3 * ) ≤ 0 R ( u − , u 2 * ; ( 0 , 0 ) ) ∪ S l r ( u 2 * , u 3 * ; ( 0 , 0 ) ) , if ω ( u 2 * , u 3 * ) > 0 and f ′ ( u − ) ≥ 0 R ( u 5 * , u 2 * ; ( 0 , 0 ) ) ∪ S l r ( u 2 * , u 3 * ; ( 0 , 0 ) ) , if ω ( u 2 * , u 3 * ) > 0 and f ′ ( u − ) < 0 (24)

for the case of u + = u 3 * (see also Figure 5); and

E ¯ = { u + , if f ′ ( u + ) ≤ 0 R ( u 6 * , u + ; ( 0 , 0 ) ) , if f ′ ( u + ) > 0 and ω ( u 2 * , u 3 * ) ≤ 0 R ( u − , u 2 * ; ( 0 , 0 ) ) ∪ S l r ( u 2 * , u 3 * ; ( 0 , 0 ) ) ∪ R ( u 3 * , u + ; ( 0 , 0 ) ) , A0; if f ′ ( u + ) > 0 and ω ( u 2 * , u 3 * ) > 0 and f ′ ( u − ) ≥ 0 R ( u 5 * , u 2 * ; ( 0 , 0 ) ) ∪ S l r ( u 2 * , u 3 * ; ( 0 , 0 ) ) ∪ R ( u 3 * , u + ; ( 0 , 0 ) ) , A0; if f ′ ( u + ) > 0 and ω ( u 2 * , u 3 * ) > 0 and f ′ ( u − ) < 0 (25)

for the case of u + > u 3 * (see also Figure 6); and

E ¯ = { u + , if ω ( u 4 * , u + ) ≤ 0 R ( u − , u 4 * ; ( 0 , 0 ) ) ∪ S l ( u 4 * , u + ; ( 0 , 0 ) ) , if ω ( u 4 * , u + ) > 0 and f ′ ( u − ) ≥ 0 R ( u 5 * , u 4 * ; ( 0 , 0 ) ) ∪ S l ( u 4 * , u + ; ( 0 , 0 ) ) , if ω ( u 4 * , u + ) > 0 and f ′ ( u − ) < 0 (26)

for the case of u d < u + < u 3 * (see also Figure 7).

When u − = u 2 * ,

E ( u − , u + ) = { S l r ( u 2 * , u 3 * ; ( 0 , 0 ) ) , if u + = u 3 * S l r ( u 2 * , u 3 * ; ( 0 , 0 ) ) ∪ R ( u 3 * , u + ; ( 0 , 0 ) ) , if u + > u 3 * R ( u 2 * , u 4 * ; ( 0 , 0 ) ) ∪ S l ( u 4 * , u + ; ( 0 , 0 ) ) , if u d < u + < u 3 * (27)

and this expansion wave solution of Riemann problem (12) can be written as:

v ( x , t ) = { u − , x < f ′ ( u 3 * ) t u + , x > f ′ ( u 3 * ) t (28)

for u + = u 3 * , and for u + > u 3 * ,

v ( x , t ) = { u − , x < f ′ ( u 3 * ) t ( f ′ ) − 1 ( x t ) , f ′ ( u 3 * ) t < x ≤ f ′ ( u + ) t u + , x > f ′ ( u + ) t (29)

and for u d < u + < u 3 * ,

v ( x , t ) = { u − , x ≤ f ′ ( u − ) t ( f ′ ) − 1 ( x t ) , f ′ ( u − ) t < x < f ′ ( u 4 * ) t u + , x > f ′ ( u 4 * ) t (30)

When u 2 * < u − < u d ,

E ( u − , u + ) = { S r ( u − , u 7 * ; ( 0 , 0 ) ) , if u + = u 7 * S r ( u − , u 7 * ; ( 0 , 0 ) ) ∪ R ( u 7 * , u + ; ( 0 , 0 ) ) , if u + > u 7 * S ( u − , u + ; ( 0 , 0 ) ) , if u d < u + < u 7 * , (31)

where u 7 * ∈ ( u d , u 3 * ) satisfies ω ( u − , u 7 * ) = f ′ ( u 7 * ) . The weak entropy solution v ( x , t ) of Riemann problem (12) v ( x , t ) can be expressed as follows:

v ( x , t ) = { u − , x < f ′ ( u 7 * ) t u + , x > f ′ ( u 7 * ) t (32)

for u + = u 7 * , and for u + > u 7 * ,

v ( x , t ) = { u − , x < f ′ ( u 7 * ) t ( f ′ ) − 1 ( x t ) , f ′ ( u 7 * ) t < x ≤ f ′ ( u + ) t u + , x > f ′ ( u + ) t , (33)

and for u d < u + < u 7 * ,

v ( x , t ) = { u − , x < ω ( u − , u + ) t u + , x > ω ( u − , u + ) t (34)

When u − = u 2 * or u 2 * 0 , then similar to the case of u − < u 2 * , we can give the expression for u ( 0 + , t ) , and by which and Lemma 2, it is easy to be verified that u ( x , t ) is the global weak entropy solution of the problem (2)-(4).

3.2. Case (II): u m = u − ≠ u +

Similar to sub-section 3.1, we divide this case into two sub-cases: 1) u + < u d < u − ; 2) u − < u d < u + .

3.2.1. u + < u d < u −

Consider the following initial value problem

{ v t + f ( v ) x = 0 , − ∞ < x < ∞ , t > 0 v ( x , 0 ) = { u − , x < a u + , x > a (35)

Let u i * ( i = 1 , 2 , ⋯ , 7 ) be defined in sub-section 3.1.

If u + ≤ u 1 * , the weak entropy solution v ( x , t ) of Riemann problem (35) includes only a shock wave S ( u − , u + ; ( a , 0 ) ) starting at point ( a , 0 ) , and this shock wave solution can be expressed as follows:

v ( x , t ) = { u − , x < ω ( u − , u + ) t u + , x > ω ( u − , u + ) t (36)

Let u ( x , t ) = v ( x , t ) | x , t > 0 , then as ω ( u − , u + ) ≥ 0 , u ( 0 + , t ) = u − ( t > 0 ), and as ω ( u − , u + ) < 0 ,

u ( 0 + , t ) = { u − , 0 < t < − a ( ω ( u − , u + ) ) − 1 u + , t > − a ( ω ( u − , u + ) ) − 1 (37)

where t = − a ( ω ( u − , u + ) ) − 1 is the intersection time of S ( u − , u + ; ( a , 0 ) ) and t-axis. Thus by Lemma 2, we can derive that u ( x , t ) is the global weak entropy solution of the initial-boundary problem (2)-(4). u ( x , t ) includes only a shock wave S ( u − , u + ; ( a , 0 ) ) , which will be far away from the boundary x = 0 as ω ( u − , u + ) ≥ 0 or will interact with the boundary and be absorbed by the boundary at time t = t 1 (see Figure 8).

If u 1 * < u + < u d , a compression wave C ( u − , u + ) , which includes two shock waves S ( u − , u d ; ( a , 0 ) ) and S ( u d , u + ; ( a , 0 ) ) , i.e., C ( u − , u + ) = S ( u − , u d ; ( a , 0 ) ) ∪ S ( u d , u + ; ( a , 0 ) ) , appears in the weak entropy solution v ( x , t ) of the Riemann problem (35), this compression wave solution v ( x , t ) can be written as:

v ( x , t ) = { u − , x < a + ω ( u − , u d ) t u d , a + ω ( u − , u d ) t < x < a + ω ( u d , u + ) t u + , x > a + ω ( u d , u + ) t (38)

We take u ( x , t ) = v ( x , t ) | x , t > 0 , then the expression for u ( 0 + , t ) can be given as follows. As ω ( u − , u d ) ≥ 0 , u ( 0 + , t ) = u − ( t > 0 ); as ω ( u − , u d ) < 0 and ω ( u d , u + ) ≥ 0 ,

u ( 0 + , t ) = { u − , 0 < t < − a / ω ( u − , u d ) u d , t > − a / ω ( u − , u d ) (39)

as ω ( u d , u + ) < 0 ,

u ( 0 + , t ) = { u − , 0 < t < − a / ω ( u − , u d ) u d , − a / ω ( u − , u d ) < t < − a / ω ( u d , u + ) u + , t > − a / ω ( u d , u + ) (40)

Therefore, from Lemma 2, it follows that u ( x , t ) is the global weak entropy solution of the initial-boundary problem (2)-(4). u ( x , t ) includes only a compression wave C ( u − , u + ) | x , t > 0 , which will be far away from the boundary x = 0 (if ω ( u − , u d ) ≥ 0 ) or will interact with the boundary and be partially absorbed (if ω ( u − , u d ) < 0 and ω ( u d , u + ) ≥ 0 ) or completely absorbed (if ω ( u d , u + ) < 0 ) by the boundary (see Figure 9).

3.2.2. u − < u d < u +

The weak entropy solution v ( x , t ) of Riemann problem (35) also includes only an expansion wave E ( u − , u + ) .

When u − < u 2 * , E ( u − , u + ) can be expressed as follows:

E ( u − , u + ) = { R ( u − , u 2 * ; ( a , 0 ) ) ∪ S l r ( u 2 * , u 3 * ; ( a , 0 ) ) , if u + = u 3 * R ( u − , u 2 * ; ( a , 0 ) ) ∪ S l r ( u 2 * , u 3 * ; ( a , 0 ) ) ∪ R ( u 3 * , u + ; ( a , 0 ) ) , if u + > u 3 * R ( u − , u 4 * ; ( a , 0 ) ) ∪ S l ( u 4 * , u + ; ( a , 0 ) ) , if u d < u + < u 3 * (41)

Thus this expansion wave solution can be written as:

v ( x , t ) = { u − , x ≤ a + f ′ ( u − ) t ( f ′ ) − 1 ( x − a t ) , a + f ′ ( u − ) t < x < a + f ′ ( u 2 * ) t u + , x > a + f ′ ( u 2 * ) t (42)

for u + = u 3 * , and for u + > u 3 * ,

v ( x , t ) = { u − , x ≤ a + f ′ ( u − ) t ( f ′ ) − 1 ( x − a t ) , a + f ′ ( u − ) t < x < a + f ′ ( u 2 * ) t or a + f ′ ( u 3 * ) t < x ≤ a + f ′ ( u + ) t u + , x > a + f ′ ( u + ) t (43)

and for u d < u + < u 3 * ,

v ( x , t ) = { u − , x ≤ a + f ′ ( u − ) t ( f ′ ) − 1 ( x − a t ) , a + f ′ ( u − ) t < x < a + f ′ ( u 4 * ) t u + , x > a + f ′ ( u 4 * ) t (44)

When u − = u 2 * ,

E ( u − , u + ) = { S l r ( u 2 * , u 3 * ; ( a , 0 ) ) , if u + = u 3 * S l r ( u 2 * , u 3 * ; ( a , 0 ) ) ∪ R ( u 3 * , u + ; ( a , 0 ) ) , if u + > u 3 * R ( u 2 * , u 4 * ; ( a , 0 ) ) ∪ S l ( u 4 * , u + ; ( a , 0 ) ) , if u d < u + < u 3 * (45)

This expansion wave solution can be written as:

v ( x , t ) = { u − , x < a + f ′ ( u 3 * ) t u + , x > a + f ′ ( u 3 * ) t (46)

for u + = u 3 * , and for u + > u 3 * ,

v ( x , t ) = { u − , x < a + f ′ ( u 3 * ) t ( f ′ ) − 1 ( x − a t ) , a + f ′ ( u 3 * ) t < x ≤ a + f ′ ( u + ) t u + , x > a + f ′ ( u + ) t (47)

and for u d < u + < u 3 * ,

v ( x , t ) = { u − , x ≤ a + f ′ ( u − ) t ( f ′ ) − 1 ( x − a t ) , a + f ′ ( u − ) t < x < a + f ′ ( u 4 * ) t u + , x > a + f ′ ( u 4 * ) t (48)

When u 2 * < u − < u d ,

E ( u − , u + ) = { S r ( u − , u 7 * ; ( a , 0 ) ) , if u + = u 7 * S r ( u − , u 7 * ; ( a , 0 ) ) ∪ R ( u 7 * , u + ; ( a , 0 ) ) , if u + > u 7 * S ( u − , u + ; ( a , 0 ) ) , if u d < u + < u 7 * , (49)

The weak entropy solution v ( x , t ) of Riemann problem (35) v ( x , t ) can be expressed as follows:

v ( x , t ) = { u − , x < a + f ′ ( u 7 * ) t u + , x > a + f ′ ( u 7 * ) t (50)

for u + = u 7 * , and for u + > u 7 * ,

v ( x , t ) = { u − , x < a + f ′ ( u 7 * ) t ( f ′ ) − 1 ( x − a t ) , a + f ′ ( u 7 * ) t < x ≤ a + f ′ ( u + ) t u + , x > a + f ′ ( u + ) t (51)

and for u d < u + < u 7 * ,

v ( x , t ) = { u − , x < a + ω ( u − , u + ) t u + , x > a + ω ( u − , u + ) t (52)

Let u ( x , t ) = v ( x , t ) | x , t > 0 , then by using of Lemma 2 and the expression of v ( x , t ) , we can also verify that u ( x , t ) is the global weak entropy solution of the problem (2)-(4). In what follows, we write the expression for u ( 0 + , t ) and state the interaction of the elementary wave and the boundary only for the case of u − < u 2 * , and we can deal with the other cases similarly.

If u − 0 ) as f ′ ( u − ) ≥ 0 , and

u ( 0 + , t ) = { u − , 0 < t ≤ − a / f ′ ( u − ) ( f ′ ) − 1 ( x − a t ) , t > − a / f ′ ( u − ) (53)

as f ′ ( u − ) < 0 and f ′ ( u 2 * ) ≥ 0 , and

u ( 0 + , t ) = { u − , 0 < t ≤ − a / f ′ ( u − ) ( f ′ ) − 1 ( x − a t ) , − a / f ′ ( u − ) < t < − a / f ′ ( u + ) u + , t > − a / f ′ ( u + ) (54)

as f ′ ( u 2 * ) < 0 . In this case, u ( x , t ) includes only a expansion wave E ( u − , u + ) | x , t > 0 . It will be far away from the boundary as f ′ ( u − ) ≥ 0 , or one part R ( u − , u 5 * ; ( a , 0 ) ) of rarefaction wave in E ( u − , u + ) | x , t > 0 will be absorbed by the boundary as f ′ ( u − ) < 0 and f ′ ( u 2 * ) > 0 , or the whole of E ( u − , u + ) | x , t > 0 will be completely absorbed by the boundary as f ′ ( u 2 * ) ≤ 0 (see Figure 10).

If u − u 3 * , then u ( 0 + , t ) = u − ( t > 0 ) as f ′ ( u − ) ≥ 0 , and

u ( 0 + , t ) = { u − , 0 < t ≤ − a / f ′ ( u − ) ( f ′ ) − 1 ( x − a t ) , t > − a / f ′ ( u − ) (55)

as f ′ ( u − ) < 0 and f ′ ( u 2 * ) ≥ 0 , and

u ( 0 + , t ) = { u − , 0 < t ≤ − a / f ′ ( u − ) ( f ′ ) − 1 ( x − a t ) , t > − a / f ′ ( u − ) , t ≠ − a / ω ( u 2 * , u 3 * ) (56)

as f ′ ( u 2 * ) < 0 and f ′ ( u + ) ≥ 0 , and

u ( 0 + , t ) = { u − , 0 < t ≤ − a / f ′ ( u − ) ( f ′ ) − 1 ( x − a t ) , − a / f ′ ( u − ) < t ≤ − a / f ′ ( u + ) , t ≠ − a / ω ( u 2 * , u 3 * ) u + , t > − a / f ′ ( u + ) (57)

as f ′ ( u + ) < 0 . In this case, u ( x , t ) includes only a expansion wave E ( u − , u + ) | x , t > 0 . It will be far away from the boundary as f ′ ( u − ) ≥ 0 , or one part R ( u − , u 5 * ; ( a , 0 ) ) of rarefaction wave in E ( u − , u + ) | x , t > 0 will be absorbed by the boundary as f ′ ( u − ) < 0 and f ′ ( u 2 * ) ≥ 0 , or one part R ( u − , u 2 * ; ( a , 0 ) ) ∪ S l r ( u 2 * , u 3 * ; ( a , 0 ) ) ∪ R ( u 3 * , u 6 * ; ( a , 0 ) ) of E ( u − , u + ) | x , t > 0 will be absorbed by the boundary as f ′ ( u 2 * ) < 0 and f ′ ( u + ) > 0 , or the whole of E ( u − , u + ) | x , t > 0 will be completely absorbed by the boundary f ′ ( u + ) ≤ 0 (see Figure 11).

If u − 0 ) as f ′ ( u − ) ≥ 0 , and

u ( 0 + , t ) = { u − , 0 < t ≤ − a / f ′ ( u − ) ( f ′ ) − 1 ( x − a t ) , t > − a / f ′ ( u − ) (58)

as f ′ ( u − ) < 0 and f ′ ( u 4 * ) ≥ 0 , and

u ( 0 + , t ) = { u − , 0 < t ≤ − a / f ′ ( u − ) ( f ′ ) − 1 ( x − a t ) , − a / f ′ ( u − ) < t < − a / f ′ ( u 4 * ) u + , t > − a / f ′ ( u 4 * ) (59)

as f ′ ( u 4 * ) < 0 . We now state the interaction of the unique elementary wave E ( u − , u + ) | x , t > 0 in u ( x , t ) with the boundary for this case. E ( u − , u + ) | x , t > 0 will be far away from the boundary as f ′ ( u − ) ≥ 0 , or one part R ( u − , u 5 * ; ( a , 0 ) ) of rarefaction wave in E ( u − , u + ) | x , t > 0 will be absorbed by the boundary as f ′ ( u − ) < 0 and f ′ ( u 4 * ) > 0 or the whole of E ( u − , u + ) | x , t > 0 will be completely absorbed by the boundary f ′ ( u 4 * ) ≤ 0 (see Figure 12).

4. Conclusion

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 f ′ − ( u d ) > f ′ + ( u d ) . 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. Compared with the case of f ′ − ( u d ) < f ′ + ( u d ) , the weak entropy solution of the problem (2)-(4) with the case of f ′ − ( u d ) > f ′ + ( u d ) includes different wave type: contact discontinuity.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11731008) and the Guangdong Natural Science Foundation of China (2018A030313906).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Li, X.Q. and Zhang, J. (2018) Global Solution of a Nonlinear Conservation Law with Weak Discontinuous Flux in the Half Space. American Journal of Computational Mathematics, 8, 326-342. https://doi.org/10.4236/ajcm.2018.84026

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