^{1}

^{2}

This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R
^{+}= {
x |
x
>
0}
^{+}
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.

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:

(A_{1}) Its derivative function f ′ is piecewise C^{1}-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;

(A_{2}) f ″ ( u ) > 0 for u ≠ u d .^{ }

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}) and (A_{2}). 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}-smooth flux, some results have been obtained in this regard. The authors in papers [

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 [_{1}) and (A_{2}) 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.

Following the papers [

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 [_{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 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 [

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 [

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

and

u ( x , t ) = { u l , x < b + f ′ + ( u l ) ( t − c ) ( f ′ ) − 1 ( x − b t − c ) , b + f ′ + ( u l ) ( t − c ) < x < b + f ′ − ( u r ) ( t − c ) u + , 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 [

According to the discussion framework in [

{ 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 + .

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

If u + ≤ u 1 * , similar to the discussion in [

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

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

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

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

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

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

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

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

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

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

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 * < u − < u d , if we set u ( x , t ) = v ( x , t ) | x , t > 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).

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

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

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 − < u 2 * and 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 − ) (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

If u − < u 2 * and 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

If u − < u 2 * and u d < 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 − ) (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

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.

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

The authors declare no conflicts of interest regarding the publication of 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