^{1}

^{*}

^{2}

This paper is concerned with the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function. Under the condition that the flux function has a finite number of weak discontinuous points, by using the structure of weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the global weak entropy solution for this initial-boundary value problem, and by investigating the interaction of elementary waves and the boundary, we clarify the geometric structure and the behavior of boundary for the weak entropy solution.

Consider the following initial-boundary value problem for scalar conservation laws:

{ u t + f ( u ) x = 0 , x > 0 , t > 0 u ( x , 0 ) = u 0 ( x ) , x > 0 u ( 0 , t ) = u b ( t ) , t > 0 , (1)

where u 0 ( ⋅ ) and u b ( ⋅ ) are two bounded and local bounded variation functions on [ 0, + ∞ ) , and the flux f is assumed to be locally Lipschitz continuous.

The initial-boundary value problem for scalar conservation laws plays an important role in mathematical modelling and simulation of practical problem of the one-dimensional sedimentation processes and traffic flow on highways [

For the geometric structure and regularity and large time behavior of solution of the initial value problem for scalar conservation laws, see [^{2}- smooth function class, some results have been obtained in this regard. The authors in [

The purpose of our present paper is devoted to constructing the global weak entropy solution of the initial-boundary value problem (1) for scalar conservation laws with two pieces of constant initial data and constant boundary data under the condition that the flux function has a finite number of weak discontinuous points, and clarifying the geometric structure and the behavior of boundary for 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 (1), and give a lemma to be used to construct the piecewise smooth solution of (1). In Section 3, basing on the analysis method in [

In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different classes of equations. About the definition of weak solution for the equation of scalar conservation laws, see [

Definition 1. A bounded and local bounded variable function u ( x , t ) on [ 0, ∞ ) × [ 0, ∞ ) is called a weak entropy solution of the initial-boundary problems (1), 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 + s g n ( u − k ) ( f ( u ) − f ( k ) ) ϕ x ] d x d t + ∫ 0 ∞ | u 0 ( x ) − k | ϕ ( x ,0 ) d x + ∫ 0 ∞ s g n ( u b ( t ) − k ) ( f ( u ( 0, t ) ) − f ( k ) ) ϕ ( 0, t ) d t ≥ 0, (2)

where

s g n ( x ) = { 1, x ≥ 0 − 1, x < 0

For the initial-boundary value problems (1) whose initial data and bounded data are general bounded variation functions, the existence and uniqueness of the global weak entropy solution in the sense of (2) has been obtained, and the global weak entropy solution satisfies the following boundary entropy condition (3) (see [

Lemma 1. If u ( x , t ) is a weak entropy solution of (1), then,

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, (3)

where I ( u ( 0 + , t ) , u b ( t ) ) = [ m i n { u ( 0 + , t ) , u b ( t ) } , m a x { u ( 0 + , t ) , u b ( t ) } ] .

In what follows, we give a lemma for the piecewise smooth solution to (1), which will be employed to construct the piecewise smooth solution of (1).

Before stating the lemma, we make the following assumptions to the flux f :

(A_{1}) f ∈ C ;

(A_{2}) Its derivative function f ′ is a piecewise C^{1}-smooth function with a finite number of discontinuous points u d i , and there exist f ′ ± ( u d i ) such that f ′ − ( u d i ) < f ′ + ( u d i ) , where f ′ − and f ′ + represent the left and right derivatives of f respectively;

(A_{3}) f ″ ( u ) > 0 , u ≠ u d i .

Lemma 2. Under the assumptions (A_{1})-(A_{3}), a piecewise smooth function u ( x , t ) with piecewise smooth discontinuity curves is a weak entropy solution of (1) in the sense of (2), if and only if the following conditions are satisfied:

(1) u ( x , t ) satisfies Equation (1)_{1} on its smooth domains;

(2) If x = x ( t ) is a weak discontinuity of u ( x , t ) , then when u ( x ( t ) , t ) is

not the discontinuous point of f ′ , then 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 of u ( x , t ) , then the Rankine-Hugoniot discontinuity condition

d x ( t ) d t = f ( u − ) − f ( u + ) u − − u + (4)

and the Oleinik entropy condition

f ( u − ) − f ( u + ) u − − u + ≤ f ( u − ) − f ( u ) u − − u (5)

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

(3) The boundary entropy condition (3) is valid;

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

Lemma 2 is easily to be proved by Definition 1 and Lemma 1 (see [

Notations. For the convenience of our construction work, we introduce some notations. Let R ( u − , u + ; ( a , b ) ) denote a rarefaction wave connecting u − and u + from the left to the right centered at point ( a , b ) in the x − t plane, abbreviated as R ( u − , u + ) , and S ( u − , u + ; ( a , b ) ) denote a shock wave x = x ( t ) connecting u − and u + from the left to the right starting at point ( a , b ) in the x − t plane, abbreviated as S ( u − , u + ) , whose speed x ′ ( t ) is also denoted by

s ( u − , u + ) , i.e., x ′ ( t ) = s ( u − , u + ) = f ( u − ) − f ( u + ) u − − u + , where x = x ( t ) satisfies the

Rankine-Hugoniot condition (4) and the Oleinik entropy condition (5).

It is well known that the solution of the shock wave S ( u − , u + ) centered at point ( a , b ) and the solution of the central rarefaction wave R ( u − , u + ) starting at point ( a , b ) in the x − t plane are respectively expressed as:

u ( x , t ) = { u − , x < a + f ( u − ) − f ( u + ) u − − u + ( t − b ) u + , x > a + f ( u − ) − f ( u + ) u − − u + ( t − b )

and

u ( x , t ) = { u − , x < a + f ′ ( u − ) ( t − b ) ( f ′ ) − 1 ( x − a t − b ) , a + f ′ ( u − ) ( t − b ) < x < a + f ′ ( u + ) ( t − b ) u + , x > a + f ′ ( u + ) ( t − b )

where t > b .

In this section, with the aid of the analysis method in [^{2}-smooth flux function. This analysis method is basing on the tracing of the position of elementary waves (especially the shock wave) in the weak entropy solution v ( x , t ) for the corresponding initial value problem and the boundary entropy condition (3). According to [

Consider the following initial-boundary problem:

{ u t + f ( u ) x = 0 , x > 0 , t > 0 u ( x , 0 ) = { u m , 0 < x < a u + , x > a u ( 0 , t ) ≡ u − , t > 0 (6)

where u ± , u m are constant, for x > 0 and x ≠ a , a > 0 is a constant. We first consider the case that f has only one weak discontinuous point, and then the case that f has finitely many weak discontinuities.

Throughout this sub-section, the flux f is assumed to satisfy (A_{1}) and the following conditions:

(A_{2})' f ′ is a piecewise C^{1}―smooth function with one weak discontinuous point u d , and there exist f ′ ± ( u d ) such that f ′ − ( u d ) < f ′ + ( u d ) ;

(A_{3})' f ″ ( u ) > 0 , u ≠ u d .

We first discuss the initial boundary value problem (6) for the case of u m = u + ≠ u − , which is called Riemann initial-boundary problem, written as

{ u t + f ( u ) x = 0 , x > 0 , t > 0 u ( x , 0 ) ≡ u + , x > 0 u ( 0 , t ) ≡ u − , t > 0 , (7)

where u − ≠ u + . And then investigate (6) with u m ≠ u + . For definiteness, it has no harm to assume that f ( 0 ) = f ( 0 ) ′ = 0 and u d < 0 in this sub-section. The other cases can be dealt with similarly.

When ( u − − u d ) ⋅ ( u + − u d ) ≥ 0 , (7) is degenerated into a corresponding problem with f ∈ C 2 (see [

Case (1) u − < u d < u + .

Consider the following Riemann problem corresponding to (7):

{ v t + f ( v ) x = 0 , − ∞ < x < ∞ , t > 0 v ( x , 0 ) = v 0 ( x ) : = { u − , x < 0 u + , x > 0. (8)

In this case, since the flux function has a weak discontinuity point u = u d , the Riemann problem (8) includes only a rarefaction wave R = R ( u − , u d ) ∪ R ( u d , u + ) centered at point ( 0,0 ) of the x − t plane. This rarefaction wave solution can be written as:

v ( x , t ) = { u − , x < f ′ ( u − ) t ( f ′ ) − 1 ( x t ) , f ′ ( u − ) t ≤ x < f ′ − ( u d ) t u d , f ′ − ( u d ) t ≤ x < f ′ + ( u d ) t ( f ′ ) − 1 ( x t ) , f ′ + ( u d ) t ≤ x < f ′ ( u + ) t u + , x ≥ f ′ ( u + ) t

Let u ( x , t ) = v ( x , t ) | x , t > 0 , then u ( 0 + , t ) = m i n { u + ,0 } , hence, it holds the boundary entropy condition:

f ( u ( 0 + , t ) ) − f ( k ) u ( 0 + , t ) − k ≤ 0 ( ∀ k ∈ [ u − , u ( 0 + , t ) ] , k ≠ u ( 0 + , t ) ) .

It is easy to verify that u ( x , t ) also satisfies all other conditions in Lemma 2. Therefore, by Lemma 2, u ( x , t ) is the global weak entropy solution of (7).

Case (2) u + < u d < u − .

In this case, (8) includes only a shock wave S ( u − , u + ) at point ( 0,0 ) in the x − t plane. This shock wave solution can be expressed as follows:

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

where s ( u − , u + ) is the speed of the shock S ( u − , u + ) . Let u ( x , t ) = v ( x , t ) | x , t > 0 , then

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

From Lemma 2, we can easily verify that u ( x , t ) is the global weak entropy solution of (7).

Consider the following initial value problem corresponding to (6):

{ v t + f ( v ) x = 0 , − ∞ < x < ∞ , t > 0 v ( x , 0 ) = v 0 ( x ) : = { u − , x < 0 u m , 0 < x < a u + , x > a . (9)

According to the solution structure of (9), we construct the global weak entropy solution of (6) with u m ≠ u + by dividing our problem into five cases: (1) u − = u m ≠ u + ; (2) u − < u m < u + ; (3) u + < u m < u − ; (4) u − , u + < u m ; (5) u m < u − , u + .

Case (1) u − = u m ≠ u + .

In fact, when u − = u m , ( u − − u d ) ⋅ ( u + − u d ) ≥ 0 , (6) becomes a problem with f ∈ C 2 , which was discussed in [

If u − = u m < u + , only a rarefaction wave R = R ( u − , u d ) ∪ R ( u d , u + ) , centered at ( a ,0 ) of the x − t plane, appears in the weak entropy solution of (9). This rarefaction wave solution of (9) 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 d ) t u d , a + f ′ − ( u d ) t ≤ x < a + f ′ + ( u d ) t ( f ′ ) − 1 ( x − a t ) , a + f ′ + ( u d ) t ≤ x < a + f ′ ( u + ) t u + , x ≥ a + f ′ ( u + ) t .

Let u ( x , t ) = v ( x , t ) | x , t > 0 , where v ( x , t ) is the weak entropy solution of (9). It is easy to verify u ( x , t ) satisfies all conditions in Lemma 2, thus u ( x , t ) is the global weak entropy solution of (6). It includes only a rarefaction wave R | x , t > 0 , which will interact with the boundary x = 0 and be completely absorbed (if u + ≤ 0 ) (see

If u − = u m > u + , the weak entropy solution v ( x , t ) of (9) includes only a shock wave emanating at point ( a ,0 ) of the x − t plane, which can be expressed as follow:

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

Let u ( x , t ) = v ( x , t ) | x , t > 0 , then by Lemma 2, it is also easy to verify that u ( x , t ) is the global weak entropy solution of (6). It includes only a shock wave S ( u − , u + ; ( a ,0 ) ) , which will interact with the boundary x = 0 and be absorbed (if s ( u − , u + ) ≤ 0 ) (see

Case (2) u − < u m < u + .

If u ± , u m ≥ u d , or u ± , u m ≤ u d , (6) becomes a problem with f ∈ C 2 , see [

When u − < u m < u d < u + , two rarefaction waves R ( u − , u m ) and

R 1 = R ( u m , u d ) ∪ R ( u d , u + ) , centered at point ( 0,0 ) and ( a ,0 ) , respectively, appear in the weak entropy solution v ( x , t ) of (9); when u − < u d < u m < u + ,

two rarefaction waves R 2 = R ( u − , u d ) ∪ R ( u d , u m ) and R ( u m , u + ) , centered at point ( 0,0 ) and ( a ,0 ) , respectively, appear in the weak entropy solution v ( x , t ) of (9); when u − < u m = u d < u + , two centered rarefaction waves R ( u − , u d ) and R ( u d , u + ) , centered at point ( 0,0 ) and ( a ,0 ) , respectively, appear in the weak entropy solution v ( x , t ) of (9). The two rarefaction waves in v ( x , t ) , centered at point ( 0,0 ) and ( a ,0 ) , respectively, will not overtake each other since the propagating speed of the wave front in the first wave is not greater than that of the wave back in the second wave. Let u ( x , t ) = v ( x , t ) | x , t > 0 , from Lemma 2, we can easily verify that u ( x , t ) is the global weak entropy solution of (6).

When u − < u m < u d < u + , u ( x , t ) includes only a rarefaction wave R | x , t > 0 , which will interact with the boundary x = 0 and be partially absorbed (if u + > 0 ) or absorbed (if u + ≤ 0 ) by the boundary.

When u − < u d < u m < u + , if f ′ ( u m ) = 0 , u ( x , t ) includes only the central rarefaction wave R ( u m , u + ) far away from the boundary x = 0 ; if f ′ ( u m ) > 0 , u ( x , t ) includes two central rarefaction waves R 2 | x , t > 0 and R ( u m , u + ) far away from the boundary; if f ′ ( u m ) < 0 , u ( x , t ) includes only the central rarefaction wave R ( u m , u + ) | x , t > 0 , which will interact with the boundary and be partially absorbed (if u + > 0 ) or completely absorbed (if u + ≤ 0 ) by the boundary.

When u − < u m = u d < u + , u ( x , t ) includes only the central rarefaction wave R ( u d , u + ) | x , t > 0 , which will interact with the boundary and be partially absorbed (if u + > 0 ) or completely absorbed (if u + ≤ 0 ) by the boundary.

Case (3) u + < u m < u − .

The discussion for this case is the same as that of the corresponding case in [

Case (4) u − , u + < u m .

When u − , u + ≥ u d , or u m ≤ u d , then (6) is degenerated into the problem with f ∈ C 2 . When u − ≥ u d , then the discussion on this problem is the same as that of the case f ∈ C 2 . Hence, we only investigate the case of u − < u d < u m , u + < u m .

In this case, an initial rarefaction wave R = R ( u − , u d ) ∪ R ( u d , u m ) centered at point ( 0,0 ) and an initial shock wave S ( u m , u + ) starting at point ( a ,0 ) appear in the weak entropy solution v ( x , t ) of (9). In what follows, similar to [

We now state the interaction of the elementary and the boundary x = 0 for the global weak entropy solution of (6). When the shock x = x ( t ) in v ( x , t ) is in the first quadrant of the x − t plane, the elementary wave in the solution u ( x , t ) of (6) does not interact with the boundary x = 0 ; when the shock wave x = x ( t ) of v ( x , t ) enters the second quadrant from the first quadrant including the t-axis and then keeps staying in the second quadrant, the shock wave x = x ( t ) in u ( x , t ) interacts with the initial rarefaction wave R | x , t > 0 on its right at t = t 1 , and then crosses R | x , t > 0 at its right at t > t 1 , finally it collides with the boundary x = 0 and is absorbed by the boundary (see

Case (5) u m < u − , u +

If u − , u + ≤ u d , or u m ≥ u d , then (6) becomes a problem with f ∈ C 2 (see [

In this case, an initial shock wave S ( u − , u m ) starting at point ( 0,0 ) and an initial rarefaction wave R ( u m , u d ) ∪ R ( u d , u + ) centered at point ( a ,0 ) in the

x − t plane appear in the weak entropy solution v ( x , t ) of (9). We denote this initial shock wave curve by x = x ( t ) . As in [

In sub-case (a) and (b), let u ( x , t ) = v ( x , t ) | x , t > 0 , then from Lemma 2, we can verify that u ( x , t ) is the global weak entropy solution of (6). The interaction of the elementary wave and the boundary x = 0 in the solution u ( x , t ) of (6) is stated as follows: For sub-case (a), when s ( u − , u m ) > 0 , the weak entropy

solution of (6) does not include the interaction of elementary waves and the boundary x = 0 ; when s ( u − , u m ) = 0 , the rarefaction wave R = R ( u m , u d ) ∪ R ( u d , u + ) collides with the boundary x = 0 at time t = t 1 , and the boundary x = 0 reflects a new shock wave tangent to the boundary itself at point ( 0, t 1 ) , which is just the restriction of x = x ( t ) at t > t 1 and will penetrate R after t = t 1 . For sub-case (b), the weak entropy solution of (6) only includes the rarefaction wave R = R ( u m , u d ) ∪ R ( u d , u + ) | x , t > 0 , which interacts with the boundary x = 0 at some time and is absorbed completely by the boundary.

For sub-case (c), there exists u * ∈ ( u m ,0 ) such that f ( u * ) = f ( u − ) . Furthermore, there is t * > t 1 such that

x ′ ( t ) = s ( u − , u ( x ( t ) + 0 , t ) ) { < 0 , 0 < t < t * = 0 , t = t * > 0 , t > t *

for u * ≠ u d and there are exist t * , t * ¯ > t 1 ( t * ¯ < t * ) such that

x ′ ( t ) = s ( u − , u ( x ( t ) + 0 , t ) ) { < 0 , 0 < t < t * ¯ = 0 , t * ¯ ≤ t ≤ t * > 0 , t > t *

for u * = u d , where x ′ ( t ) is the speed function of the shock wave x = x ( t ) in the weak entropy solution of (9). If we construct the solution of (6) by taking u ( x , t ) = v ( x , t ) | x , t > 0 as in sub-cases (a), (b), then this u ( x , t ) does not satisfy the boundary entropy condition (4) for t > t 2 , where t 2 = − a / f ′ + ( u * ) ( < t * ) is the time at which the characteristic line with speed f ′ + ( u ( x ( t * ) + 0, t * ) ) from the point ( x ( t * ) , t * ) backward to x-axis intersects the t-axis (see

v ( x , t 2 ) = { u − , x < 0 v ( x , t 2 − 0 ) , x > 0 (10)

as the new initial value of (9)_{1}, then the solution v ¯ ( x , t ) of the initial value problem (9)_{1}, (10) in ( 0, ∞ ) × ( t 2 , ∞ ) includes only a new shock wave x = x + ( t ) starting at point ( 0, t 2 ) , whose original speed is zero and the left state is u − . When t > t 2 , this shock crosses the rarefaction wave R | ( x − a ) / t > f ′ + ( u * ) on its right with a varying positive speed of propagation in the first quadrant. Let

V ( x , t ) = { v ( x , t ) , 0 < t < t 2 v ¯ ( x , t ) , t ≥ t 2 ,

then, from Lemma 2, this u ( x , t ) = v ¯ ( x , t ) | x > 0 , t > 0 is the global weak entropy solution of (6). Now we give the statement of the interaction of the elementary and the boundary x = 0 in the solution u ( x , t ) of (6) (see

In this sub-section, the flux f is supposed to satisfy the conditions (A_{1})-(A_{3}). As an example, we discuss the case that f ′ has only two discontinuous points, and we can similarly deal with the case that f ′ has n discontinuous points. It has no harm to assume that f ( 0 ) = f ′ ( 0 ) = 0 and u d 1 < u d 2 < 0 as in above sub-section.

We now construct the global weak entropy solution of (7) under the condition that u d 1 , u d 2 are located between u − and u + . If not so, see [

Case (1) u − < u d 1 , u d 2 < u + .

In this case, since the flux function f has two weak discontinuous points u d 1 , u d 2 , (8) includes only a rarefaction wave R = R ( u − , u d 1 ) ∪ R ( u d 1 , u d 2 ) ∪ R ( u d 2 , u + ) centered at point ( 0,0 ) of the x − t plane. We can express this rarefaction wave solution as:

v ( x , t ) = { u − , x < f ′ ( u − ) t ( f ′ ) − 1 ( x t ) , f ′ ( u − ) t ≤ x < f ′ − ( u d 1 ) t u d 1 , f ′ − ( u d 1 ) t ≤ x < f ′ + ( u d 1 ) t ( f ′ ) − 1 ( x t ) , f ′ + ( u d 1 ) t ≤ x < f ′ − ( u d 2 ) t u d 2 , f ′ − ( u d 2 ) t ≤ x < f ′ + ( u d 2 ) t ( f ′ ) − 1 ( x t ) , f ′ + ( u d 2 ) t ≤ x < f ′ ( u + ) t u + , x ≥ f ′ ( u + ) t .

Let u ( x , t ) = v ( x , t ) | x , t > 0 then u ( 0 + , t ) = m i n { u + ,0 } . It is easy to verify that u ( x , t ) is the global weak entropy solution of (7).

Case (2) u + < u d < u + .

In this case, only a shock wave S ( u − , u + ) starting at point ( 0,0 ) appears in the weak entropy solution of (8). This shock wave solution can be denoted as:

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

where s ( u − , u + ) is the speed of the shock wave S ( u − , u + ) . Let u ( x , t ) = v ( x , t ) | x , t > 0 , then

u ( 0 + , t ) = { u + , if x < s ( u − , u + ) ≤ 0 u − , if x > s ( u − , u + ) > 0

By Lemma 2, one can verify that u ( x , t ) is the global weak entropy solution of (7).

For the initial boundary value problem (6) with u m ≠ u + , we only investigate the case of u m < u d 1 < u d 2 < 0 < u − , u + , f ( u m ) > f ( u − ) , which is the most typical and complicated case.

In this case, an initial shock wave S ( u − , u m ) , emanating at point ( 0,0 ) , and an initial rarefaction wave R = R ( u m , u d 1 ) ∪ R ( u d 1 , u d 2 ) ∪ R ( u d 2 , u + ) , centered at point ( a ,0 ) , appear in the weak entropy solution v ( x , t ) of the corresponding initial value problem (9). We denote this shock by x = x ( t ) , whose original speed of propagation is negative. The shock x = x ( t ) will interact with the rarefaction wave R on its right at some finite time t = t 1 . This interaction will generate a new shock, still denoted by x = x ( t ) . The left state of the resulting shock wave is u − . If u − > u + , the shock x = x ( t ) is able to cross the whole of R at finite time (see

non-decreasing speed of propagation. The shock x = x ( t ) ( t > 0 ) is a piecewise smooth curve.

During the process of its penetrating the rarefaction wave R, the shock wave x = x ( t ) ( t > 0 ) first crosses the leftmost part R ( u m , u d 1 ) of R with a varying speed of propagation, and then crosses the constant state u = u d 1 with constant speed. When the shock wave x = x ( t ) encounters the characteristic line of the leftmost characteristic line of the rarefaction wave R ( u d 1 , u d 2 ) , it again begins to cross the rarefaction wave R ( u d 1 , u d 2 ) with a varying speed. And then it crosses the constant state u = u d 2 with constant speed. Finally, it crosses the rightmost part R ( u d 2 , u − ) of R with a varying speed of propagation.

In view of u m < u d 1 < u d 2 < 0 < u − , u + and f ( u m ) > f ( u − ) , there exists u * ∈ ( u m ,0 ) such that f ( u * ) = f ( u − ) . Furthermore, there is t * > t 1 such that

x ′ ( t ) = s ( u − , u ( x ( t ) + 0 , t ) ) { < 0 , 0 < t < t * = 0 , t = t * > 0 , t > t *

for u * ≠ u d 1 , u d 2 and there are exist t 0 * , t * > t 1 ( t 0 * < t * ) , such that

x ′ ( t ) = s ( u − , u ( x ( t ) + 0 , t ) ) { < 0 , 0 < t < t 0 * = 0 , t 0 * ≤ t ≤ t * > 0 , t > t *

for u * = u d 1 or u d 2 , where x ′ ( t ) is the speed function of the shock wave x = x ( t ) in the weak entropy solution v ( x , t ) of (9). The position of the shock x = x ( t ) ( t > 0 ) is stated as follows: the shock x = x ( t ) lies in the second quadrant of the x − t plane as t ∈ ( 0, t * * ) , and cross the t-axis at t = t * * , and then enter and keep staying in the first quadrant (see

In what follows, we construct the global weak entropy solution of (6). Let t 2 denote the intersection time of the t-axis and the characteristic line with speed f ′ + ( u ( x ( t * ) + 0, t * ) ) from the point ( x ( t * ) , t * ) backward to x-axis, namely, t 2 = − a / f ′ + ( u * ) ( < t * ) . First take u ( x , t ) = v ( x , t ) | x , t > 0 , then by lemma 2, this u ( x , t ) is the local weak entropy solution of (6) on ( 0, + ∞ ) × ( 0, t 2 ) . Next we will extend this u ( x , t ) to ( 0, + ∞ ) × ( 0, + ∞ ) . Consider the following Cauchy problem:

{ ( v ¯ ) t + f ( v ¯ ) x = 0 , − ∞ < x < + ∞ , t > t 2 v ¯ ( x , t 2 ) = { u − , x < 0 v ( x , t 2 − 0 ) , x > 0 (11)

The weak entropy solution v ¯ ( x , t ) of (11) in ( 0, + ∞ ) × ( t 2 , + ∞ ) includes only a shock wave x = x + ( t ) starting at ( 0, t 2 ) , whose original speed of propagation is zero. The shock x = x + ( t ) will cross this part of the rarefaction wave on its right: R | ( x − a ) / t > f ′ + ( u * ) with a varying positive speed of propagation during the penetration in the first quadrant as t > t 2 . Then by Lemma 2,

u ( x , t ) : = v ¯ ( x , t ) | x > 0 , t > t 2 is the weak entropy solution of (6) on ( 0, + ∞ ) × ( t 2 , + ∞ ) .

Thus we accomplish the construction of the solution to (6) (see

R = R ( u m , u d 1 ) ∪ R ( u d 1 , u d 2 ) ∪ R ( u d 2 , u + ) centered at point ( a ,0 ) collides with the boundary x = 0 , then the boundary reflects a new shock wave tangent to the boundary x = 0 at time t = t 2 , which will penetrate another part R | ( x − a ) / t > f ′ + ( u * ) of R with a varying positive speed of propagation in the first quadrant. This shock is just that one in v ¯ ( x , t ) .

This paper is mainly concerned about the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function. Under this condition, the flux function has a finite number of weak discontinuous points, by using the structure of weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec. We give a construction method to the global weak entropy solution for this initial-boundary value problem, and by investigating the interaction of elementary waves and the boundary. We clarify the geometric structure and the behavior of boundary for the weak entropy solution.

This work was supported by National Natural Science Foundation of China (No. 11271160).

Dai, Y.H. and Zhang, J. (2017) Construction of Global Weak Entropy Solution of Initial-Boundary Value Problem for Scalar Conservation Laws with Weak Discontinuous Flux. American Journal of Computational Mathematics, 7, 451-468. https://doi.org/10.4236/ajcm.2017.74033