Construction of Global Weak Entropy Solution of Initial-Boundary Value Problem for Scalar Conservation Laws with Weak Discontinuous Flux

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.


Introduction
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 [1] [2] [3] [4] [5].The existence and uniqueness of global weak entropy solution in the BV-setting were first established by Bardos-Leroux-Nedelec [6] for the initial-boundary value problem of scalar conservation laws with several space variables by vanishing viscosity method and by Kruzkovs method [7], respectively.The core of studying the initial-boundary value problem of conservation laws is the boundary entropy condition which requires only that the boundary data and the boundary value of solution satisfy an inequality.This makes it very interesting to study the initial-boundary value problems of hyperbolic conservation laws.The interested reader is referred to [8]- [14] about other results of existence and uniqueness for the initial-boundary value problem of scalar conservation laws.For the initial-boundary value problem of systems of conservation laws, some progresses have been made in the past: Dubotis-Le Floch [10] discussed the boundary entropy condition, the authors in [15] [16] [17] [18] studied the boundary layers, Chen-Frid [19] proved the existence of global weak entropy solution for the system of isentropic gas dynamics equations by using the method of Compensated compactness and vanishing viscosity.
For the geometric structure and regularity and large time behavior of solution of the initial value problem for scalar conservation laws, see [20] [21] [22] [23] [24] [25] etc. Due to the occurrence of boundary, the geometric structure of the solution of ( 1) is much more difficult than that of corresponding initial value problem.In recent years, for the case of the flux function belonging to C 2smooth function class, some results have been obtained in this regard.The authors in [1] [3] [26] constructed the global 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 [27]  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 [27], 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 (1) 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 state the geometric structure and the behavior of boundary for the weak entropy solution.

Definition of Weak Entropy Solution and Related Lemmas
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 [31].Generally speaking, there is no uniqueness for the weak solution of scalar conservation laws.Since the equation of scalar conservation laws arises in the physical sciences, we must have some mechanism to pick out the physically relevant solution.Thus, we are led to impose an a-priori condition on solutions which distinguishes the correct one from the others.
where ( ) 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 [3] [6] [10] [12]).Lemma 1.If ( ) , u x t is a weak entropy solution of (1), then, where 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 : , where f − ′ and f + ′ represent the left and right derivatives of f respectively; 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 ( ) is a weak discontinuity of ( ) and when and the Oleinik entropy condition hold, where , and u is any number between u − and u + ; (3) The boundary entropy condition (3) is valid; American Journal of Computational Mathematics Lemma 2 is easily to be proved by Definition 1 and Lemma 1 (see [12] [32]).
Notations.For the convenience of our construction work, we introduce some notations.Let is also denoted by ( ) − where ( ) satisfies the Rankine-Hugoniot condition (4) and the Oleinik entropy condition (5).
It is well known that the solution of the shock wave ( ) ( ) where t b > .

Construction of Global Weak Entropy Solutions
In this section, with the aid of the analysis method in [27], the authors in [27] used the truncation method to construct the global weak entropy solution ( ) , u x t of initial-boundary value problem for scalar conservation laws with C 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 [27], if ( ) , v x t does not include any shock wave or includes a shock wave whose position is not the following case: the shock wave lies in the second quadrant and the sign of the shock speed is changed from negative to positive before a finite time, then ( ) ( ) 0, 0 , , otherwise we need to find some time Consider the following initial-boundary problem: where , m u u ± are constant, for 0 x > and , 0 x a a ≠ > 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.

The Case That f Has Only One Weak Discontinuous Point
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 d u , and there exist ( ) We first discuss the initial boundary value problem (6) for the case of m u u u where u u − + ≠ .And then investigate (6) with m u u + ≠ .For definiteness, it has no harm to assume that ( ) ( ) and 0 d u < in this sub-section.The other cases can be dealt with similarly.

Riemann Initial-Boundary Problem
When ( ) ( ) [27]).We now construct the weak entropy solution of (7) only for the case of ( ) ( ) We divide our problem into two cases: (1) Case (1) Consider the following Riemann problem corresponding to (7): In this case, since the flux function has a weak discontinuity point , hence, it holds the boundary entropy condition: 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).
In this case, (8) includes only a shock wave ( ) This shock wave solution can be expressed as follows: , , where ( ) is the speed of the shock ( ) , as , 0 0 , , as , 0 From Lemma 2, we can easily verify that ( ) , u x t is the global weak entropy solution of (7).

The General Problem with m u u + ≠
Consider the following initial value problem corresponding to (6): , .
According to the solution structure of (9), we construct the global weak entropy solution of ( 6) with m u u + ≠ by dividing our problem into five cases: (1) , In fact, when 6) becomes a problem with 2 f C ∈ , which was discussed in [27].We now investigate the case of ( ) ( )

( ) ( )
, , , centered American Journal of Computational Mathematics at ( ) ,0 a of the x t − plane, appears in the weak entropy solution of ( 9).This rarefaction wave solution of ( 9) can be written as: x a f u t , 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 , 0 x t R > , which will interact with the boundary 0 x = and be completely absorbed (if 0 u + ≤ ) (see Figure 1 a s u u t v x t u x a s u u t , 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  f C ∈ , see [27].
We now consider the following three cases:   ,0 a , 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 ( ) ,0 a , 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 ( ) ( ) , 0 , , from Lemma 2, we can easily verify that ( ) , u x t is the global weak entropy solution of (6).R u u + > , which will interact with the boundary and be partially absorbed The discussion for this case is the same as that of the corresponding case in [27].
Case (4) , − ≥ , then the discussion on this problem is the same as that of the case ,0 a appear in the weak entropy solution ( ) , v x t of (9).In what follows, similar to American Journal of Computational Mathematics [33], we give the statement of interaction of the initial rarefaction wave R and the initial shock wave ( ) ( ) is a piecewise smooth curve.First, the shock wave ( ) ( ) cross the right part ( ) x x t = enters the second quadrant from the first quadrant including the t-axis at some finite time and then keeps staying in the second quadrant.Let ( ) ( ) , 0 , , , then by Lemma 2, ( ) , u x t is the global weak entropy solution of (6).
We now state the interaction of the elementary and the boundary 0 x = for the global weak entropy solution of (6).When the shock ( ) 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 0 x = ; when the shock wave ( ) 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 ( ) , u x t interacts with the initial rarefaction wave Case (5) , (see [27]).If  x t − plane appear in the weak entropy solution ( ) , v x t of (9).We denote this initial shock wave curve by ( ) As in [33], the shock ( ) , m S u u − interacts with the rarefaction wave R on its right at some time    For sub-case (c), there exists Furthermore, there is * 1 ′ is the speed function of the shock wave ( )

The Case That f Has Finitely Many Weak Discontinuous Points
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 ( ) ( ) < as in above sub-section.

Riemann Initial-Boundary Problem
We now construct the global weak entropy solution of (7) under the condition that Consider the following initial-boundary value problem for scalar conservation laws: two bounded and local bounded variation functions on [ ) 0, +∞ , and the flux f is assumed to be locally Lipschitz continuous.
[28] [29] studied the initial-boundary problem with piecewise smooth initial dada and constant boundary data for scalar convex conservation laws, they gave a construction method to the global weak entropy solution of this initial-boundary value problem and clarified the structure and boundary behavior of the weak entropy solution.Moreover, Liu-Pan also constructed the global weak entropy solution of the initial-boundary value problem for scalar non-convex conservation laws under the condition that the initial dada is a function with two pieces of constant and the boundary data is a constant function in [30] and by investigating the interaction of elementary waves and the boundary, they discovered some different behaviors of elementary waves nearby the boundary from the corresponding initial-boundary value problem for scalar convex conservation American Journal of Computational Mathematics laws.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.

, a b and the solution of the central rarefaction
the x t − plane are respectively expressed as: local solution to this time, and then take the time 0 t t = as the new initial time to extend this local solution to t → ∞ .We will construct the global weak entropy solution of (1) 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 by employing Lemma 2 and the structure of weak entropy solution to the corresponding initial value problem.Moreover, we will also describe the interaction of elementary waves with the boundary and clarify the behaviors of the global weak entropy solution near the boundary.
Riemann problem(8) includes only a rarefaction wave centered at point ( ) 0,0 of the x t − plane.This rarefaction wave solution can be written as: ) is degenerated into a Riemann problem.If m u u u − + = < , only a rarefaction wave Figure 1(c))by the boundary.If (see Figure2(a)) or be far away from the boundary (if

Figure 1 .
Figure 1.The interaction of the rarefaction wave , 0 x t R > with the boundary 0 x = .

When
with the boundary 0 x = and be partially absorbed (if 0 u + > ) or absorbed (if 0 u + ≤ ) by the boundary.+ > , which will interact with the boundary and be partially absorbed (if 0 u + > ) or completely absorbed (if 0 u + ≤ ) by the boundary.
and an initial shock wave ( ) , m S u u + starting at point ( ) with a varying speed of propagation during the penetration.If it is able to cross the whole of ( ) , d m R u u completely at some finite time, then it crosses the domain of constant state d u u = with a constant speed of propagation.When the shock Figure 3(b)).
the discussion of the problem is the same as that of the case this case, an initial shock wave ( ) , m S u u − starting at point ( ) 0,0 and an initial rarefaction wave ( R u u +  centered at point ( ) ,0 a in the American Journal of Computational Mathematics

Figure 3 .
Figure 3.The interaction of the shock wave After

Figure 4 .
Figure 4.The interaction of the shock wave

Figure 4 (
Figure 4(d)).For the problem (6), an initial rarefaction wave ( ) ( ) , , m d d R R u u R u u + =  emanates from the point ( ) ,0 a on the x-axis.One part of R collides with the boundary 0 x = , and then the boundary 0 x = reflects a new shock wave located between u − and u + .If not so, see[27] or sub-, since the flux function f has two weak discontinuous points centered at point ( ) 0,0 of the x t − plane.We can express this rarefaction wave solution as: American Journal of Computational Mathematics

Figure 5 .
Figure 5.The interaction of the shock wave The rarefaction wave R interacts with the In what follows, we construct the global weak entropy solution of (6).Let 2 t denote the intersection time of the t-axis and the characteristic line with speed > =, then by lemma 2, this