MiS Preprint Repository

In this paper we study the following scalar conservation law: $$\frac{\partial u}{\partial t} + \frac{\partial}{\partial x} (F(x,u))=0,\ \ \ x \in \mathbb{R}, \ \ t>0,$$ $$u(x,0) = u_0 (x),\ \ \ x \in \mathbb{R} \ \ (0,1) $$ where the flux function F (x, u) is a discontinuous function of x given by $F(x,u)=H(x)f(u)+(1-H(x))g(u),H$ is the Heaviside function, f and g are smooth functions on $\mathbb{R}$.
It is easy to see that weak solution of (1.1) is the weak solution of the following problem:
$u_t +f(u)_x =0\ \ for \ \ x>0, \ \ t>0, $
$u_t g(u)_x = 0\ \ for\ \ x0,\ \ t>0, $
$u(x,0)=u_0(x) $
and at $x =0, u$ satisfies the Rankin-Hugoniot condition i.e., for allmost all t > 0,
$f(u(0+,t))=g(u(o-t)) \ \ \ (0,3) $
where $u(0+,t)=lim_{z \to 0} + u (x,t) $ and $u(0-,t=lim_{x \to 0.} - u(x,t)$.
Kruzkov proved that if F is continuous in u and $\frac{\partial F}{\partial x}$ is bounded, then (1.1) admits a weak solution. If F is discontinuous in x, Kruzkov\'s method does not gaurantee a solution. The discontinuity of the flux function at x=0 causes a discontinuity of a solution which in general not uniquely determined by the initial data. When there is no discontinuity of a flux function at x=0,that is f=g and strictly convex, this problem was studied by Lax and Olenik. Using the Hamilton-Jacobi equation they obtain an explicit formula for the solution and derive an entropy condition so that the solution they obtained is unique. For a general f,Kruzkov proves the uniqueness of an entroy solution. Kruzkov and Keyfitz showed that the entropy solution can be represented by $L^1$- contraction semigroup.
When $f \ne g$,this problem is studied by Gimse and Risebro and Diehl .In the case of two phase flow problem,Gimse and Risebro obtain a unique solution of the Riemann problem for (1.2) and (1.3) by minimizing |u(0+ ,t) - u(0-,t)|.Using this they construct a sequence of approximate solutions converging to a weak solution for bounded initial data. Later it was pointed out by Diehl that " to minimize |u(0+,t)-u(0-,t)| " may not be a suitable choice. Instead of this one has to look for the solution which has smaller variation(he puts a condtion called $\Gamma$ condition).In this class Diehl gives an explicit formula for a solution in the case of a Riemann problem and proves the uniqueness.
Now the question is " Whether the solution obtained from Diehl can be represented by a contraction semigroup in $L^1$ norm in the sense of Kruzkov and Keyitz? ".
By looking at Diehl\'s work it is not clear that solution can be represented by a contraction semigroup.The main difficulty is to obtain a proper entropy condition at x = 0.
In this paper under the following condition:
(H) f, g are strictly convex and super linear growth, we settle this question affirmitively for arbitrary bounded initial data. Also we show that ,in general,our solution differ from the solution obtained by Diehl. Here we give an explicit formula for the solution of (1.2) satisfying (1.3). This agrees with the Lax-Olenik formula when f = g. Also we give a correct entropy condition at x = 0 so that the problem (1.2) and (1.3) admits a unique solution determined by the initial condition like in Kruzkov.