EXISTENCE AND UNIQUENESS OF THE ENTROPY SOLUTION TO A NONLINEAR HYPERBOLIC EQUATION

ＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＯＦＴＨＥＥＮＴＲＯＰＹＳＯＬＵＴＩＯＮＴＯＡＮＯＮＬＩＮＥＡＲＨＹＰＥＲＢＯＬＩＣＥＱＵＡＴＩＯＮ￥Ｒ．ＥＹＭＡＲＤ；Ｔ．ＧＡＬＬＯＵＥＴ；Ｒ．ＨＥＲＢＩＮ（ＬａｂｏｒａｔｏｉｒｅＣｅｎｔｒａｌｄｅｓＰｏ...

EXISTENCE AND UNIQUENESS OF THE ENTROPY SOLUTION TO A NONLINEAR HYPERBOLIC EQUATION

This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: $u_t+\roman{div}({\bold v}f(u))=0$ in $\BR^N\times 0,T],$ with initial data $u(\cdot, 0)=u_0(\cdot)$ in $\BR^N$, where $u_0\in L^{\infty}(\BR^N)$ is a given function, $\bold v$ is a divergence-free bounded function of class $C^1$ from $\BR^N\times 0,T]$ to $\BR^N,$ and $f$ is a function of class $C^1$ from $\BR$ to $\BR.$ It also gives a result of convergence of a numerical scheme for the discretization of this equation. The authors first show the existence of a ``process' solution (which generalizes the concept of entropy weak solutions, and can be obtained by passing to the limit of solutions of the numerical scheme). The uniqueness of this entropy process solution is then proven; it is also proven that the entropy process solution is in fact an entropy weak solution. Hence the existence and uniqueness of the entropy weak solution are proven.