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| EXISTENCE AND UNIQUENESS OF THE ENTROPY SOLUTION TO A NONLINEAR HYPERBOLIC EQUATION |
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Citation： |
R. Eymard,T. Gallouet,R. Herbin.EXISTENCE AND UNIQUENESS OF THE ENTROPY SOLUTION TO A NONLINEAR HYPERBOLIC EQUATION[J].Chinese Annals of Mathematics B,1995,16(1):1~14 |
| Page view： 1310
Net amount： 742 |
Authors： |
R. Eymard; T. Gallouet;R. Herbin |
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| Abstract： |
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. |
Keywords： |
Nonlinear hyperbolic equation, Process solution,Existence and uniqueness,Convergence of finite volume scheme. |
Classification： |
35A05,35A40,35L60,65M12. |
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