Uniqueness for discontinuous O.D.E. and conservation laws
Alberto Bressan and Wen Shen
Consider a scalar O.D.E. of the form $\dot x=f(t,x),$ where $f$ is possibly
discontinuous w.r.t. both variables $t,x$. Under suitable assumptions, we
prove that the corresponding Cauchy problem admits a unique solution, which
depends H\"older continuously on the initial data.
Our result applies in particular to the case where $f$ can be written in the
form $f(t,x)\doteq g\big( u(t,x)\big)$, for some function $g$ and some
solution $u$ of a scalar conservation law, say $u_t+F(u)_x=0$. In turn, this
yields the uniqueness and continuous dependence of solutions to a class of
$2\times 2$ strictly hyperbolic systems, with initial data in $\L^\infty$.
- Available as PostScript
- Uniqueness for discontinuous O.D.E. and conservation laws
- Alberto Bressan,
- Wen Shen,
- Publishing information:
- SISSA preprint 1997. Accepted for publication in "Nonlinear Analysis,
Theory, Methods, and Applications".
- Revised version resubmitted September 10 1997.
- Submitted by:
February 28 1997.
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Last modified: Thu Sep 11 09:14:20 1997