### Uniqueness for discontinuous O.D.E. and conservation laws

Alberto Bressan and Wen Shen

Abstract: 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 $\L^\infty$.

Paper:
Available as PostScript
Title:
Uniqueness for discontinuous O.D.E. and conservation laws
Author(s):
Alberto Bressan, <bressan@sissa.it>
Wen Shen, <wens@ifi.uio.no>
Publishing information:
SISSA preprint 1997. Accepted for publication in "Nonlinear Analysis, Theory, Methods, and Applications".