Uniqueness Domains for ${\mathbf{L}}^{\infty }$ Solutions of $2\times 2$ Hyperbolic Conservation Laws

Abstract

For a genuinely nonlinear $2\times 2$ hyperbolic system of conservation laws, assuming that the initial data have small ${\mathbf{L}}^{\infty }$ norm but possibly unbounded total variation, the existence of global solutions was proved in a classical paper by Glimm and Lax (1970). In general, the total variation of these solutions decays like $t^{-1}$. Motivated by the theory of fractional domains for linear analytic semigroups, we consider here solutions with faster decay rate: $Tot.Var.\{u(t,\cdot )\}\le C{t}^{\alpha -1}$. For these solutions, a uniqueness theorem is proved. Indeed, as the initial data range over a domain of functions with $\Vert \bar{u}{\Vert }_{{\mathbf{L}}^{\infty }}\le {\epsilon }_1$ small enough, solutions with fast decay yield a Hölder continuous semigroup. The Hölder exponent can be taken arbitrarily close to $1$ by further shrinking the value of ${\epsilon }_1>0$. An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation.

arXiv comment24 pages, 12 figures