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

TL;DR

This paper investigates solutions to 2x2 hyperbolic conservation laws with faster decay of total variation, establishing uniqueness and regularity results for solutions with decay rates faster than the classical $t^{-1}$.

Contribution

It introduces a class of solutions with accelerated decay of total variation and proves their uniqueness and H"older continuity of the solution semigroup.

Findings

Solutions with decay rate $t^{eta-1}$ for $eta>0$ are unique.

The solution semigroup is H"older continuous with exponent approaching 1.

A class of initial data leading to rapid decay of total variation is identified.

Abstract

For a genuinely nonlinear $2\times 2$ hyperbolic system of conservation laws, assuming that the initial data have small ${\bf L}^\infty$ 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: $\mathrm{Tot. Var. }\bigl\{u(t,\cdot)\bigr\}\leq C t^{\alpha-1}$. For these solutions, a uniqueness theorem is proved. Indeed, as the initial data range over a domain of functions with $\|\bar u\|_{{\bf L}^\infty} \leq\varepsilon_1$ small enough, solutions with fast decay yield a H\"older continuous semigroup. The H\"older exponent can be taken arbitrarily close to $1$ by further shrinking the value of $\varepsilon_1>0$. An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation.