Intermediate Domains for Scalar Conservation Laws

Abstract

For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variation of a solution with initial data $\overline{u}\in {\mathbf{L}}^{\infty }(\mathbb{R})$ decays like $t^{-1}$. This paper introduces a class of intermediate domains ${\mathcal{P}}_{\alpha }$, $0<\alpha <1$, such that for $\overline{u}\in {\mathcal{P}}_{\alpha }$ a faster decay rate is achieved: $Tot.Var.\{u(t,\cdot )\}\sim t^{\alpha -1}$. A key ingredient of the analysis is a ``Fourier-type" decomposition of $\overline{u}$ into components which oscillate more and more rapidly. The results aim at extending the theory of fractional domains for analytic semigroups to an entirely nonlinear setting.