Intermediate Domains for Scalar Conservation Laws

For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variation of a solution with initial data u L^ (R) decays like t^-1. This paper introduces a class of intermediate domains P_, 0<<1, such that for u P_ a faster decay rate is achieved: Tot. Var. \ u (t, ) \ t^-1. A key ingredient of the analysis is a ``Fourier-type" decomposition of 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.