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_α, 0<α<1, such that for u‾∈P_α a faster decay rate is achieved: Tot.Var.{u(t,⋅)}∼t^α−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.