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.