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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.
Ancona et al. (Tue,) studied this question.
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