Parabolic Systems with Nowhere Smooth Solutions


Stefan Müller
Max Planck Institute for Mathematics in the Sciences
Ineselstr 22-26
04103 Leipzig, Germany
sm@mis.mpg.de

Marc Oliver Rieger
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rieger@sns.it

Vladimir Sverak
Department of Mathematics
University of Minnesota
Minneapolis, MN 55455
sverak@math.umn.edu



Abstract

We construct smooth $2 \times 2$ parabolic systems with smooth initial data and $C^{\alpha}$ right hand side which admit solutions that are nowhere $C^1$. The elliptic part is in variational form and the corresponding energh $\phi$ is strongly quasiconvex and n particular satisfies a uniform Legendre-Hdamard (or sttrong ellipticity) condition.

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