Archive for Rational Mechanics and Analysis

, Volume 231, Issue 1, pp 591–636 | Cite as

Partial Regularity for Type Two Doubly Nonlinear Parabolic Systems

  • Ryan HyndEmail author


We consider weak solutions v : \({U \times (0, T ) \rightarrow \mathbb{R}^{m}}\) of the nonlinear parabolic system
$${D\psi({v}_{t} ) = {\rm div} DF({D}_{v}),}$$
where \({\psi}\) and F are convex functions. This is a prototype for more general doubly nonlinear evolutions which arise in the study of structural properties of materials. Under the assumption that the second derivatives of F are Hölder continuous, we show that D2v and vt are locally Hölder continuous except for possibly on a lower dimensional subset of \({U \times (0, T )}\). Our approach relies on two integral identities, decay of the local space-time energy of solutions, and fractional time derivative estimates for D2v and vt.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA

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