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Partial Regularity for Type Two Doubly Nonlinear Parabolic Systems

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Abstract

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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Authors and Affiliations

  1. Department of Mathematics, MIT, Cambridge, USA

    Ryan Hynd

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  1. Ryan Hynd
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Correspondence to Ryan Hynd.

Additional information

Communicated by F. Lin

Partially supported by NSF Grant DMS-1554130 and an MLK visiting professorship.

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Hynd, R. Partial Regularity for Type Two Doubly Nonlinear Parabolic Systems. Arch Rational Mech Anal 231, 591–636 (2019). https://doi.org/10.1007/s00205-018-1286-5

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