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We consider weak solutions v : $${U \times (0, T ) \rightarrow \mathbb{R}^{m}}$$U×(0,T)→Rm of the nonlinear parabolic system $${D\psi({v}_{t} ) = {\rm div} DF({D}_{v}),}$$Dψ(vt)=divDF(Dv), where $${\psi}$$ψ and F are convex… Click to show full abstract
We consider weak solutions v : $${U \times (0, T ) \rightarrow \mathbb{R}^{m}}$$U×(0,T)→Rm of the nonlinear parabolic system $${D\psi({v}_{t} ) = {\rm div} DF({D}_{v}),}$$Dψ(vt)=divDF(Dv), where $${\psi}$$ψ 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 $${U \times (0, T )}$$U×(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.
Journal Title: Archive for Rational Mechanics and Analysis
Year Published: 2017
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