Abstract
We consider the regularity for weak solutions of second-order nonlinear parabolic systems under a natural growth condition when , and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, we get the optimal regularity by the method of A-caloric approximation introduced by Duzaar and Mingione.
Highlights
1 Introduction Electrorheological fluids are special viscous liquids, that are characterized by their ability to undergo significant changes in their mechanical properties when an electric field is applied
We find it a bit difficult to handle, in many points of the paper, we shall use: (H ) For β ∈ (, ) and K : [, ∞) → [L, ∞) monotone nondecreasing such that
2 The A-caloric approximation technique and preliminaries we introduce the A-caloric approximation lemma [ ] and some preliminaries
Summary
Electrorheological fluids are special viscous liquids, that are characterized by their ability to undergo significant changes in their mechanical properties when an electric field is applied. Since the material function p, which essentially determines S, depends on the magnitude of the electric field |E| , we have to deal with an elliptic or parabolic system of partial differential equations with the so-called non-standard growth conditions, i.e., the elliptic operator S satisfies. ), one needs to impose some regularity conditions and constructer conditions to Aαi and Bi. For a vector field Aαi : QT × RN × RnN , we shall denote the We assume that coefficients by Aαi (z, u, p) = Aαi (x, t, the functions (z, u, p) → Aαi (z, u, p); u, p) (z, u, if z = p) →. We shall specify the regularity assumptions on Aαi (z, u, p) with respect to the ‘coefficient’ (z, u) and assume that the function (z, u) →
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