The Nonlinear Stokes Problem with General Potentials Having Superquadratic Growth

Abstract

We discuss partial regularity results concerning local minimizers u :defined on appropriate classes of solenoidal fields, where h is a N -function of rather general type. As a byproduct we obtain a theorem on partial C 1 -regularity for weak solutions of certain non-uniformly elliptic Stokes-type systems modelling generalized Newtonian fluids. Mathematics Subject Classification (2000). 76M30, 76D07, 49N60, 35J50.

Korn-Type Inequalities in Orlicz-Sobolev Spaces Involving the Trace-Free Part of the Symmetric Gradient and Applications to Regularity Theory

“…Corresponding results are shown by the first author and Fuchs [8] in the context of the nonlinear Stokes problem, where the density of the functional depends on the symmetric gradient. Let us give a detailed formulation of our results: Assume that h : [ 0, ∞) → [ 0, ∞) is a function of class C 2 that satisfies the conditions (H1) h is strictly increasing and convex (H2) h (0) > 0 and lim…”

“…The corresponding versions of the Korn inequalities, which are already proved in [7], [8] and [13], are the essential tools in the study of variational problems for generalized Newtonian fluids. vi) Part b) is also true with E D v replaced by Ev if χ is a suitable rigid motion.…”

“…Although, they just consider a special case, they provide the necessary tools to deal with much more general situation. Moreover, the more general result (1.3) is proved in [13], [21], [8], and [7] with the result: (1.3) holds if h satisfies ∆ 2 -and ∇ 2 -conditions (a precise definition is given below). In this note we prove Korn inequalities of the above type, where Ev is replaced by its tracefree part E .3) (and variants of this).…”

“…Let us give some remarks on the above conditions; the details can be found in [3] and [8]. Examples of functions h satisfying (H1)-(H4) are given in [22].…”

“…Remark 1.4 i) In the proof of part b) of the above theorem we use a blow-up argument, which generalizes the approach used in [8], where an E-version of the above theorem is proved, but only in case n = 3. We are able to extend this result to arbitrary dimensions owing to our Korn-type inequalities.…”