Natural second-order regularity for parabolic systems with operators having $$(p,\delta )$$-structure and depending only on the symmetric gradient

Abstract

In this paper we consider parabolic problems with stress tensor depending only on the symmetric gradient. By developing a new approximation method (which allows to use energy-type methods typical for linear problems) we provide an approach to obtain global regularity results valid for general potential operators with $$(p,\delta )$$ ( p , δ ) -structure, for all $$p>1$$ p > 1 and for all $$\delta >0$$ δ > 0 . In this way we prove “natural” second order spatial regularity—up to the boundary—in the case of homogeneous Dirichlet boundary conditions. The regularity results, are presented with full details for the parabolic setting in the case $$p>2$$ p > 2 . However, the same method also yields regularity in the elliptic case and for $$1<p\le 2$$ 1 < p ≤ 2 , thus proving in a different way results already known.

Convergence analysis of a Local Discontinuous Galerkin approximation for nonlinear systems with Orlicz-structure

“…The assertion (ii) is proved in [7] (cf. [8,Lemma 2.10]). Assertion (iii) follows from (ii), since for N-functions satisfying the ∆ 2 -condition, uniformly with respect to t > 0, there holds…”

A Local Discontinuous Galerkin approximation for the $p$-Navier-Stokes system, Part I: Convergence analysis

“…It is well-known that ϕ is balanced (cf. [53,11]), since min {1, p − 1} (δ + t) p−2 ≤ ϕ (t) ≤ max {1, p − 1}(δ + t) p−2 for all t, δ ≥ 0. Moreover, ϕ satisfies, independent of δ ≥ 0, the ∆ 2 -condition with ∆ 2 (ϕ) ≤ c 2 max {2,p} .…”