Campanato estimates for the generalized Stokes System

Abstract

We study interior regularity of solutions of a generalized stationary Stokes problem in the plane. The main, elliptic part of the problem is given in the form div(A(Du)), where D is the symmetric part of the gradient. The model case is A(Du)=(kappa+|Du|)^{p-2}Du. We show optimal BMO and Campanato estimates for A(Du). Some corollaries for the generalized stationary Navier-Stokes system and for its evolutionary variant are also mentioned

On Regularity of the Time Derivative for Degenerate Parabolic Systems

“…For the definition and properties of L ϕ (Ω), W 1,ϕ (Ω) and for the analysis on divergence free Orlicz spaces we refer to [9,11]. The following estimate is important for us.…”

Nonlinear Potential Estimates for Generalized Stokes System

“…In the sequel, we will use the following self-improving property of reverse Hölder type inequality. And then, combining the aforementioned Lemma 2.2 with Lemma 3.4 in [10], one obtains a reserve Hölder type estimate for the symmetric gradient Du to (1.1) . Lemma 2.3.…”

“…We may assume without loss of generality that ρ ∈ (0, 1], since (4.1) obviously holds for 1 < ρ ≤ 3 2 . By virtue of Theorem 3.8 in [10], we have…”

Higher differentiability for solutions of stationary p‐Stokes systems

“…In this case, the problem becomes more involved because the system (1.1) contains only information on the symmetric part of the gradient, and, moreover, because of the presence of the convective term. For systems without convective term and without coefficients, higher differentiability of solutions was established by Naumann [32], and by Diening & Kaplický [14, 15] for related problems with a more general growth condition. With an additional convective term, but again without coefficients, Breit [7] proved the existence of a weak solution with higher differentiability properties.…”