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Spatial $C^1$, $C^2$, and Schauder estimates for nonstationary Stokes equations with Dini mean oscillation coefficients

Hongjie Dong, Hyunwoo Kwon

TL;DR

This work analyzes the nonstationary Stokes system with spatially variable viscosity that satisfies a $L_2$-Dini mean oscillation condition. Employing a Campanato-type framework centered on decay of vorticity mean oscillations and polynomial velocity-approximation on shrinking cylinders, the authors establish spatial $C^1_x$ regularity for divergence-form solutions and $C^2_x$ regularity for nondivergence-form solutions under $L_2$-DMO$_x$ hypotheses. When the coefficients and data possess Hölder continuity in space, the results yield local spatial Schauder estimates and further regularity for the velocity gradient and vorticity. The approach directly handles Dini-type oscillations without heavy inductive schemes, yielding robust spatial regularity results applicable to variable-viscosity and density-dependent fluid models as well as Stokes on manifolds.

Abstract

We establish the spatial differentiability of weak solutions to nonstationary Stokes equations in divergence form with variable viscosity coefficients having $L_2$-Dini mean oscillations. As a corollary, we derive local spatial Schauder estimates for such equations if the viscosity coefficient belongs to $C^α_x$. Similar results also hold for strong solutions to nonstationary Stokes equations in nondivergence form.

Spatial $C^1$, $C^2$, and Schauder estimates for nonstationary Stokes equations with Dini mean oscillation coefficients

TL;DR

This work analyzes the nonstationary Stokes system with spatially variable viscosity that satisfies a -Dini mean oscillation condition. Employing a Campanato-type framework centered on decay of vorticity mean oscillations and polynomial velocity-approximation on shrinking cylinders, the authors establish spatial regularity for divergence-form solutions and regularity for nondivergence-form solutions under -DMO hypotheses. When the coefficients and data possess Hölder continuity in space, the results yield local spatial Schauder estimates and further regularity for the velocity gradient and vorticity. The approach directly handles Dini-type oscillations without heavy inductive schemes, yielding robust spatial regularity results applicable to variable-viscosity and density-dependent fluid models as well as Stokes on manifolds.

Abstract

We establish the spatial differentiability of weak solutions to nonstationary Stokes equations in divergence form with variable viscosity coefficients having -Dini mean oscillations. As a corollary, we derive local spatial Schauder estimates for such equations if the viscosity coefficient belongs to . Similar results also hold for strong solutions to nonstationary Stokes equations in nondivergence form.
Paper Structure (17 sections, 10 theorems, 235 equations)

This paper contains 17 sections, 10 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Notation and main results
  3. Notation
  4. Main results
  5. Preliminaries
  6. Basic estimates
  7. Caccioppoli estimates for Stokes equations
  8. Estimating the decay rate of the mean oscillation of vorticity
  9. Estimating the convergence rate of approximation to velocity
  10. Reduction
  11. Proof of Proposition \ref{['prop:velocity-approximation']} (i)
  12. Proof of Proposition \ref{['prop:velocity-approximation']} (ii)
  13. Proof of Theorems \ref{['thm:A']} and \ref{['thm:B']}
  14. Solvability of Stokes equations with simple coefficients
  15. Approximation argument
  16. ...and 2 more sections

Key Result

Theorem 2.4

Let $\mathbf{F}\in L_{2}(Q_2)$ and $g\in L_{2}(Q_2)$. Suppose that $A$ and $\mathbf{F}$ satisfy the $L_{2}$-$\mathrm{DMO}_x$ in small cylinders and $g$ satisfies $L_{2}$-$\mathrm{DMO}_x$ in small balls. If $u \in L_{\infty}L_{2}(Q_1)\cap L_{2}{H}^{1}(Q_1)$ is a weak solution to eq:div-form in $Q_1$, and for $t\in (-1/64,0)$ and $x,y \in B_{1/8}$ with $|x-y|<1/8$, where the implicit constant depen

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1
  • Remark 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 21 more