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Nonstationary Stokes equations on a domain with curved boundary under slip boundary conditions

Hongjie Dong, Hyunwoo Kwon

TL;DR

This work analyzes local regularity for the time-dependent Stokes system in nondivergence form with variable viscosity $a^{ij}(t,x)$ under generalized Navier slip boundary conditions with slip tensor $\mathcal{A}$ on curved boundaries. By employing a boundary-flattening strategy that preserves impermeability and studying perturbed Stokes equations close to the identity, the authors prove local Hessian estimates near curved boundary portions and, when $\mathcal{A}$ is the shape operator, Hessian estimates whose right-hand side is pressure-free. They also establish local boundary mixed-norm estimates under slip conditions with $\mathcal{A}\in C^{1,2}_{par}$, extending prior flat-boundary results to curved geometries. The results hold for variable coefficients with small mean oscillation (VMO) and are novel even in the constant-viscosity case, leveraging weighted spaces, maximal regularity, boundary flattening, and Agmon-type absorption techniques to control time derivatives and pressure terms.

Abstract

We consider nonstationary Stokes equations in nondivergence form with variable viscosity coefficients and generalized Navier slip boundary conditions with slip tensor $\mathcal{A}$ in a domain $Ω$ in $\mathbb{R}^d$. First, under the assumption that slip matrix $\mathcal{A}$ is sufficiently smooth, we establish a priori local regularity estimates for solutions near a curved portion of the domain boundary. Second, when $\mathcal{A}$ is the shape operator, we derive local boundary estimates for the Hessians of the solutions, where the right-hand side does not involve the pressure. Notably, our results are new even if the viscosity coefficients are constant.

Nonstationary Stokes equations on a domain with curved boundary under slip boundary conditions

TL;DR

This work analyzes local regularity for the time-dependent Stokes system in nondivergence form with variable viscosity under generalized Navier slip boundary conditions with slip tensor on curved boundaries. By employing a boundary-flattening strategy that preserves impermeability and studying perturbed Stokes equations close to the identity, the authors prove local Hessian estimates near curved boundary portions and, when is the shape operator, Hessian estimates whose right-hand side is pressure-free. They also establish local boundary mixed-norm estimates under slip conditions with , extending prior flat-boundary results to curved geometries. The results hold for variable coefficients with small mean oscillation (VMO) and are novel even in the constant-viscosity case, leveraging weighted spaces, maximal regularity, boundary flattening, and Agmon-type absorption techniques to control time derivatives and pressure terms.

Abstract

We consider nonstationary Stokes equations in nondivergence form with variable viscosity coefficients and generalized Navier slip boundary conditions with slip tensor in a domain in . First, under the assumption that slip matrix is sufficiently smooth, we establish a priori local regularity estimates for solutions near a curved portion of the domain boundary. Second, when is the shape operator, we derive local boundary estimates for the Hessians of the solutions, where the right-hand side does not involve the pressure. Notably, our results are new even if the viscosity coefficients are constant.
Paper Structure (17 sections, 19 theorems, 226 equations, 1 figure)

This paper contains 17 sections, 19 theorems, 226 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and main results
  3. Notation
  4. Main results
  5. Preliminaries
  6. Function spaces
  7. Trace operators and solvability of the inhomogeneous heat equation
  8. Boundary flattening map preserving impermeability condition
  9. Solvability results for Stokes equations with variable coefficients
  10. Hessian estimates for Stokes equations near the flat boundary
  11. Perturbed Stokes equations
  12. Weighted solvability results for perturbed Stokes systems
  13. A priori regularity estimates for perturbed Stokes systems
  14. Proof of Theorem \ref{['thm:main-theorem-perturb']}
  15. Proof of Theorem \ref{['thm:A']}
  16. ...and 2 more sections

Key Result

Theorem 2.4

Let $s,q_0,q\in (1,\infty)$, $r\in (0,R)$, $x_0\in\partial \Omega,$ and $z_0=(0,x_0)$. Then there exists a constant $\gamma>0$ depending on $d$, $s$, $q$, $q_0$, $\nu$ such that under Assumption assump:VMO$(\gamma)$, if $(u,p) \in {W}^{1,2}_{q_0}(\widehat{Q}_R(z_0))^d \times {W}^{0,1}_{1}(\widehat{Q for some constant $N=N(d,s,q,\nu,\Omega,r,R,R_0)>0$.

Figures (1)

  • Figure 1:

Theorems & Definitions (34)

  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 24 more