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The local regularity theory for the Stokes and Navier--Stokes equations near the curved boundary

Hui Chen, Su Liang, Tai-Peng Tsai

TL;DR

This work develops local regularity theory for the nonstationary Stokes and Navier–Stokes equations near curved boundaries under no-slip and Navier boundary conditions. The authors adapt boundary-straightening, mollification, and a normal-form framework to transfer horizontal-derivative control into vertical-derivative and temporal-to-spatial estimates, addressing higher-order perturbations from curvature. They establish gradient and second-derivative estimates for the Stokes problem, and extend gradient, second- and third-derivative bounds to the Navier–Stokes setting, including an application defining boundary regular points that guarantee higher spatial regularity. The results require only modest starting regularity and general boundary smoothness, thus broadening the boundary-regularity theory from flat to curved geometries with concrete PDE techniques that are amenable to further NS analysis.

Abstract

In this paper, we study local regularity of the solutions to the Stokes equations near a curved boundary under no-slip or Navier boundary conditions. We extend previous boundary estimates near a flat boundary to that near a curved boundary, under very low starting regularity assumptions. Compared with the flat case, the proof for the curved case is more complicated and we adapt new techniques such as the ``normal form" after the mollification, recovering vertical derivative estimates from horizontal derivative estimates, and transferring temporal derivatives to spatial derivatives, to deal with the higher order perturbation terms generated by boundary straightening. As an application, we propose a new definition of boundary regular points for the incompressible Navier--Stokes equations that guarantees higher spatial regularity.

The local regularity theory for the Stokes and Navier--Stokes equations near the curved boundary

TL;DR

This work develops local regularity theory for the nonstationary Stokes and Navier–Stokes equations near curved boundaries under no-slip and Navier boundary conditions. The authors adapt boundary-straightening, mollification, and a normal-form framework to transfer horizontal-derivative control into vertical-derivative and temporal-to-spatial estimates, addressing higher-order perturbations from curvature. They establish gradient and second-derivative estimates for the Stokes problem, and extend gradient, second- and third-derivative bounds to the Navier–Stokes setting, including an application defining boundary regular points that guarantee higher spatial regularity. The results require only modest starting regularity and general boundary smoothness, thus broadening the boundary-regularity theory from flat to curved geometries with concrete PDE techniques that are amenable to further NS analysis.

Abstract

In this paper, we study local regularity of the solutions to the Stokes equations near a curved boundary under no-slip or Navier boundary conditions. We extend previous boundary estimates near a flat boundary to that near a curved boundary, under very low starting regularity assumptions. Compared with the flat case, the proof for the curved case is more complicated and we adapt new techniques such as the ``normal form" after the mollification, recovering vertical derivative estimates from horizontal derivative estimates, and transferring temporal derivatives to spatial derivatives, to deal with the higher order perturbation terms generated by boundary straightening. As an application, we propose a new definition of boundary regular points for the incompressible Navier--Stokes equations that guarantees higher spatial regularity.
Paper Structure (18 sections, 16 theorems, 253 equations)

This paper contains 18 sections, 16 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Notations
  4. Boundary straightening, cut-off functions and mollification
  5. Various definitions of weak solutions
  6. Proof of Theorem \ref{['thm_0BC']}
  7. Heat equations of the approximations of velocity
  8. Gradient estimate
  9. Second derivative estimate
  10. Proof of Theorem \ref{['thm_navierBC']}
  11. Heat equations of the approximations of vorticity
  12. Gradient estimate with an extra term
  13. Removing the extra term
  14. Proof of Theorem \ref{['thm2_navierBC']}
  15. Stokes equations of the approximations of velocity
  16. ...and 3 more sections

Key Result

Theorem 1.1

Assume q_*. Suppose that the boundary portion $\Gamma \in C^{1,1}$, $\bm{u}, p, \mathbb{F}\in L^{q,r}(Q^+_1)$, $\bm{f}\in L^{q_*,r}(Q^+_1)$, and $(\bm{u},p)$ is a very weak solution pair (see Definition def:vwsp-0BC) to the Stokes equations Stokes-Eqn with no-slip boundary condition 0-BC0, then $(\b If in addition $\bm{f}+\mathop{\rm div}\nolimits \mathbb{F} \in L^{q,r}(Q^+_1)$, we have $\partial_

Theorems & Definitions (32)

  • Theorem 1.1: Derivative estimates for no-slip BC
  • Theorem 1.2: Boundary gradient estimate for Navier BC
  • Theorem 1.3: Higher-order derivative estimates for Navier BC
  • Definition 1.4
  • Theorem 1.5: NS with no-slip BC
  • Theorem 1.6: NS with Navier BC
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 22 more