[Paper Review] Estimates of solutions to the perturbed Stokes system

Viktor Vyalov, Timofey Shilkin|arXiv (Cornell University)|Feb 28, 2014

Navier-Stokes equation solutions5 references4 citations

TL;DR

This paper establishes local L^s_l estimates for solutions to the perturbed Stokes system in a half-ball with slip boundary conditions, using a diffeomorphism to flatten curved boundaries. The key contribution is a priori estimates in anisotropic Sobolev spaces that enable the study of partial regularity of weak solutions to the Navier-Stokes equations near curved boundaries.

ABSTRACT

In this paper we derive local estimates of solutions of the Perturbed Stokes system. This system arises as a reduction of the Stokes system near a curved part of the boundary of the domain if one applies a diffeomorphism flatting the boundary. The estimates obtained in the paper play the crucial role in the investigation of partial regularity of weak solutions to the Navier-Stokes system near a curved part of the boundary of the domain.

Motivation & Objective

• To derive local a priori estimates for solutions to the perturbed Stokes system in a half-ball domain with curved boundary.
• To provide a framework for analyzing partial regularity of weak solutions to the Navier-Stokes equations near curved parts of the boundary.
• To establish estimates in anisotropic Lebesgue and Sobolev spaces that are invariant under the change of variables flattening the boundary.
• To handle the slip boundary condition in the transformed system via weighted differential operators with variable coefficients.
• To prove higher integrability and regularity of solutions through iterative refinement of integrability indices using Sobolev embedding and interpolation.

Proposed method

• Transform the original Stokes system in a domain with a curved boundary into the perturbed Stokes system via a diffeomorphism that flattens the boundary.
• Define variable-coefficient differential operators $\hat{\Delta}_{\varphi}$ and $\hat{\nabla}_{\varphi}$ to represent the Laplacian and gradient in the new coordinates.
• Use anisotropic Lebesgue and Sobolev spaces $L_{s,l}$, $W^{2,1}_{s,l}$, and $W^{1,0}_{s,l}$ to measure solution integrability in space and time.
• Apply a cutoff function technique to localize the problem and derive estimates in smaller subdomains.
• Use a compactness argument and weak convergence to pass to the limit in a sequence of approximated solutions.
• Iterate estimates across a sequence of increasing integrability indices $s_k$ using Sobolev embedding and interpolation to achieve higher regularity.

Experimental results

Research questions

• RQ1 What local regularity estimates can be derived for solutions to the perturbed Stokes system in a half-ball with a curved boundary?
• RQ2 How do variable-coefficient differential operators $\hat{\Delta}_{\varphi}$ and $\hat{\nabla}_{\varphi}$ affect the integrability and smoothness of solutions?
• RQ3 Can the solution regularity be improved through iterative application of embedding theorems and interpolation in anisotropic spaces?
• RQ4 What is the role of the slip boundary condition in the regularity theory of the perturbed Stokes system?
• RQ5 How do the estimates depend on the curvature of the boundary, quantified by the $W^3_\infty$ norm of $\varphi$?

Key findings

• The paper establishes a local a priori estimate in $W^{2,1}_{s,l}(B^{+}_{1/2} \times (-1/4, -\delta))$ for solutions $(v,p)$ of the perturbed Stokes system with $\delta > 0$.
• The estimate is uniform in $\delta$, allowing passage to the limit as $\delta \to 0^+$, which yields the full regularity in $W^{2,1}_{s,l}(B^{+}_{1/2} \times (-1/4, 0))$.
• The solution satisfies the bound $\|v\|_{W^{2,1}_{s,l}} + \|\nabla p\|_{L_{s,l}} \leq C(\|f\|_{L_{s,l}} + \|v\|_{W^{1,0}_{s,l}} + \|p\|_{L_{s,l}})$ in a smaller parabolic cylinder.
• By iterating the estimate over a sequence of increasing integrability indices $s_k$, the paper achieves the final estimate in $W^{2,1}_{m,l}$ for $m$-integrable data.
• The final estimate is of the form $\|v\|_{W^{2,1}_{s_N,l}(Q^{+}_{1/2})} + \|\nabla p\|_{L_{s_N,l}(Q^{+}_{1/2})} \leq C^N(\|f\|_{L_{m,l}} + \|v\|_{W^{1,0}_{s_0,l}} + \|p-b\|_{L_{s_0,l}})$, with $s_N = m$.
• The result confirms the higher integrability and regularity of solutions under minimal assumptions on the boundary curvature ($\varphi \in W^3_\infty$).