Local and global regularity for the Stokes and Navier-Stokes equations with boundary data in the half-space

Kyungkeun Kang; Chanhong Min

doi:10.3934/cpaa.2024015

Article Contents

2024, Volume 23, Issue 10: 1506-1561. Doi: 10.3934/cpaa.2024015

This issue Previous Article Gradient estimates for the non-stationary Stokes system with the Navier boundary condition Next Article Remark on the stability of energy maximizers for the 2D Euler equation on $ \mathbb{T}^2 $

Local and global regularity for the Stokes and Navier-Stokes equations with boundary data in the half-space

Kyungkeun Kang^, and
Chanhong Min

Yonsei University, Republic of Korea

^*Corresponding author: Kyungkeun Kang
^*Corresponding author: Kyungkeun Kang

Received: June 2023

Revised: January 2024

Early access: March 2024

Published online: March 2024

K. Kang is supported by NRF-2019R1A2C1084685; C. Min is supported by NRF-2019R1A2C1084685.

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Abstract

We study the Stokes system with the localized boundary data in the half-space. We are concerned with the local regularity of its solution near the boundary away from the support of the given boundary data which are in the product forms of each spatial variable and the temporal variable. We first show that if the boundary data are smooth in time, the corresponding solutions are also smooth in space and time near the boundary, even if the boundary data are only spatially integrable. Secondly, if the normal component of the boundary data is absent, we are able to construct a solution such that the second normal derivatives of its tangential components become singular near the boundary. Perturbation argument enables us to construct solutions of the Navier-Stokes equations with similar singular behaviors near the boundary in the half-space as the case of Stokes system. Lastly, we provide specific types of the localized boundary data to obtain the pointwise bounds of the solutions up to second derivatives. It turns out that such solutions are globally strong, while the second normal derivatives are unbounded near the boundary. These results can be compared to the previous works in which only the normal component is present. In fact, the non-smooth tangential boundary data in time can also cause spatially singular behaviors of the solution near the boundary, although such behaviors are milder than those caused by the normal boundary data of the same type.

Keywords:

Stokes equations,
Navier–Stokes equations,
local and global regularity near boundary.

Mathematics Subject Classification: Primary: 35Q30, 35B65.

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Figure 1. Figure of $\mathcal{G}, \mathcal{G}', \mathcal{G}''$, $\delta: = \frac{1}{20\sqrt{n-1}}$

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Figure 2. (Case $i\neq k$) Graph of the domain for $n = 3$, $i = 1$ and $k = 2$

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Figure 3. (Case $i = k$) Graph of the domain for $n = 3$ and $k = 1$

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