Description
- Abstract:
- This dissertation presents new regularity criteria of the incompressible Navier-Stokes equations and derive further results, including extensions of the Ladyzhenskaya-Prodi-Serrin condition in high dimensional critical case. In the first part, we establish several boundary $\varepsilon$-regularity criteria for suitable weak solutions for the 3D incompressible Navier-Stokes equations in a half cylinder with the Dirichlet boundary condition on the flat boundary. Our concise proofs are based on delicate iteration arguments and interpolation techniques. These results extend and provide alternative proofs for the earlier interior results by Vasseur , Choi-Vasseur, and Phuc-Guevara. In the second part, we study regularity criteria for the $d$-dimensional incompressible Navier-Stokes equations. We prove if $u\in L_{\infty}^tL_d^x((0,T)\times\R^d_+)$ is a Leray-Hopf weak solution vanishing on the boundary, then $u$ is regular up to the boundary in $(0,T)\times \R^d_+$. Furthermore, with a stronger uniform local condition on the pressure $p$, we prove that $u$ is unique and tends to zero as $t\rightarrow \infty$ if $T=\infty$. This generalizes a result by Escauriaza, Seregin, and \v{S}ver\'{a}k \cite{Seregin1} to higher dimensions and domains with boundary. We also study the local problem in half unit cylinder $Q^+$ and prove that if $u\in L^t_{\infty}L^x_d(Q^+)$ and $ p\in L_{2-1/d}(Q^+)$, then $u$ is H\"{o}lder continuous in the closure of the set $Q^+(1/4)$.
- Notes:
- Thesis (Ph. D.)--Brown University, 2020
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Citation
Wang, Kunrui,
"The Incompressible Navier-Stokes Equations: Two Regularity Criteria"
(2020).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://repository.library.brown.edu/studio/item/bdr:1129508/
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Applied Mathematics Theses and Dissertations
Theses and Dissertations for the Applied Mathematics department....