Browsing by Subject "Navier-Stokes equations"
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Item Asymptotic integration of Navier-Stokes equations with potential forces. II. An explicit Poincaré-Dulac normal form(University of Minnesota. Institute for Mathematics and Its Applications, 2011-01) Foias, Ciprian; Hoang, Luan; Saut, Jean-ClaudeItem Global well-posedness of the 3D primitive equations with partial vertical turbulence mixing heat diffusion(University of Minnesota. Institute for Mathematics and Its Applications, 2010-10) Cao, Chongsheng; Titi, Edriss S.Item Incompressible fluids in thin domains with Navier friction boundary conditions (I)(University of Minnesota. Institute for Mathematics and Its Applications, 2008-07) Hoang, Luan ThachItem Incompressible fluids in thin domains with Navier friction boundary conditions (II)(University of Minnesota. Institute for Mathematics and Its Applications, 2012-08) Hoang, Luan ThachItem Navier-Stokes equations with Navier boundary conditions in nearly flat domains(University of Minnesota. Institute for Mathematics and Its Applications, 2008-11) Hoang, Luan Thach; Sell, George R.Item On solutions to Navier-Stokes equations in critical spaces.(2010-07) Rusin, Walter MieczyslawIn this thesis, we consider solutions to the incompressible Navier-Stokes equa- tions in three spatial dimensions in the critical homogenous Sobolev space. We attempt to unify the theory of mild solutions and the theory of suitable weak solutions to show that assuming the existence of initial data leading to a finite time singularity the set of such initial is closed in the weak topology and sequen- tially compact modulo translations and dilations. This result is motivated by a theorem of Gallagher, Iftimie and Planchon which states that this set is closed in the strong topology. We present two proofs of our result. The first one is based on the theory of suitable weak solutions and partial regularity for the Navier-Stokes equaitons. The second approach is rooted in the profile decomposition developed by Gallagher.Item Regularity aspects of the Navier-Stokes equations in critical spaces(2020-08) Albritton, DallasFor better or for worse, our current understanding of the Navier-Stokes regularity problem is intimately connected with certain dimensionless quantities known as critical norms. In this thesis, we concern ourselves with one of the most basic questions about Navier-Stokes regularity: How must the critical norms behave at a potential Navier-Stokes singularity? In Chapter 2, we give a broad overview of the Navier-Stokes theory necessary to answer this question. This chapter is suitable for newcomers to the field. Next, we present two of our published papers [4,5] which answer this question in the context of homogeneous Besov spaces. In Chapter 3, we demonstrate that the critical Besov norms $\| u(\cdot,t) \|_{\dot B^{-1+3/p}_{p,q}(\R^3)}$, $p,q \in (3,+\infty)$, must tend to infinity at a potential singularity. Our proof has been streamlined from the published version [4]. In Chapter 4 (joint work with Tobias Barker), we develop a framework of global weak Besov solutions with initial data belonging to $\dot B^{-1+3/p}_{p,\infty}(\R^3)$, $p \in (3,+\infty)$. To illustrate this framework, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. This chapter has been reproduced from the published version [5].