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Register a new userGlobal wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations
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In this paper, we consider a global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations (textit{ANS}). In order to do so, we first introduce the scaling invariant Besov-Sobolev type spaces, $B^{-1+frac{2}{p},{1/2}}_{p}$ and $B^{-1+frac{2}{p},{1/2}}_{p}(T)$, $pgeq2$. Then, we prove the global wellposedness for (textit{ANS}) provided the initial data are sufficient small compared to the horizontal viscosity in some suitable sense, which is stronger than $B^{-1+frac{2}{p},{1/2}}_{p}$ norm. In particular, our results imply the global wellposedness of (textit{ANS}) with high oscillatory initial data.
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In this paper we study a finite-depth layer of viscous incompressible fluid in dimension $n ge 2$, modeled by the Navier-Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity-capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e. time-independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension $n ge 2$ and without surface tension in dimension $n=2$, for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well known, to the best of our knowledge this is the first construction of viscous traveling wave solutions. Our proof involves a number of novel analytic ingredients, including: the study of an over-determined Stokes problem and its under-determined adjoint, a delicate asymptotic development of the symbol for a normal-stress to normal-Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions.
In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting $varepsilon >0$ to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders $O(1)$ and $O(varepsilon^alpha)$, respectively, with $alpha>2$, for which the limiting system is the primitive equations with only horizontal viscosity as $varepsilon$ tends to zero. In particular we show that for well prepared initial data the solutions of the scaled incompressible three-dimensional Navier-Stokes equations converge strongly, in any finite interval of time, to the corresponding solutions of the anisotropic primitive equations with only horizontal viscosities, as $varepsilon$ tends to zero, and that the convergence rate is of order $Oleft(varepsilon^fracbeta2right)$, where $beta=min{alpha-2,2}$. Note that this result is different from the case $alpha=2$ studied in [Li, J.; Titi, E.S.: emph{The primitive equations as the small aspect ratio limit of the Navier-Stokes equations: Rigorous justification of the hydrostatic approximation}, J. Math. Pures Appl., textbf{124} rm(2019), 30--58], where the limiting system is the primitive equations with full viscosities and the convergence is globally in time and its rate of order $Oleft(varepsilonright)$.
In this paper, we study the decay rates for the global small smooth solutions to 3D anisotropic incompressible Navier-Stokes equations. In particular, we prove that the horizontal components of the velocity field decay like the solutions of 2D classical Navier-Stokes equations. While the third component of the velocity field decays as the solutions of 3D Navier-Stokes equations. We remark that such enhanced decay rate for the third component is caused by the interplay between the divergence free condition of the velocity field and the horizontal Laplacian in the anisotropic Navier-Stokes equations.
In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in $dot{H}^{-alpha}(mathbb{R}^{3})$ or $dot{H}^{-alpha}(mathbb{T}^{3})$ with $0<alphaleq 1/2$. This is achieved by randomizing the initial data and showing that the energy of the solution modulus the linear part keeps finite for all $tgeq0$. Moreover, the energy of the solutions is also finite for all $t>0$. This improves the recent result of Nahmod, Pavlovi{c} and Staffilani on (SIMA, [1])in which $alpha$ is restricted to $0<alpha<frac{1}{4}$.
In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in the last paper is that it varies slowly in one direction, though in some sense it is ``well prepared (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize the setting of that last paper to an ``ill prepared situation (the norm blows up as the small parameter goes to zero).The proof uses the special structure of the nonlinear term of the equation.
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