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Register a new userJustification of the Hydrostatic Approximation of the Primitive Equations in Anisotropic Space $L^infty_H L^q_{x_3}(Torus^3)$
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The primitive equations are fundamental models in geophysical fluid dynamics and derived from the scaled Navier-Stokes equations. In the primitive equations, the evolution equation to the vertical velocity is replaced by the so-called hydrostatic approximation. In this paper, we give a justification of the hydrostatic approximation by the scaled Navier-Stoke equations in anisotropic spaces $L^infty_H L^q_{x_3} (Torus^3)$ for $q geq 1$.
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Consider the anisotropic Navier-Stokes equations as well as the primitive equations. It is shown that the horizontal velocity of the solution to the anisotropic Navier-Stokes equations in a cylindrical domain of height $varepsilon $ with initial data $u_0=(v_0,w_0)in B^{2-2/p}_{q,p}$, $1/q+1/ple 1$ if $qge 2$ and $4/3q+2/3ple 1$ if $qle 2$, converges as $varepsilon to 0$ with convergence rate $mathcal{O} (varepsilon )$ to the horizontal velocity of the solution to the primitive equations with initial data $v_0$ with respect to the maximal-$L^p$-$L^q$-regularity norm. Since the difference of the corresponding vertical velocities remains bounded with respect to that norm, the convergence result yields a rigorous justification of the hydrostatic approximation in the primitive equations in this setting. It generalizes in particular a result by Li and Titi for the $L^2$-$L^2$-setting. The approach presented here does not rely on second order energy estimates but on maximal $L^p$-$L^q$-estimates for the heat equation.
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 extend some results proved in previous references for three-dimensional Navier-Stokes equations. We show that when the norm of the velocity field is small enough in $L^3({I!!R}^3)$, then a global smooth solution of the Navier-Stokes equations is ensured. We show that a similar result holds when the norm of the velocity field is small enough in $H^{frac{1}{2}}({I!!R}^3)$. The scale invariance of these two norms is discussed.
In this paper we study the $L^p$-$L^q$ boundedness of the Fourier multipliers in the setting where the underlying Fourier analysis is introduced with respect to the eigenfunctions of an anharmonic oscillator $A$. Using the notion of a global symbol that arises from this analysis, we extend a version of the Hausdorff-Young-Paley inequality that guarantees the $L^p$-$L^q$ boundedness of these operators for the range $1<p leq 2 leq q <infty$. The boundedness results for spectral multipliers acquired, yield as particular cases Sobolev embedding theorems and time asymptotics for the $L^p$-$L^q$ norms of the heat kernel associated with the anharmonic oscillator. Additionally, we consider functions $f(A)$ of the anharmonic oscillator on modulation spaces and prove that Linsku iis trace formula holds true even when $f(A)$ is simply a nuclear operator.
We consider small amplitude wave packet-like solutions to the 3D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Formal multiscale calculations suggest that the modulation of such a solution is described by a profile traveling at group velocity and governed by a hyperbolic cubic nonlinear Schrodinger equation. In this paper we show that, given wave packet initial data, the corresponding solution exists and retains the form of a wave packet on natural NLS time scales. Moreover, we give rigorous error estimates between the true and formal solutions on the appropriate time scale in Sobolev spaces using the energy method. The proof proceeds by directly applying modulational analysis to the formulation of the 3D water wave problem developed by Sijue Wu.
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