From anisotropic Navier-Stokes equations to primitive equations for the ocean and atmosphere

Enhanced dissipation for the third component of 3D anisotropic Navier-Stokes equations

The purpose of this paper is twofold. First, we give a derivation of the Lagrangian averaged Euler (LAE-α) and Navier-Stokes (LANS-α) equations. This theory involves a spatial scale α and the equations are designed to accurately capture the dynamics of the Euler and Navier-Stokes equations at length scales larger than α, while averaging the motion at scales smaller than α. The derivation involves an averaging procedure that combines ideas from both the material (Lagrangian) and spatial (Eulerian) viewpoints. This framework allows the use of a variant of G. I. Taylor's turbulence hypothesis as the foundation for the model equations; more precisely, the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow. In this article, we use this hypothesis to derive the averaged Lagrangian for the theory, and all the terms up to and including order α^2 are accounted for. The equations come in both an isotropic and anisotropic version. The anisotropic equations are a coupled system of PDEs (partial differential equations) for the mean velocity field and the Lagrangian covariance tensor. In earlier works by Foias, Holm & Titi [10], and ourselves [16], an analysis of the isotropic equations has been given. In the second part of this paper, we establish local in time well-posedness of the anisotropic LANS-α equations using quasilinear PDE type methods.

Stability and optimal decay for the 3D Navier-Stokes equations with horizontal dissipation

Continuous dependence on initial data for the solutions of 3-D anisotropic Navier-Stokes equations

In this paper, we consider a global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations (ANS). We prove the global wellposedness for ANS provided the initial horizontal data are sufficient small in the scaling invariant Besov-Sobolev type space \({B^{0,\frac{1}{2}}}\) . In particular, the result implies the global wellposedness of ANS with large initial vertical velocity.

In this paper, we consider a global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations (ANS). We prove the global wellposedness for ANS provided the initial horizontal data are sufficient small in the scaling invariant Besov-Sobolev type space \({B^{0,\frac{1}{2}}}\) . In particular, the result implies the global wellposedness of ANS with large initial vertical velocity.

We prove the global well-posedness of the 3-D anisotropic Navier-Stokes equations for a class of large initial data, which slowly varies in the vertical direction. The proof uses the analytical-type estimates and the special structure of the nonlinear term of the equation.

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations (NSν) with initial data in the scaling invariant Besov space, \(\mathcal{B}^{-1+\frac3p}_{p,\infty},\) here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (ANSν), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, \(\mathcal{B}^{-\frac12,\frac12}_4\) and \(\mathcal{B}^{-\frac12,\frac12}_4(T).\) Then with initial data in the scaling invariant space \(\mathcal{B}^{-\frac12,\frac12}_4,\) we prove the global wellposedness for (ANSν) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (ANSν) with high oscillatory initial data (1.2).

Considering 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 ɛ with initial data , 1/q + 1/p ⩽ 1 if q ⩾ 2 and 4/3q + 2/3p ⩽ 1 if q ⩽ 2, converges as ɛ → 0 with convergence rate 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 which allow us to conclude that local in-time convergence already implies global in-time convergence, where moreover the convergence rate is independent of p and q.

The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

Considering the stochastic 3-D incompressible anisotropic Navier-Stokes equations, we prove the local existence of strong solution in \begin{document}$H^2(\mathbb{T}^3)$\end{document} . Moreover, we express the probabilistic estimate of the random time interval for the existence of a local solution in terms of expected values of the initial data and the random noise, and establish the global existence of strong solution in probability if the initial data and the random noise are sufficiently small.

The global existence for the strong solution of the stochastic 3-D incompressible anisotropic Naiver-Stokes equations

A finite element method for the numerical solution of the anisotropic Navier-Stokes equations in shallow domain is presented. This method take into account the low regularity of the vertical component of the velocity in the hydrostatic approximation of the Navier-Stokes equations [2, 3, 5]. A projection method [8] is used for the time discretization. The linear systems are solved via a some preconditioned conjugate algorithm, well adapted to massively parallel computers [4]. Some results are presented for the wind driven water circulation in lakes Geneva and Neuchâtel.

In this paper, we investigate the compressible primitive equations (CPEs) with density-dependent viscosity for large initial data. The CPE model can be derived from the 3D compressible and anisotropic Navier–Stokes equations by hydrostatic approximation. Motivated by the work of Vasseur and Yu [SIAM J. Math. Anal. 48, 1489–1511 (2016); Invent. Math. 206, 935–974 (2016)], in which the global existence of weak solutions to the compressible Navier–Stokes equations with degenerate viscosity was obtained, we construct approximate solutions and prove the global existence of weak solutions to the CPE in this paper. In our proof, we first present the vertical velocity as a function of density and horizontal velocity, which plays a role in using the Faedo–Galerkin method to obtain the global existence of the approximate solutions. Then, we obtain the key estimates of lower bound of the density, the Bresch–Desjardins entropy on the approximate solutions. Finally, we apply compactness arguments to obtain global existence of weak solutions by vanishing the parameters in our approximate system step-by-step.

In this paper, we study the asymptotic behavior for the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition in a half space of \({\mathbb{R}^3}\) when the vertical viscosity goes to zero. Firstly, by multi-scale analysis, we formally deduce an asymptotic expansion of the solution to the problem with respect to the vertical viscosity, which shows that the boundary layer appears in the tangential velocity field and satisfies a nonlinear parabolic–elliptic coupled system. Also from the expansion, it is observed that away from the boundary the solution of the anisotropic Navier–Stokes equations formally converges to a solution of a degenerate incompressible Navier–Stokes equation. Secondly, we study the well-posedness of the problems for the boundary layer equations and then rigorously justify the asymptotic expansion by using the energy method. We obtain the convergence results of the vanishing vertical viscosity limit, that is, the solution to the incompressible anisotropic Navier–Stokes equations tends to the solution to degenerate incompressible Navier–Stokes equations away from the boundary, while near the boundary, it tends to the boundary layer profile, in both the energy space and the L∞ space.