Anisotropic Navier-Stokes Equations Research Articles

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

In this paper, we prove the continuous dependence on the initial data for the solutions of 3-D incompressible anisotropic Navier-Stokes equations in the functional space Xs(T)=def{u:u∈C([0,T];H0,s)with∇hu∈L2(]0,T[;H0,s)} for s>12. We also show the non-uniform continuity of the data-to-solution map in C([0,T];H0,s) for s>12, which makes sharp contrast with the corresponding result for the classical 3-D Navier-Stokes equations.

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

This paper considers the stochastic 3-D incompressible anisotropic Navier-Stokes equations. We establish the global existence of a strong solution in probability when either the horizontal component of the initial data and external force is sufficiently small or the horizontal viscosity coefficient is sufficiently large.

3D anisotropic Navier–Stokes equations in : stability and large-time behaviour

The study on the large-time behaviour of solutions to the 3D incompressible anisotropic Navier–Stokes (ANS) equations is very recent. Powerful tools designed for the Navier–Stokes equations with full Laplacian dissipation such as the Fourier splitting method no longer apply to the case when there is only horizontal dissipation. For the whole space , as , solutions of the ANS equations converge to the trivial solution and the convergence rate is algebraic. This paper is devoted to the case when the spatial domain Ω is . Our results reveal that the large-time behaviour for is quite different from that for . We show that any small initial velocity field leads to a unique global solution u that remains small in . More importantly, as , the velocity field u converges to a nontrivial steady state. The first two components of the steady state are given by the horizontal average of the first two components of u 0 while the third component vanishes. In addition, this convergence is exponentially fast.

Optimal decay for the 2D anisotropic Navier–Stokes equations with mixed partial dissipation

This paper studies the stability and large time behavior to the 2D anisotropic Navier–Stokes equations with mixed partial dissipation. We establish the uniform upper bounds and the global stability of solutions, and obtain the optimal decay properties to these global solutions and their higher order derivatives without any small assumptions on the initial data.

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

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.

Stability and Exponential Decay for the 2D Anisotropic Navier–Stokes Equations with Horizontal Dissipation

Solutions to the 2D Navier–Stokes equations with full dissipation in the whole space $${\mathbb {R}}^2$$ always decay to zero in Sobolev spaces. In particular, any perturbation near the trivial solution is always asymptotically stable. In contrast, solutions to the 2D Euler equations for inviscid flows can grow rather rapidly. An intermediate situation is when the dissipation is anisotropic and only one-directional. The stability and large-time behavior problem for the 2D Navier–Stokes equations with only one-directional dissipation is not well-understood. When the spatial domain is the whole space $${\mathbb {R}}^2$$ , this problem is widely open. This paper solves this problem when the domain is $${\mathbb {T}}\times {\mathbb {R}}$$ with $${\mathbb {T}}$$ being a 1D periodic box. The idea here is to decompose the velocity u into its horizontal average $${\bar{u}}$$ and the corresponding oscillation $${\widetilde{u}}$$ . By making use of special properties of $${\widetilde{u}}$$ , we establish a uniform upper bound and the stability of u in the Sobolev space $$H^2$$ , and show that $${\widetilde{u}}$$ in $$H^1$$ decays to zero exponentially in time.

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

Stability and large-time behavior are essential properties of solutions to many partial differential equations (PDEs) and play crucial roles in many practical applications. When there is full Laplacian, many techniques such as the Fourier splitting method have been created to obtain the large-time decay rates. However, when a PDE is anisotropic and involves only partial dissipation, these methods no longer apply and no effective approach is currently available. This paper aims at the stability and large-time behavior of the 3D anisotropic Navier-Stokes equations. We present a systematic approach to obtain the optimal decay rates of the stable solutions emanating from a small data. We establish that, if the initial velocity is small in the Sobolev space H4(R3)∩Hh−σ(R3), then the anisotropic Navier-Stokes equations have a unique global solution, and the solution and its first-order derivatives all decay at the optimal rates. Here Hh−σ with σ>0 denotes a Sobolev space with negative horizontal index.

Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier–Stokes equations* *This work was partly supported by the DFG International Research Training Group IRTG 1529 and the JSPS Japanese-German Graduate Externship on Mathematical Fluid Dynamics. The second author is partly supported by JSPS through grant Kiban S (No. 26220702), Kiban A (No. 19H00639), Kaitaku (No. 18H05323), Kiban

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.

Remark on the Lifespan of Solutions to 3-D Anisotropic Navier Stokes Equations

Remark on the Lifespan of Solutions to 3-D Anisotropic Navier Stokes Equations

Global existence of weak solutions to 3D compressible primitive equations with degenerate viscosity

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.

Local and global strong solution to the stochastic 3-D incompressible anisotropic 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.

Global regularity of [formula omitted]-dimensional generalized MHD system with anisotropic dissipation and diffusion

Motivated by the anisotropic Navier–Stokes equations that have much applications in studying fluids in thin domains, in particular meteorology and oceanography, we study the magnetohydrodynamics system with generalized dissipation and diffusion, considering different exponents of the fractional Laplacians with logarithmic worsening applied to different directions and components of the solution vector fields. The results indicate that it is possible to regularize the flows by anisotropic dissipation, some components in various directions allowed to be below the critical exponents in the expense of others being above.

Global regularity for some classes of large solutions to the 3-D anisotropic Navier-Stokes equations

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.

The inviscid limit of the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition

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.

Global regularity for generalized anisotropic Navier–Stokes equations

We consider the global Cauchy problem for the three-dimensional generalized anisotropic Navier–Stokes system with the viscosity term \documentclass[12pt]{minimal}\begin{document}$\Delta _h u-M^2u$\end{document}Δhu−M2u, where M is a Fourier multiplier whose symbol \documentclass[12pt]{minimal}\begin{document}$m:\mathbb {R}\rightarrow \mathbb {R}^+$\end{document}m:R→R+ is non-negative. The case \documentclass[12pt]{minimal}\begin{document}$m(\xi _3)=|\xi _3|$\end{document}m(ξ3)=|ξ3| is essentially the Navier–Stokes system, while the case m = 0 is the anisotropic Navier–Stokes system. We obtain the global regularity in the case that \documentclass[12pt]{minimal}\begin{document}$m(\xi _3)\ge {|\xi _3|^{\frac{3}{2}}}/{g(|\xi _3|)}$\end{document}m(ξ3)≥|ξ3|32/g(|ξ3|) for all sufficiently large \documentclass[12pt]{minimal}\begin{document}$|\xi _3|$\end{document}|ξ3| where \documentclass[12pt]{minimal}\begin{document}$g:=\mathbb {R}^+\rightarrow \mathbb {R}^+$\end{document}g:=R+→R+ is a nondecreasing function such that \documentclass[12pt]{minimal}\begin{document}$\int ^\infty _1\frac{ds}{ g^2(s) s\ln s}=+\infty$\end{document}∫1∞dsg2(s)slns=+∞. Then, using the method of descent, we can obtain the classical global regularity result of the two-dimensional Navier–Stokes system.

Erratum to: Global Wellposed Problem for the 3-D Incompressible Anisotropic Navier-Stokes Equations in an Anisotropic Space

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.

Global Wellposed Problem for the 3-D Incompressible Anisotropic Navier-Stokes Equations in an Anisotropic Space

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.

Global wellposed problem for the 3-D incompressible anisotropic Navier–Stokes equations

In this paper, we consider a global wellposed problem for the 3-D incompressible anisotropic Navier–Stokes equations (ANS). In order to do so, we first introduce the scaling invariant Besov–Sobolev type spaces, B p − 1 + 2 p , 1 2 and B p − 1 + 2 p , 1 2 ( T ) , p ⩾ 2 . Then, we prove the global wellposedness for (ANS) provided the initial data are sufficient small compared to the horizontal viscosity in some suitable sense, which is stronger than B p − 1 + 2 p , 1 2 norm. In particular, our results imply the global wellposedness of (ANS) with high oscillatory initial data.

On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations

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).

Finite element solution of Navier-Stokes equations in shallow domains

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.