Navier-Stokes equations: almost $L_{3,\infty}$-case

On the hydrostatic approximation of the Navier-Stokes equations in a thin strip

In this paper, we first prove the global well-posedness of a scaled anisotropic Navier-Stokes system and the hydrostatic Navier-Stokes system in a 2-D striped domain with small analytic data in the tangential variable. Then we justify the limit from the anisotropic Navier-Stokes system to the hydrostatic Navier-Stokes system with analytic data.

Global Solution for the incompressible Navier-Stokes equations] { Global Solution for the incompressible Navier-Stokes equations with a class of large data in $BMO^{-1}(\mathbb{R}^3)$

In this paper, we shall establish the global well-posedness, the space-time analyticity of the Navier-Stokes equations for a class of large periodic data $u_0 \in BMO^{-1}(\mathbb{R}^3)$. This improves the classical result of Koch \& Tataru \cite{koch-tataru}, for the global well-posedness with small initial data $u_0 \in BMO^{-1}(\mathbb{R}^n)$.

Stokes and Navier-Stokes equations with Navier boundary condition

We study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain $Ω\subset\mathbb{R}^{3}$ of class $\mathcal{C}^{1,1}$. We prove existence, uniqueness of weak and strong solutions in $\mathbf{W}^{1,p}(Ω)$ and $\mathbf{W}^{2,p}(Ω)$ for all $1<p<\infty$ considering minimal regularity on the friction coefficient $α$. Moreover, we deduce uniform estimates on the solution with respect to $α$ which enables us to analyze the behavior of the solution when $α\rightarrow \infty$.

On the global regularity of axisymmetric Navier-Stokes-Boussinesq system

In this paper we prove a global well-posedness result for tridimensional Navier-Stokes-Boussinesq system with axisymmetric initial data. This system couples Navier-Stokes equations with a transport equation governing the density.

On hydrodynamic limits of the Vlasov-Navier-Stokes system

We introduce a framework to justify hydrodynamic limits of the Vlasov-Navier-Stokes system. We specifically study high friction regimes, which take into account the fact that particles of the dispersed phase are light (resp. small) compared to the fluid part, and lead to the derivation of Transport-Navier-Stokes (resp. Inhomogeneous Navier-Stokes) systems.

Stabilization of Navier - Stokes equations by oblique boundary feedback controllers

One designs a linear stabilizable boundary feedback controller for the Navier-Stokes equations which is oblique to boundary.

Local Regularity of Axisymmetric Solutions to the Navier-Stokes Equations}

In the note, a local regularity condition for axisymmetric solutions to the non-stationary 3D Navier-Stokes equations is proven. It reads that axially symmetric energy solutions to the Navier-Stokes equations have no Type I blowups.

Space-time holomorphic solutions of Navier-Stokes equations

We introduce a concept of space-time holomorphic solutions of partial differential equations and construct a meromorphic solution of Navier-Stokes equations.

The energy equality for the Navier-Stokes equations in bounded domains

In this paper, we provide a sufficient condition of the energy equality for the incompressible Navier-Stokes equations in bounded domains.

Estimates of time-periodic fundamental solutions to the linearized Navier-Stokes equations

Fundamental solutions to the time-periodic Stokes and Oseen linearizations of the Navier-Stokes equations in dimension $n\geq 2$ are investigated. Integrability properties and pointwise estimates are established.

Sturmian Multiple Zeros for Stokes and Navier--Stokes Equations in $\re^3$ via Solenoidal Hermite Polynomials

Multiple spatial zero formations for Stokes and Navier-Stokes equations in three dimensions are shown to occur according to nodal sets of solenoidal Hermite polynomials. Extensions to well-posed Burnett equations with the bi-harmonic viscosity operator are also discussed.

Gevrey stability of hydrostatic approximate for the Navier-Stokes equations in a thin domain

In this paper, we justify the limit from the Navier-Stokes system in a thin domain to the hydrostatic Navier-Stokes/Prandtl system for the convex initial data with Gevrey 9/8 regualrity in $x$.

Partially congested propagation fronts in one-dimensional Navier-Stokes equations

These notes are dedicated to the analysis of the one-dimensional free-congested Navier-Stokes equations. After a brief synthesis of the results obtained in [4] related to the existence and the asymptotic stability of partially congested profiles associated to the soft congestion Navier-Stokes system, we present a first local well-posedness result for the one-dimensional free-congested Navier-Stokes equations.

Enstrophy cascade in physical scales for the 3D Navier-Stokes equations

An enstrophy cascade is exhibited for the Navier-Stokes equations in physical scales independently of boundary conditions under physically reasonable assumptions on the flow.

Liouville Theorem for 2D Navier-Stokes equations in Half Space

In the paper, a Liouville theorem for mild bounded ancient solutions to the 2D Navier-Stokes equations in half space has been proven.

Existence of Smooth Solutions of the Navier-Stokes Equations

In this paper, we prove existence of smooth solutions of the Navier-Stokes equations that gives a positive answer to the problem proposed by Fefferman [3].

Asymptotic behavior of solutions to 3D incompressible Navier-Stokes equations with damping

In this paper, we study the upper bound of the time decay rate of solutions to the Navier-Stokes equations and generalized Navier-Stokes equations with damping term $|u|^{β-1}u$ ($β>1$) in $\mathbb{R}^3$.

2D Constrained Navier-Stokes Equations

We study 2D Navier-Stokes equations with a constraint on $L^2$ energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier-Stokes equation on $\R^2$ and $\T$, by a fixed point argument. We also show that the solution of constrained Navier-Stokes converges to the solution of Euler equation as viscosity $ν$ vanishes.

Remark on Wolf's condition for boundary regularity of Navier-Stokes equations

We prove Wolf's regularity condition up to the boundary for solutions to the Navier-Stokes equations satisfying non-slip boundary condition.

A global scheme for the incompressible Navier-Stokes equation on compact Riemannian manifolds

We propose a global scheme for a controlled Navier-Stokes equation system on compact smooth Riemannian manifolds.

Holder Continuity of Solutions of 2D Navier-Stokes Equations with Singular Forcing

We discuss the regularity of solutions of 2D incompressible Navier-Stokes equations forced by singular forces. The problem is motivated by the study of complex fluids modeled by the Navier-Stokes equations coupled to a nonlinear Fokker-Planck equation describing microscopic corpora embedded in the fluid. This leads naturally to bounded added stress and hence to $W^{-1,\infty}$ forcing of the Navier-Stokes equations.

On the non-homogeneous Navier-Stokes system with Navier friction boundary conditions

We address the issue of existence of weak solutions for the non-homogeneous Navier-Stokes system with Navier friction boundary conditions allowing the presence of vacuum zones and assuming rough conditions on the data. We also study the convergence, as the viscosity goes to zero, of weak solutions for the non-homogeneous Navier-Stokes system with Navier friction boundary conditions to the strong solution of the Euler equations with variable density, provided that the initial data converge in $L^{2}$ to a smooth enough limit.

Singular solutions of Navier Stokes equations with time-dependent external force terms in L2

It is shown that Navier Stokes equation models with time dependent external forces in L2 can have singular solutions.

二维随机系数Navier--Stokes方程的随机均匀化

Stochastic homogenization for two dimensional Navier--Stokes equations with random coefficients

本文利用弱收敛与Stratonovich--Khasminskii平均原理，证明二维随机系数Navier--Stokes方程的解在分布上收敛到常系数方程的解，实现随机均匀化。

Decay Estimate for some Toy-models related to the Navier-Stokes system

We prove in this paper a decay estimate for scaling invariant local energy solutions for some toy-models related to the incompressible Navier-Stokes system.

Global well-posedness for the two dimensional Navier-Stokes-Vlasov Equations

The global well-posedness for the incompressible Navier-Stokes-Vlasov equations in two spatial dimensions is established by a priori estimates, the characteristic method and the semigroup analysis.

A Liouville Type Theorem for Steady-State Navier-Stokes Equations

A Liouville type theorem is proven for the steady-state Navier-Stokes equations. It follows from the corresponding theorem on the Stokes equations with the drift. The drift is supposed to belong to a certain Morrey space.

A Slightly Supercritical Condition of Regularity of Axisymmetric Solutions to the Navier-Stokes Equations

In the note, a new regularity condition for axisymmetric solutions to the non-stationary 3D Navier-Stokes equations is proven. It is slightly supercritical.

Low Mach number limit of full compressible Navier-Stokes equations with revised Maxwell law

In this paper, we study the low Mach number limit of the full compressible Navier-Stokes equations with revised Maxwell law. By applying the uniform estimation of the error system, we prove that the solutions of the full compressible Navier-Stokes equations with time relaxation converge to that of the incompressible Navier-Stokes equations as the Mach number tends to zero. Moreover, the convergence rates are also obtained.

A remark on weak-strong uniqueness for suitable weak solutions of the Navier-Stokes equations

We extend Barker's weak-strong uniqueness results for the Navier--Stokes equations and consider a criterion involving Besov spaces and weighted Lebesgue spaces.

Liouville Type Theorem for Stationary Navier-Stokes Equations

It is shown that any smooth solution to the stationary Navier-Stokes system in $R^3$ with the velocity field, belonging globally to $L_6$ and $BM0^{-1}$, must be zero.

Existence and uniqueness of the global solution to the Navier-Stokes boundary problem

A proof is given of the global existence and uniqueness of a weak solution to Navier-Stokes boundary problem. The proof is short and essentially self-contained.

Forward Discretely Self-Similar Solutions of the Navier-Stokes Equations

Extending the work of Jia and Sverak on self-similar solutions of the Navier-Stokes equations, we show the existence of large, forward, discretely self-similar solutions.

The Holder continuity of the swirl for the Navier-Stokes motions

Applying the DeGiorgi Method the Holder continuity of the swirl is proved for solutions to the Navier-Stokes equations from Lq(0,T;Lp) with 3/p+2/q < 1.

Global regularity for the 3D Navier-Stokes and the 3D Euler equations

The article `Global regularity for the 3D Navier-Stokes and the 3D Euler equations'(arXiv:0711.2453) is withdrawn due to a serious error in the proof.

On the Nernst-Planck-Navier-Stokes system

We consider ionic electrodiffusion in fluids, described by the Nernst-Planck-Navier-Stokes system in bounded domains, in two dimensions, with Dirichlet boundary conditions for the Navier-Stokes and Poisson equations, and blocking (vanishing normal flux) or selective (Dirichlet) boundary conditions for the ionic concentrations. We prove global existence and stability results for large data.

Uniform regularity for the Navier-Stokes equation with Navier boundary condition

We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier-Stokes equation with Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in $L^\infty$. This allows to get the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.

Characterisation of the pressure term in the incompressible Navier-Stokes equations on the whole space

We characterise the pressure term in the incompressible 2D and 3D Navier-Stokes equations for solutions defined on the whole space.

On self-similar solutions to degenerate compressible Navier-Stokes equations

We study cavitating self-similar solutions to compressible Navier-Stokes equations with degenerate density-dependent viscosity. We prove both existence of expanders and non-existence of small shrinkers.

Topological Properties of Strong Solutions for the 3D Navier-Stokes Equations

In this note we give a criterion for the existence of global strong solutions for the 3D Navier-Stokes system for any regular initial data.

Existence and uniqueness of solutions to the damped Navier-Stokes equations with Navier boundary conditions for three dimensional incompressible fluid

In this article, we study the solutions of the damped Navier--Stokes equation with Navier boundary condition in a bounded domain $Ω$ in $\mathbb{R}^3$ with smooth boundary. The existence of the solutions is global with the damped term $\vartheta |u|^{β-1}u, \vartheta >0.$ The regularity and uniqueness of solutions with Navier boundary condition is also studied. This extends the existing results in literature.

The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions

We obtain existence and conormal Sobolev regularity of strong solutions to the 3D compressible isentropic Navier-Stokes system on the half-space with a Navier boundary condition, over a time that is uniform with respect to the viscosity parameters when these are small. These solutions then converge globally and strongly in $L^2$ towards the solution of the compressible isentropic Euler system when the viscosity parameters go to zero.

Vanishing viscosity limit of navier-stokes equations in gevrey class

In this paper we consider the inviscid limit for the periodic solutions to Navier-Stokes equation in the framework of Gevrey class. It is shown that the lifespan for the solutions to Navier-Stokes equation is independent of viscosity, and that the solutions of the Navier-Stokes equation converge to that of Euler equation in Gevrey class as the viscosity tends to zero. Moreover the convergence rate in Gevrey class is presented.

Ancient Solutions to Navier-Stokes Equations in Half Space

The goal of the paper is to understand properties of the so-called ancient (backward) solutions to the Navier-Stokes equations. We focus on the case of the half space.

Point singularities of 3D stationary Navier-Stokes flows

This article characterizes the singularities of very weak solutions of 3D stationary Navier-Stokes equations in a punctured ball which are sufficiently small in weak $L^3$.

On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations

We prove new \emph{a priori} estimates for the 3D Euler, the 3D Navier-Stokes and the 2D quasi-geostrophic equations by the method of similarity transforms.

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 equat…

Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain

This paper is concerned with the existence and uniqueness of the strong solution to the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a two-dimensional strip domain where the slip coefficients may not have defined sign. In the meantime, we also establish a number of Gagliardo-Nirenberg inequalities in the corresponding Sobolev spaces which will be applicable to other similar situations.

Analytical solutions for Navier-Stokes equations with $ψ$-Caputo fractional derivative

This work aims to use the homotopy analysis method to obtain analytical solutions of linear time-fractional Navier-Stokes equations with cylindrical coordinates and of a system of nonlinear time-fractional Navier-Stokes equations with Cartesian coordinates. These equations are described in the $ψ$-Caputo time-fractional derivative. The solutions obtained for time-fractional Navier-Stokes equations are graphically presented.

Structure of Helicity and Global Solutions of Incompressible Navier-Stokes Equation

In this paper we derive a new energy identity for the three-dimensional incompressible Navier-Stokes equations by a special structure of helicity. The new energy functional is critical with respect to the natural scalings of the Navier-Stokes equations. Moreover, it is conditionally coercive. As an application we construct a family of finite energy smooth solutions to the Navier-Stokes equations whose critical norms can be arbitrarily large.

Self-similar solutions for compressible Navier-Stokes equations

We construct forward self-similar solutions (expanders) for the compressible Navier-Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity do to cavitation at the origin.

Remarks on Liouville type theorems for the 3D steady axially symmetric Navier-Stokes equations

In this note, we investigate the 3D steady axially symmetric Navier-Stokes equations, and obtained Liouville type theorems if the velocity or the vorticity satisfies some a priori decay assumptions.

Estimates of suitable weak solutions to the Navier-Stokes equations in critical Morrey spaces

We prove some estimates for suitable weak solutions to the non-stationary three-dimensional Navier-Stokes equations under assumptions that certain invariant functionals of the velocity are bounded.

On Smoothness of $L_{3,\infty}$-Solutions to the Navier-Stokes Equations up to Boundary

We show that $L_{3,\infty}$-solutions to the three-dimensional Navier-Stokes equations near a flat part of the boundary are smooth.