On a variational approach to the Navier-Stokes Equations

Inertial forces in the Navier-Stokes equation

We estimate the inertial force field in the 3d Euler/Navier-Stokes equation in function of |u|_\infty and |\nabla u |_\infty and provide an application to the well-posed problem

Analytical Solutions to the Navier-Stokes Equations

With the previous results for the analytical blowup solutions of the N-dimensional Euler-Poisson equations, we extend the similar structure to construct an analytical family of solutions for the isothermal Navier-Stokes equations and pressureless Navier-Stokes equations with density-dependent viscosity.

On convergence criteria for incompressible Navier-Stokes equations with Navier boundary conditions and physical slip rates

We prove some criteria for the convergence of weak solutions of the 2D incompressible Navier-Stokes equations with Navier slip boundary conditions to a strong solution of incompressible Euler. The slip rate depends on a power of the Reynolds number, and it is increasingly apparent that the power 1 may be critical for L^2 convergence, as hinted at in [hal-01093331].

Pressure Regularity Criterion for the Three-dimensional Navier-Stokes Equations in Infinite Channel

In this paper we consider the three-dimensional Navier-Stokes equations in an infinite channel. We provide a sufficient condition, in terms of $\partial_z p$, where $p$ is the pressure, for the global existence of the strong solutions to the three-dimensional Navier-Stokes equations.

Global Attractors and Determining Modes for the 3D Navier-Stokes-Voight Equations

We investigate the long-term dynamics of the three-dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid. Specifically, we derive upper bounds for the number of determining modes for the 3D Navier-Stokes-Voight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view we consider the Navier-Stokes-Voight model as a non-viscous (inviscid) regularization of the three-dimensional Navier-Stokes equations. Furthermore, we also show that the weak solutions of the Navier-Stokes-Voight equat…

A maximum modulus estimate for solutions of the Navier-Stokes system in domains of polyhedral type

The authors prove a maximum modulus estimate for solutions of the stationary Navier-Stokes system in bounded domains of polyhedral type.

Analyticity estimates for the Navier-Stokes equations

We study spatial analyticity properties of solutions of the Navier-Stokes equations and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier-Stokes equations with data in H^r, r greater or equal 1/2, and prove a stability result for the analyticity radius.

Regularity criteria in weak spaces for Navier-Stokes equations in R3

In this paper we establish a Serrin type regularity criterion on the gradient of pressure in weak spaces for the Leray-Hopf weak solutions of the Navier-Stokes equations in R3.

Navier-Stokes equations and Banach spaces of local measures

This paper was removed by arXiv admin because it plagiarizes "Kozono, Hideo; Shimada, Yukihiro Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations. Math. Nachr. 276 (2004), 63--74."

Analytical Blowup Solutions to the 4-dimensional Pressureless Navier-Stokes-Poisson Equations with Density-dependent Viscosity

We study the 4-dimensional pressureless Navier--Stokes-Poisson equations with density-dependent viscosity. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed in this paper.

Regularity of Leray-Hopf solutions to Navier-Stokes equations (I)-Critical interior regularity in weak spaces

We consider the interior regularity of Leray-Hopf solutions to Navier-Stokes equations on critical case L^2_w(0,T;L^\infty(R^3)). Particularly, an open problem proposed in [KK] was solved.

On the energy equality for weak solutions of the 3D Navier-Stokes equations

We prove that the energy equality holds for weak solutions of the 3D Navier-Stokes equations in the functional class $L^3([0,T);V^{5/6})$, where $V^{5/6}$ is the domain of the fractional power of the Stokes operator $A^{5/12}$.

Regularity of Leray-Hopf solutions to Navier-Stokes equations (II)-Blow up rate with small L^2(R^3) data

An upper bound of blow up rate for the Navier-Stokes equations with small data in L^2(R^3) is obtained.

Inviscid Limit for 2D Stochastic Navier-Stokes equations

We consider stochastic Navier-Stokes equations in a 2D-bounded domain with the Navier with friction boundary condition. We establish the existence and the uniqueness of the solutions and study the vanishing viscosity limit. More precisely, we prove that solutions of stochastic Navier-Stokes equations converge, as the viscosity goes to zero, to solutions of the corresponding stochastic Euler equations.

Geometric Regularity Criteria for Incompressible Navier--Stokes Equations with Navier Boundary Conditions

We study the regularity criteria for weak solutions to the $3D$ incompressible Navier--Stokes equations in terms of the geometry of vortex structures, taking into account the boundary effects. A boundary regularity theorem is proved on regular domains with a class of oblique derivative boundary conditions, providing that the vorticity of the fluid is coherently aligned. In particular, we establish the boundary regularity on round balls, half-spaces and right circular cylindrical ducts, subject to the classical Navier and kinematic boundary conditions.

On the Dynamics of Navier-Stokes and Euler Equations

This is a rather comprehensive study on the dynamics of Navier-Stokes and Euler equations via a combination of analysis and numerics. We focus upon two main aspects: (a). zero viscosity limit of the spectra of linear Navier-Stokes operator, (b). heteroclinics conjecture for Euler equation, its numerical verification, Melnikov integral, and simulation and control of chaos. Besides Navier-Stokes and Euler equations, we also study two models of them.

Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers

We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain corresponding constants of the motion.

Rigorous derivation of the compressible Navier-Stokes equations from the two-fluid Navier-Stokes-Maxwell equations

In this paper, we rigorously derive the compressible one-fluid Navier-Stokes equation from the scaled compressible two-fluid Navier-Stokes-Maxwell equations locally in time under the assumption that the initial data are well prepared. We justify the singular limit by proving the uniform decay of the error system, which is obtained by elaborate energy estimates.

Attractor Dimensions of Three-Dimensional Navier-Stokes-$α$ Model for Fast Rotating Fluids on Generic-Period Domains: Comparison with Navier-Stokes Equations

The three-dimensional Navier-Stokes-$α$ model for fast rotating geophysical fluids is considered. The Navier-Stokes-$α$ model is a nonlinear dispersive regularization of the exact Navier-Stokes equations obtained by Lagrangian averaging and tend to the Navier-Stokes equations as $α\rightarrow 0^+$. We estimate upper bounds for the dimensions of global attractors and study the dependence of the dimensions on the parameter $α$. All the estimates are uniform in $α$, and our estimate of attractor dimensions remain finite when $α\rightarrow 0^+$.

On Caccioppoli's inequalities of Stokes equations and Navier-Stokes equations near boundary

We study Caccioppoli's inequalities of the non-stationary Stokes equations and Navier-Stokes equations. Our analysis is local near boundary and we prove that, in contrast to the interior case, the Caccioppoli's inequalities of the Stokes equations and the Navier-Stokes equations, in general, fail near boundary.

Reductions of the Navier-Stokes-Allen-Cahn and the Navier-Stokes-Cahn-Hilliard equations

This paper studies two well-known models for two-phase fluid flow at constant temperature, the isothermal Navier-Stokes-Allen-Cahn and the isothermal Navier-Stokes-Cahn-Hilliard equations, both of which consist of equations for the (total) fluid density rho, the (mass-averaged)velocity u and the concentration (of one of the phases,) c. Assuming in either case that both phases are incompressible with different densities, each of the models is shown to reduce to a system of evolution equations in rho and u alone. In the case of the Navier-Stokes-Allen-Cahn model, this reduced system is the class…

Algebraic decay rates for 3D Navier-Stokes and Navier-Stokes-Coriolis equations in $ \dot{H}^{\frac{1}{2}}$

An algebraic upper bound for the decay rate of solutions to the Navier-Stokes and Navier-Stokes-Coriolis equations in the critical space $\dot{H} ^{\frac{1}{2}} (\mathbb{R} ^3)$ is derived using the Fourier Splitting Method. Estimates are framed in terms of the decay character of initial data, leading to solutions with algebraic decay and showing in detail the roles played by the linear and nonlinear parts.

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.

Convergence of the stochastic Navier-Stokes-$α$ solutions toward the stochastic Navier-Stokes solutions

Loosely speaking, the Navier-Stokes-$α$ model and the Navier-Stokes equations differ by a spatial filtration parametrized by a scale denoted $α$. Starting from a strong two-dimensional solution to the Navier-Stokes-$α$ model driven by a multiplicative noise, we demonstrate that it generates a strong solution to the stochastic Navier-Stokes equations under the condition $α$ goes to 0. The initially introduced probability space and the Wiener process are maintained throughout the investigation, thanks to a local monotonicity property that abolishes the use of Skorokhod's theorem. High spatial re…

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.

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

A sufficient condition of regularity for solutions to the Navier-Stokes equations is proved. It generalizes the so-called $L_{3,\infty}$-case.

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.