Asymptotic energy concentration in the phase space of the weak solutions to the Navier-Stokes equations

We study the asymptotic behavior of the energy of weak solutions of Navier-Stokes equations as t→∞. We characterize the space of the initial data which causes a concentration of the kinetic energy in the phase space. Moreover, an explicit convergence rate is obtained.     

A removable isolated singularity theorem for the stationary Navier-Stokes equations

We show that an isolated singularity at the origin 0 of a smooth solution (u,p) of the stationary Navier-Stokes equations is removable if the velocity u satisfies u∈Lⁿ or |u(x)|=o(|x|^-1) as x→0. Here n?3 denotes the dimension. As a byproduct of the proof, we also obtain a new interior regularity theorem.     

On partial regularity for weak solutions to the Navier-Stokes equations

In this paper, we study the partial regularity of the general weak solution u∈L∞(0,T;L2(Ω))∩L2(0,T;H1(Ω)) to the Navier-Stokes equations, which include the well-known Leray-Hopf weak solutions. It is shown that there is a absolute constant ε such that for the weak solution u, if either the scaled local L^q(1?q?2) norm of the gradient of the solution, or the scaled local ) norm of u is less than ε, then u is locally bounded. This implies that the one-dimensional Hausdorff measure is zero for the possible singular point set, which extends the corresponding result due to Caffarelli et al. (Comm. Pure Appl. Math. 35 (1982) 717) to more general weak solution.     

On the Navier problem for the stationary Navier-Stokes equations

The Navier problem is to find a solution of the steady-state Navier-Stokes equations such that the normal component of the velocity and a linear combination of the tangential components of the velocity and the traction assume prescribed value a and s at the boundary. If Ω is exterior it is required that the velocity converges to an assigned constant vector u₀ at infinity. We prove that a solution exists in a bounded domain provided ‖a_‖L²(∂Ω) is less than a computable positive constant and is unique if ‖a_‖W1/2,2(∂Ω)+‖s_‖L²(∂Ω) is suitably small. As far as exterior domains are concerned, we show that a solution exists if ‖a_‖L²(∂Ω)+‖a−u₀⋅n_‖L²(∂Ω) is small.     

Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces

First, the existence and structure of uniform attractors in H is proved for nonautonomous 2D Navier-Stokes equations on bounded domain with a new class of distribution forces, termed normal in (see Definition 3.1), which are translation bounded but not translation compact in . Then, the properties of the kernel section are investigated. Last, the fractal dimension is estimated for the kernel sections of the uniform attractors obtained.     

Solvability of the Navier-Stokes equations on manifolds with boundary

This paper deals with the solvability of the Navier-Stokes equations on manifolds with boundary. In particular, we concentrate on the inhomogeneous slip boundary condition. Our formulation of the equations takes into account a curvature term which results from a proper derivation of the Navier-Stokes equations. This term has not been considered in prior work. During the work on this version, the author received technical support through a fellowship of the DFG     

Existence of global weak solutions for Navier-Stokes equations with large flux

Global existence of weak solutions to the Navier-Stokes equations in a cylindrical domain under boundary slip conditions and with inflow and outflow is proved. To prove the energy estimate, crucial for the proof, we use the Hopf function. This makes it possible to derive an estimate such that the inflow and outflow need not vanish as t→∞. The proof requires estimates in weighted Sobolev spaces for solutions to the Poisson equation. Our result is the first step towards proving the existence of global regular special solutions to the Navier-Stokes equations with inflow and outflow.     

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

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

On regularity criteria for the n-dimensional Navier-Stokes equations in terms of the pressure

We study the Cauchy problem for the n-dimensional Navier-Stokes equations (n?3), and prove some regularity criteria involving the integrability of the pressure or the pressure gradient for weak solutions in the Morrey, Besov and multiplier spaces.     

A priori estimates in terms of the maximum norm for the solutions of the Navier-Stokes equations

In this paper, we consider the Cauchy problem for the incompressible Navier-Stokes equations with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial velocity field. For illustrative purposes, we first derive corresponding a priori estimates for certain parabolic systems. Because of the pressure term, the case of the Navier-Stokes equations is more difficult, however.     

Stochastic Navier-Stokes Equations

The purpose of this article is to survey some results related to the theory of stochastic Navier-Stokes equations (SNSE). The interest of SNSE arises from modelling turbulence. We begin to show how SNSE can be introduced intuitively from the random motion of particles. We then review briefly the deterministic theory and present the main core of existence theory for NSE. We also discuss uniqueness issues. We end up by showing how the splitting-up method provides a useful constructive approach to existence, and by presenting some extensions, like weakening assumptions or considering the special case of small initial data.These lectures have been delivered in the Frontiers Lectures program of Texas A&M University. The author thanks Prof. Goong Chen for his invitation.     

A new proof of partial regularity of solutions to Navier-Stokes equations

In this paper we give a new proof of the partial regularity of solutions to the incompressible Navier-Stokes equation in dimension 3 first proved by Caffarelli, Kohn and Nirenberg. The proof relies on a method introduced by De Giorgi for elliptic equations. This work was supported in part by NSF Grant DMS-0607953.     

Attractors for nonautonomous 2D Navier-Stokes equations with less regular symbols

We introduce a new class of functions satisfying normal Condition (C*), denoted by , which are translation bounded but not translation compact — in particular, which are more general than normal functions (see [S.S. Lu, H.Q. Wu, C.K. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005) 701-719] for the definition), denoted by . Furthermore, we prove the existence of uniform attractors for 2D Navier-Stokes equations with external forces belonging to in .     

Tuning the mesh of a mixed method for the stream function Vorticity formulation of the Navier-Stokes equations

Summary In this article we study a new mixed method for the Stokes and Navier-Stokes equations. The method uses two meshes, one very fine for and a coarser one for . Error estimates show that boundary layers do not require to refine the mesh for the stream function as much as for the vorticity when the Reynolds number is large. We prove estimates and study implementation problems.     

Stochastic Navier-Stokes equations

We construct a solution to stochastic Navier-Stokes equations in dimension n4 with the feedback in both the external forces and a general infinite-dimensional noise. The solution is unique and adapted to the Brownian filtration in the 2-dimensional case with periodic boundary conditions or, when there is no feedback in the noise, for the Dirichlet boundary condition. The paper uses the methods of nonstandard analysis.The research of this author was supported by an SERC Grant.     

Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum

This paper is concerned with existence of global weak solutions to a class of compressible Navier-Stokes equations with density-dependent viscosity and vacuum. When the viscosity coefficient μ is proportional to ρθ with , a global existence result is obtained which improves the previous results in Fang and Zhang (2004) [4], Vong et al. (2003) [27], Yang and Zhu (2002) [30]. Here ρ is the density. Moreover, we prove that the domain, where fluid is located on, expands outwards into vacuum at an algebraic rate as the time grows up due to the dispersion effect of total pressure. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., θ=1, γ=2).