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.     

Blow-up criteria of smooth solutions to the 3D Boussinesq system with zero viscosity in a bounded domain

We prove that a smooth solution of the 3D Boussinesq system with zero viscosity in a bounded domain breaks down, if a certain norm of vorticity blows up at the same time. Here this norm is weaker than the bmo-norm.     

Optimal weighted estimates of the flows in exterior domains

In this paper, we prove some decay properties of global solutions for the Navier-Stokes equations in an exterior domain Ω⊂Rⁿ, n=2,3.When a domain has a boundary, the pressure term is troublesome since we do not have enough information on the pressure near the boundary. To overcome this difficulty, by multiplying a special form of test functions, we obtain an integral equation. He-Xin (2000) [12] first introduced this method and then Bae-Jin (2006, 2007) [1] and [13] modified their method to obtain better decay rates. Also, Bae-Roh (2009) [11] improved Bae-Jin’s results. Unfortunately, their results were not optimal, because there exists an unpleasant positive small δ in their rates.In this paper, we obtain the following optimal rate without δ,

     

On weighted-norm estimates for nonstationary incompressible Navier-Stokes flows in a 3D exterior domain

We study the time-decay of weighted norms of weak and strong solutions to the Navier-Stokes equations in a 3D exterior domain. Moment estimates for weak solutions and weighted Lq-estimates for strong solutions are deduced, both of which seem to be optimal. The relation is discussed between the space-time decay and the vanishing of the total net force exerted by the fluid to the body. A class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish.     

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.     

Asymptotic behavior for the Navier-Stokes equations with nonzero external forces

We estimate the asymptotic behavior for the Stokes solutions, with external forces first. We found that if there are external forces, then the energy decays slowly even if the forces decay quickly. Then, we also obtain the asymptotic behavior in the temporal-spatial direction for weak solutions of the Navier-Stokes equations. We also provide a simple example of external forces which shows that the Stokes solution does not decay quickly.     

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.     

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.     

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.     

Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations

We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. The coupling arises from a drag force exerted by each other. We establish existence of global weak solutions for the system in two and three dimensions. Furthermore, we obtain the existence and uniqueness result of global smooth solutions for dimension two. In case of three dimensions, we also prove that strong solutions exist globally in time for the Vlasov-Stokes system.     

Asymptotic stability of Oseen vortices for a density-dependent incompressible viscous fluid

In the analysis of the long-time behavior of two-dimensional incompressible viscous fluids, Oseen vortices play a major role as attractors of any homogeneous solution with integrable initial vorticity [T. Gallay, C.E. Wayne, Global stability of vortex solutions of the two-dimensional Navier–Stokes equation, Commun. Math. Phys. 255 (1) (2005) 97–129]. As a first step in the study of the density-dependent case, the present paper establishes the asymptotic stability of Oseen vortices for slightly inhomogeneous fluids with respect to localized perturbations.     

Local well-posedness and regularity criterion for the density dependent incompressible Euler equations

In this paper, local well-posedness for the density dependent incompressible Euler equations is established in Besov spaces. We also obtain a blow-up criterion for the corresponding solution.     

Fractional Navier-Stokes equations and a Hölder-type inequality in a sum of singular spaces

In this paper we study the local well-posedness of the fractional Navier-Stokes system with initial data belonging to a sum of two pseudomeasure-type spaces denoted by PM^a,b:=PM^a+PM^b. The proof requires showing a Hölder-type inequality in PM^a,b, as well as establishing estimates of the semigroup generated by the fractional power of Laplacian (−Δ)^γ on these spaces.     

Stability to the global large solutions of 3-D Navier-Stokes equations

In this paper, we consider the stability to the global large solutions of 3-D incompressible Navier-Stokes equations in the anisotropic Sobolev spaces. In particular, we proved that for any , given a global large solution v∈C([0,∞);H0,s₀(R³)∩L³(R³)) of (1.1) with and a divergence free vector satisfying for some sufficiently small constant depending on , v, and , (1.1) supplemented with initial data v(0)+w₀ has a unique global solution in u∈C([0,∞);H0,s₀(R³)) with ∇u∈L²(R⁺,H0,s₀(R³)). Furthermore, uh is close enough to vh in C([0,∞);H0,s(R³)).     

Regularity criteria for the 3D generalized MHD equations in terms of vorticity

We prove regularity criteria for the 3D generalized MHD equations. These criteria impose assumptions on the vorticity only. In addition, we also prove a result of global existence for smooth solution under some special conditions.