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.     

The 2D Boussinesq equations with vertical viscosity and vertical diffusivity

This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the Lr-norm of the vertical velocity v for any 1<r<∞ is globally bounded and that the L^∞-norm of v controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace δ(−Δ) for δ>0 would guarantee the global regularity of classical solutions.     

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.     

Large time behavior for the incompressible magnetohydrodynamic equations in half‐spaces

The weighted L^r‐asymptotic behavior of the strong solution and its first‐order spacial derivatives to the incompressible magnetohydrodynamic (MHD) equations is established in a half‐space. Further, the L^∞‐decay rates of the second‐order spatial derivatives of the strong solution are derived by using the Stokes solution formula and employing a decomposition for the nonlinear terms in MHD equations. Copyright © 2014 John Wiley & Sons, Ltd.     

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)).     

Liouville type theorems for the Euler and the Navier–Stokes equations

We prove Liouville type theorems for weak solutions of the Navier–Stokes and the Euler equations. In particular, if the pressure satisfies p∈L¹(0,T;L¹(RN)) with , then the corresponding velocity should be trivial, namely v=0 on RN×(0,T). In particular, this is the case when p∈L¹(0,T;Hq(RN)), where Hq(RN), q∈(0,1], the Hardy space. On the other hand, we have equipartition of energy over each component, if p∈L¹(0,T;L¹(RN)) with . Similar results hold also for the magnetohydrodynamic equations.     

Local regularity criteria of a suitable weak solution to MHD equations

We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are H¨older continuous near boundary provided that the scaled mixed L_(x,t)~(p,q) -norm of the velocity vector field with 3/p + 2/q ≤ 2,2 q ∞ is sufficiently small near the boundary. Also, we will investigate that for this solution u ∈ L_(x,t)~(p,q) with 1≤3/p+2/q≤3/2, 3 p ∞, the Hausdorff dimension of its singular set is no greater than max{p, q}(3/p+2/q-1).     

Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations in R³. We prove the global existence of the smooth solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H³-framework. Moreover, if additionally the initial data belong to L^p with , the optimal convergence rates of the solutions in L^q-norm with 2≤q≤6 and its spatial derivatives in L²-norm are obtained.     

On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Ω=Ω₀×(0,L)∈R³. We show existence of a solution , p>3, where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of . The term u⋅∇w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (vn,ρn) that is bounded in and satisfies the Cauchy condition in a larger space L_∞(0,L;L₂(Ω₀)) what enables us to deduce that the weak limit of a subsequence of (vn,ρn) is in fact a strong solution to our problem.     

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 existence of local strong solutions for the Navier-Stokes equations in completely general domains

There are only very few results on the existence of unique local in time strong solutions of the Navier-Stokes equations for completely general domains Ω⊆R³, although domains with edges and corners, bounded or unbounded, are very important in applications. The reason is that the L^q-theory for the Stokes operator A is available in general only in the Hilbert space setting, i.e., with q=2. Our main result for a general domain Ω is optimal in a certain sense: Consider an initial value and a zero external force. Then the condition is sufficient and necessary for the existence of a unique local strong solution u∈L⁸(0,T;L⁴(Ω)) in some interval [0,T), 0<T≤∞, with u(0)=u₀, satisfying Serrin’s condition . Note that Fujita-Kato’s sufficient condition u₀∈D(A^1/4) is strictly stronger and therefore not optimal.     

Local well-posedness for the homogeneous Euler equations

We introduce Triebel-Lizorkin-Lorentz function spaces, based on the Lorentz L^p,q-spaces instead of the standard L^p-spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of inviscid incompressible fluid in Rⁿ,n≥2. As a corollary we obtain global existence of solutions to the 2D Euler equations in the Triebel-Lizorkin-Lorentz space. For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator estimates in our spaces. The key methods of proof used are the Littlewood-Paley decomposition and the paradifferential calculus by J.M. Bony.     

On strong convergence to 3D steady vortex sheets

In this paper, we first establish a strong convergence criterion of approximate solutions for the 3D steady incompressible Euler equations. For axisymmetric flows, under the assumption that the vorticity is of one sign and uniformly bounded in L¹ space, we obtain a sufficient and necessary condition for the strong convergence in of approximate solutions. Furthermore, for one-sign and L¹-bounded vorticity, it is shown that if a sequence of approximate solutions concentrates at an isolated point in (r,z)-plane, then the concentration point can appear neither in the region near the axis (including the symmetry axis itself) nor in the region far away from the axis. Finally, we present an example of approximates solutions which converge strongly in by using Hill's spherical vortex.     

Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid

There are two results within this paper. The one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non-Newtonian fluid on 2D bounded domains, for which the solution to each initial value could be non-unique. The other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from Ωm to Ω=R×(−L,L), where is an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that Ωm→Ω as m→+∞. That is, let A and Am be the global attractors of the fluid corresponding to Ω and Ωm, respectively, we establish that for any neighborhood O(A) of A, the global attractor Am enters O(A) if m is large enough.     

A sufficiency class for global (in time) solutions to the 3D Navier-Stokes equations

Let Ω be an open domain of class C² contained in R³, let L²(Ω)³ be the Hilbert space of square integrable functions on Ω and let H[Ω]?H be the completion of the set, , with respect to the inner product of L²(Ω)³. A well-known unsolved problem is that of the construction of a sufficient class of functions in H which will allow global, in time, strong solutions to the three-dimensional Navier-Stokes equations. These equations describe the time evolution of the fluid velocity and pressure of an incompressible viscous homogeneous Newtonian fluid in terms of a given initial velocity and given external body forces. In this paper, we use the analytic nature of the Stokes semigroup to construct an equivalent norm for H, which provides strong bounds on the nonlinear term. This allows us to prove that, under appropriate conditions, there exists a number u₊, depending only on the domain, the viscosity, the body forces and the eigenvalues of the Stokes operator, such that, for all functions in a dense set D contained in the closed ball B(Ω)?B of radius in H, the Navier-Stokes equations have unique, strong, solutions in C¹((0,∞),H).     

On the Helmholtz decomposition in general unbounded domains

It is well known that the Helmholtz decomposition of L^q-spaces fails to exist for certain unbounded smooth planar domains unless q = 2, see [2], [9]. As recently shown [6], the Helmholtz projection does exist for general unbounded domains of uniform C²-type in if we replace the space L^q, 1 < q < ∞, by L² ∩ L^q for q > 2 and by L^q + L² for 1 < q < 2. In this paper, we generalize this new approach from the three-dimensional case to the n-dimensional case, n ≥ 2. By these means it is possible to define the Stokes operator in arbitrary unbounded domains of uniform C²-type. Received: 15 February 2006     

Stability of plane Couette flows with respect to small periodic perturbations

We consider the plane Couette flow v₀=(x_n,0,…,0) in the infinite layer domain , where n≥2 is an integer. The exponential stability of v₀ in Lⁿ is shown under the condition that the initial perturbation is periodic in (x₁,…,x_n−1) and sufficiently small in the Lⁿ-norm.     

Viscosity vanishing limit of the nonlinear pipe magnetohydrodynamic flow with diffusion

We establish viscosity vanishing limit of the nonlinear pipe magnetohydrodynamic flow by the mathematical validity of the Prandtl boundary layer theory with fixed diffusion. The convergence is verified under various Sobolev norms, including the L^∞(H¹) norm.     

A splitting method for the nonlinear Schrödinger equation

We introduce a splitting method for the semilinear Schrödinger equation and prove its convergence for those nonlinearities which can be handled by the classical well-posedness L²(Rd)-theory. More precisely, we prove that the scheme is of first order in the L²(Rd)-norm for H²(Rd)-initial data.     

Pullback attractors for a non-autonomous incompressible non-Newtonian fluid

This paper studies the pullback asymptotic behavior of solutions for a non-autonomous incompressible non-Newtonian fluid in two-dimensional (2D) bounded domains. We first prove the existence of pullback attractors AV in space V (has H²-regularity, see notation in Section 2) and AH in space H (has L²-regularity) for the cocycle corresponding to the solutions of the fluid. Then we verify the regularity of the pullback attractors by showing AV=AH, which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data.