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 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.     

On cylindrical symmetric flows through pipe-like domains

We prove existence of cylindrical symmetric solutions to the steady Navier-Stokes equations in bounded pipe-like domains in with the slip boundary conditions. The result is shown for any large flows across the boundary assuming only a geometrical constraint on the shape of the domain which is independent of data. The simply connectedness of the domain is not required. The technique is based on a reformulation of the original problem and delivers us a new type of estimates in the Hölder spaces for this class of the solutions.     

Two-dimensional Navier-Stokes flow in unbounded domains

Dedicated to Professor Hiroki Tanabe on the occasion of his sixtieth birthday     

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.     

Existence and uniqueness of the solution to a free boundary-value problem with thermocapillary convection in an unbounded domain

A free boundary-value problem for a nonisothermal flow of two heavy viscous incompressible capillary liquids descibing a slot coating process is mathematically studied. The problem is stationary and two-dimensional and the Boussinesq approximation for the Navier-Stokes equations is used. The existence and uniqueness of a weak solution is proved in weighted Holder spaces for small data.     

On analyticity of the ‐Stokes semigroup for some non‐Helmholtz domains

Consider the Stokes equations in a sector‐like C ³ domain . It is shown that the Stokes operator generates an analytic semigroup in for . This includes domains where the ‐Helmholtz decomposition fails to hold. To show our result we interpolate results of the Stokes semigroup in and L ² by constructing a suitable non‐Helmholtz projection to solenoidal spaces.     

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.     

A note on the reflection principle for the biharmonic equation and the Stokes system

In this note, we discuss the reflection principle of the Stokes system in a half space for the threedimensional case, and of the biharmonic equation. Admitting different boundary conditions, we use the reflection principle to prove uniqueness of solutions of the Stokes system or the biharmonic equation in weightedLq-spaces     

On strong solutions of the Stokes equations in exterior domains

We construct strong solutionsu, p/of the general nonhomogeneous Stokes equations -u + p=f inG, ·u=g inG, u= on in an exterior domainG ⁿ (n3) with boundary of class C². Our approach uses a localization technique: With the help of suitable cut-off functions and the solution of the divergence equation ·=g inG, = 0 on , the exterior domain problem is reduced to the entire space problem and an interior problem.     

The stationary Oseen equations in an exterior domain: An approach in weighted Sobolev spaces

In this work, we study the linearized Navier–Stokes equations in an exterior domain of R³${\mathbb{R}}^3$ at the steady state, that is, the Oseen equations. We are interested in the existence and the uniqueness of weak, strong and very weak solutions in L^p$L^p$-theory which makes our work more difficult. Our analysis is based on the principle that linear exterior problems can be solved by combining their properties in the whole space R³${\mathbb{R}}^3$ and the properties in bounded domains. Our approach rests on the use of weighted Sobolev spaces.     

On Stokes operators with variable viscosity in bounded and unbounded domains

We consider a generalization of the Stokes resolvent equation, where the constant viscosity is replaced by a general given positive function. Such a system arises in many situations as linearized system, when the viscosity of an incompressible, viscous fluid depends on some other quantities. We prove that an associated Stokes-like operator generates an analytic semi-group and admits a bounded H _∞ -calculus, which implies the maximal L ^q -regularity of the corresponding parabolic evolution equation. The analysis is done for a large class of unbounded domains with -boundary for some r > d with r ≥ q, q′. In particular, the existence of an L ^q -Helmholtz projection is assumed.     

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.     

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.     

Stability of the magnetic Couette-Taylor flow

In this paper we consider the magnetic Couette-Taylor problem, that is, a conducting fluid between two infinite rotating cylinders, subject to a magnetic field parallel to the rotation axis. This configuration admits an equilibrium solution of the form It is shown that this equilibrium is Ljapounov stable under small perturbations in where provided that the parameters a, b, , are small. The methods of proof are a combination of an energy method, based on Bloch space analysis and small data techniques.Received: February 5, 2003; revised: September 29, 2003Dedicated to Prof. H. Amann on the occasion of his 65. birthday     

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.