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     

The Stokes equations in layer domains on spaces of bounded and integrable functions

The Stokes equations in a layer domain are investigated. For the case of a two‐dimensional layer it is shown by resolvent estimates that there exists a semigroup of angle on solenoidal subspaces of and L ₁. Furthermore, the stationary problem for dimension two is solvable in these spaces. It is also shown that both results cannot be extended to three and higher dimensions in the same setting.     

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.     

On a non‐stationary fluid flow problem in an infinite periodic pipe

We study linearized, non‐stationary Navier–Stokes type equations with the given flux in an infinite pipe periodic of period length L with respect to . The existence and uniqueness of the solution is proved. Moreover, the convergence of the solution in a finite pipe of length to the L‐periodic solution as is investigated.     

The Green function for the mixed problem for the linear Stokes system in domains in the plane

We construct the Green function for the mixed boundary value problem for the linear Stokes system in a two‐dimensional Lipschitz domain.     

On the method of Da Prato and Debussche for the 3D stochastic Navier Stokes equations

3D stochastic Navier-Stokes equations with a suitable nondegenerate noise are considered. Following a method introduced by Da Prato and Debussche, it is proved that every Markov process associated to the equations has a Strong Feller like continuity property with respect to initial conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Mild solutions of stochastic Navier‐Stokes equation with jump noise in ‐spaces

In this paper, we study solvability of the local mild solution of stochastic Navier‐Stokes equation with jump noise in ‐spaces. This research work has mainly carried out by exploiting some interesting works of Tosio Kato.     

Solutions to the Navier–Stokes equations with mixed boundary conditions in two‐dimensional bounded domains

In this paper we consider the system of the non‐steady Navier–Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y, respectively, to be the space of “possible” solutions of this problem and the space of its data. We define the operator and formulate our problem in terms of operator equations. Let and be the Fréchet derivative of at . We prove that is one‐to‐one and onto Y. Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. The next result proved in the Appendix of this paper is W^{2, 2}‐regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.     

The Stokes problem in a periodic layer

We construct and justify the asymptotic representation at infinity of solutions to the Stokes problem in a three‐dimensional domain with periodic geometric structure, which is bounded in one dimension, e.g., a periodically perforated layer. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Navier–Stokes equations with external forces in time‐weighted Besov spaces

We show existence theorem of global mild solutions with small initial data and external forces in the time‐weighted Besov space which is an invariant space under the change of scaling. The result on local existence of solutions for large data is also discussed. Our method is based on the ‐ estimate of the Stokes equations in Besov spaces. Since we construct the global solution by means of the implicit function theorem, as a byproduct, its stability with respect to the given data is necessarily obtained.     

Navier‐Stokes equations,Haar wavelets and Reynolds numbers

The paper deals with global solutions of Navier‐Stokes equations with infrared‐damped initial data in the context of Haar wavelets and function spaces of type where .     

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     

A justification for the thin film approximation of Stokes flow with surface tension

In the free boundary problem of Stokes flow driven by surface tension, we pass to the limit of small layer thickness. It is rigorously shown that in this limit the evolution is given by the well-known thin film equation. The main techniques are appropriate scaling and uniform energy estimates in Sobolev spaces of sufficiently high order, based on parabolicity.     

Exterior Navier–Stokes flows for bounded data

We prove unique existence of mild solutions on for the Navier–Stokes equations in an exterior domain in , subject to the non‐slip boundary condition.     

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,

     

The D incompressible Navier–Stokes equations with partial hyperdissipation

The three‐dimensional incompressible Navier–Stokes equations with the hyperdissipation always possess global smooth solutions when . Tao [6] and Barbato, Morandin and Romito [1] made logarithmic reductions in the dissipation and still obtained the global regularity. This paper makes a different type of reduction in the dissipation and proves the global existence and uniqueness in the H¹‐functional setting.     

On the uniqueness in critical spaces for compressible Navier-Stokes equations

We show some new uniqueness results for compressible flows with data having critical regularity. In the barotropic case, uniqueness is stated whenever the space dimension N satisfies N ≥ 2, and in the polytropic case, whenever N ≥ 3. The endpoints N = 2 in the barotropic case and N = 3 in the polytropic case were left open in [4], [5] and [6].     

Fractional powers of the Stokes operator with boundary conditions involving the pressure

Since the pioneer work of Leray [23] and Hopf [17], Stokes and Navier–Stokes problems have been often studied with Dirichlet boundary condition. Nevertheless, in the opinion of engineers and physicists such a condition is not always realistic in industrial and applied problems of origin. Thus arises naturally the need to carry out a mathematical analysis of these systems with different boundary conditions, which best represent the underlying fluid dynamic phenomenology. Based on the study of the complex and fractional powers of the Stokes operator with pressure boundary condition, we carry out a systematic treatment of the Stokes problem with the corresponding boundary conditions in ‐spaces.     

Remarks on Liouville type theorems for the 3D steady axially symmetric Navier–Stokes equations

In this note, we investigate the 3D steady axially symmetric Navier–Stokes equations, and obtain Liouville type theorems if the velocity or the vorticity satisfies some a priori decay assumptions, which improve the recent result of Seregin's in [15].