The Cosserat eigenfunctions for the elliptic exterior problem with applications to thermoelasticity and Stokes flow

The Cosserat eigenvalue problem for the elliptic exterior is considered. It is shown that the only Cosserat eigenvalue different from the infinite multiple eigenvalues and is which also has an infinite multiplicity. The orthogonal basis of the eigenspace corresponding to is constructed. Application to thermoelasticity and Stokes flow – extensional and shear – past a rigid elliptical cylinder are presented and agreement is obtained with classical solutions for a circle. The fact that the solutions consist of only two eigenfunctions reveals the efficiency of the method.Received: February 27, 2003     

Weak and Strong Solutions for the Stokes Approximation of Non-homogeneous Incompressible Navier-Stokes Equations

In this paper,the Dirichlet problem of Stokes approximate of non-homogeneous incompressibleNavier-Stokes equations is studied.It is shown that there exist global weak solutions as well as global andunique strong solution for this problem,under the assumption that initial density ρ_0(x)is bounded away from0 and other appropriate assumptions(see Theorem 1 and Theorem 2).The semi-Galerkin method is applied toconstruct the approximate solutions and a prior estimates are made to elaborate upon the compactness of theapproximate solutions.     

The regularity of the Stokes operator and the Fujita-Kato approach to the Navier-Stokes initial value problem in Lipschitz domains

We study the regularity of the Navier-Stokes equations in arbitrary Lipschitz domains.     

Asymptotic behavior for the Stokes flow and Navier-Stokes equations in half spaces

Using the solution formula in Ukai (1987) [27] for the Stokes equations, we find asymptotic profiles of solutions in (n?2) for the Stokes flow and non-stationary Navier-Stokes equations. Since the projection operator is unbounded, we use a decomposition for P(u⋅∇u) to overcome the difficulty, and prove that the decay rate for the first derivatives of the strong solution u of the Navier-Stokes system in is controlled by for any t>0.     

Homogenization of a non-stationary Stokes flow in porous medium including a layer

We study the behavior of the solution to the non-stationary Stokes equations in a porous medium with characteristic size of the pores ε and containing a thin fissure {0?xn?η} of width η. The limit when ε and η tend to zero gives the homogenized behavior of the flow, which depends on the comparison between ε and η.     

Lower bounds for the blow-up time in a temperature dependent Navier-Stokes flow

In this paper we consider two different initial-boundary value problems in temperature dependent viscous flow when the temperature equation has a nonlinear heat source term. When blow-up occurs we derive lower bounds for the blow-up time in each case.     

On the interior regularity of weak solutions to the non-stationary Stokes system

In this paper, we prove that any weak solution to the non-stationary Stokes system in 3D with right hand side -div f satisfying (1.4) below, belongs to C( ]0, T[; C ^α (Ω)). The proof is based on Campanato-type inequalities and the existence of a local pressure introduced in Wolf [13]. Proc. Conference “Variational analysis and PDE’s”. Intern. Centre “E. Majorana”, Erice, July 5–14, 2006.     

Exact solution of the Navier-Stokes equations for the oscillating flow in a duct of a cross-section of right-angled isosceles triangle

The Navier-Stokes equations have been solved in order to obtain an analytical solution of the fully developed laminar flow in a duct having a cross section of a right-angled, isosceles triangle. We obtained a solution for the case of oscillating pressure gradient flow. The pulsating flow is obtained by the superposition of the steady and oscillating pressure gradient solutions.     

Inverse viscosity problem for the Navier-Stokes equation

We consider an inverse problem of determining a viscosity coefficient in the Navier-Stokes equation by observation data in a neighborhood of the boundary. We prove the Lipschitz stability by the Carleman estimates in Sobolev spaces of negative order.     

On the time-periodic problem for the Stokes system in domains with cylindrical outlets to infinity

We study the time-periodic Stokes problem in the domain with cylindrical outlets to infinity in weighted function spaces. We prove that there exists a unique solution with prescribed fluxes over the sections of outlets to infinity and that, in each outlet, this solution tends to the corresponding time-periodic Poiseuille flow. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 177–195, April–June, 2007.     

Spatial behaviour of solutions in the plane Stokes flow

Some new methods are described in the study of the spatial behaviour of solutions in the slow flow of an incompressible viscous fluid along a semi-infinite strip subject to zero velocity on the lateral sides, a prescribed time-dependent specified velocity on the end and zero initial conditions. These methods provide an improved estimated decay rate over those previously predicted on the subject.     

Application of homogenization theory related to Stokes flow in porous media

We consider applications, illustration and concrete numerical treatments of some homogenization results on Stokes flow in porous media. In particular, we compute the global permeability tensor corresponding to an unidirectional array of circular fibers for several volume-fractions. A 3-dimensional problem is also considered.     

A basic inequality for the Stokes operator related to the Navier boundary condition

We show that ‖Au+Δu_‖L²(Ωε)?C(ε‖∇u_‖L²(Ωε)+‖u_‖L²(Ωε)), where Ωε is a thin domain in R³ of depth ε, the vector field u belongs to the domain of A, which is the Stokes operator for divergence-free vector fields on Ωε satisfying the Navier boundary condition.     

三维区域中的Navier-Stokes方程的初边值问题

本文在初边值适当小的假设下，建立了任意三维区域中Navier－Stokes方程初边值问题整体强解的存在性定理．     

Steady compressible Navier-Stokes flow in a square

We investigate a steady flow of compressible fluid with inflow boundary condition on the density and slip boundary conditions on the velocity in a square domain Q∈R². We show existence if a solution 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 constant flow . In order to show the existence of the solution we adapt the techniques known from the theory of weak solutions. We apply the method of elliptic regularization and a fixed point argument.     

Axial eigenfunctions and inhomogeneous solutions for the three-dimensional Stokes system with an application to natural convection in a cylinder

This paper presents two contributions to the analysis of three-dimensional slow viscous flows in cylinders of circular section. First the vector axial eigenfunctions for this geometry, namely those that satisfy homogeneous boundary conditions on the flat end walls, are derived. Secondly a method is presented to find particular solutions to the inhomogeneous Stokes equations in this geometry. These new results, together with some results obtained earlier, are used to analyse slow natural convection in a vertical cylinder completely filled with a viscous liquid. The fluid motion is generated by the differential heating of the walls of the cylinder. The natural convection flow field is shown to be a superposition of an inhomogeneous field, the fields generated by the vector eigenfunctions and a Stokes flow field. A by-product of this work has been the identification of constraints on the boundary data that have to be satisfied in order for the eigenfunction expansions to work; this knowledge will be useful when attempts are made to prove the completeness of these Stokes flow eigenfunctions.Received: June 30, 2003; revised: February 26, 2004     

Pressure projection stabilized finite element method for Stokes problem with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with the Stokes operator. The H¹ and L² error estimates for the velocity and the L² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.     

A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem

A fully discrete stabilized finite-element method is presentedfor the two-dimensional time-dependent Navier–Stokes problem.The spatial discretization is based on a finite-element spacepair (X_h, M_h) for the approximation of the velocity and thepressure, constructed by using the Q₁ – P₀ quadrilateralelement or the P₁ – P₀ triangular element; the time discretizationis based on the Euler semi-implicit scheme. It is shown thatthe proposed fully discrete stabilized finite-element methodresults in the optimal order error bounds for the velocity andthe pressure.     

Remarks on the Global Solutions of 3-D Navier-Stokes System with One Slow Variable

By applying Wiegner's method in [16 Wiegner , M. ( 1987 ). Decay results for weak solutions to the Navier-Stokes equations on ? ⁿ . J. London Math. Soc. 35 : 303 – 313 .[Crossref], [Web of Science ®] , [Google Scholar]], we first prove the large time decay estimate for the global solutions of a 2.5 dimensional Navier-Stokes system, which is a sort of singular perturbed 2-D Navier-Stokes system in three space dimension. As an application of this decay estimate, we give a simplified proof for the global wellposedness result in [6 Chemin , J.-Y. , Gallagher , I. ( 2010 ). Large, global solutions to the Navier-Stokes equations, slowly varying in one direction . Transactions of the American Mathematical Society 362 : 2859 – 2873 .[Crossref], [Web of Science ®] , [Google Scholar]] for 3-D Navier-Stokes system with one slow variable. Let us also mention that compared with the assumptions for the initial data in [6 Chemin , J.-Y. , Gallagher , I. ( 2010 ). Large, global solutions to the Navier-Stokes equations, slowly varying in one direction . Transactions of the American Mathematical Society 362 : 2859 – 2873 .[Crossref], [Web of Science ®] , [Google Scholar]], here the assumptions in Theorem 1.3 are weaker.     

A weighted L q -approach to Stokes flow around a rotating body

Considering time-periodic Stokes flow around a rotating body in or we prove weighted a priori estimates in L ^q -spaces for the whole space problem. After a time-dependent change of coordinates the problem is reduced to a stationary Stokes equation with the additional term in the equation of momentum, where ω denotes the angular velocity. In cylindrical coordinates attached to the rotating body we allow for Muckenhoupt weights which may be anisotropic or even depend on the angular variable and prove weighted L ^q -estimates using the weighted theory of Littlewood-Paley decomposition and of maximal operators. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan no. AV0Z10190503, by the Grant Agency of the Academy of Sciences No. IAA100190505, and by the joint research project of DAAD (D/04/25763) and the Academy of Sciences of the Czech Republic (D-CZ 3/05-06).