The normal form of the Navier-Stokes equations in suitable normed spaces
The problem and singular integrals on Orlicz spaces
The resolution of the Navier-Stokes equations in anisotropic spaces.
In this paper we prove global existence and uniqueness for solutions of the 3-dimensional Navier-Stokes equations with small initial data in spaces which are Hδi in the i-th direction, δ1 + δ2 + δ3 = 1/2, -1/2 < δi < 1/2 and in a space which is L2 in the first two directions and B2,11/2 in the third direction, where H and B denote the usual homogeneous Sobolev and Besov spaces.
The role of oscillations in the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations
Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations with initial data in the scaling invariant Besov space, here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, and Then with initial data in the scaling invariant space we prove the global wellposedness for provided the norm of initial data is small enough compared...
The Rothe method and time periodic solutions to the Navier-Stokes equations and equations of magnetohydrodynamics
The existence of a periodic solution of a nonlinear equation is proved. The theory developed may be used to prove the existence of a periodic solution of the variational formulation of the Navier-Stokes equations or the equations of magnetohydrodynamics. The proof of the main existence theorem is based on Rothe method in combination with the Galerkin method, using the Brouwer fixed point theorem.
The scalar Oseen operator in
This paper solves the scalar Oseen equation, a linearized form of the Navier-Stokes equation. Because the fundamental solution has anisotropic properties, the problem is set in a Sobolev space with isotropic and anisotropic weights. We establish some existence results and regularities in theory.
The stationary exterior 3 D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces.
The Stokes system in the incompressible case-revisited
The classical Stokes system is reconsidered and reformulated in a functional analytical setting allowing for low regularity of the data and the boundary. In fact the underlying domain can be any non-empty open subset Ω of ℝ³. A suitable solution concept and a corresponding solution theory is developed.
The Tartar equation for homogenization of a model of the dynamics of fine-dispersion mixtures.
The topological asymptotic for the Navier-Stokes equations
The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional...
The topological asymptotic for the Navier-Stokes equations
The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional...
The uniform attractors for the nonhomogeneous 2D Navier-Stokes equations in some unbounded domain.
The unstable spectrum of the Navier–Stokes operator in the limit of vanishing viscosity
Three-level discrete time Galerkin approximations for the non-stationary Navier-Stokes equation
Time Spectral Method for Periodic and Quasi-Periodic Unsteady Computations on Unstructured Meshes
For flows with strong periodic content, time-spectral methods can be used to obtain time-accurate solutions at substantially reduced cost compared to traditional time-implicit methods which operate directly in the time domain. However, these methods are only applicable in the presence of fully periodic flows, which represents a severe restriction for many aerospace engineering problems. This paper presents an extension of the time-spectral approach...
Time-dependent coupling of Navier–Stokes and Darcy flows
A weak solution of the coupling of time-dependent incompressible Navier–Stokes equations with Darcy equations is defined. The interface conditions include the Beavers–Joseph–Saffman condition. Existence and uniqueness of the weak solution are obtained by a constructive approach. The analysis is valid for weak regularity interfaces.
Time-dependent Stokes equations with measure data.
Transmission of obliquely incident surface waves through a narrow gap.
Trapping regions and an ODE-type proof of the existence and uniqueness theorem for Navier-Stokes equations with periodic boundary conditions on the plane.
Two component regularity for the Navier-Stokes equations.