On the regularity of the Navier–Stokes equation in a thin periodic domain (original) (raw)

Bifurcation of periodic solutions of the Navier-Stokes equation in a thin domain

Aim of this paper is to provide conditions in order to guarantee that the periodic solutions in time and in the space variables of the Navier-Stokes equations bifurcate. Specifically, we study this problem when the considered state domain has one dimension which is small with respect to the others which we let to tend to zero. The thinness of the domain represents the bifurcation parameter in our situation. ∇ · U = 0, 1991 Mathematics Subject Classification. 35B10, 35B32.

Regularity for the stationary Navier-Stokes equations in bounded domain

Archive for Rational Mechanics and Analysis, 1994

Let u, p be a weak solution of the stationary Navier-Stokes equations in a bounded domain f~ _~ IR N, 5 ~< N < oo. If u, p satisfy the additional conditions (B) fVuV(ujdx<__f(~+p)u.Vydx+ff.uTdx V7 ~ C~(a), y __> 0, they become regular. Moreover, it is proved that every weak solution u, p satisfying (A) with q = oe is regular. The existence of such solutions for N = 5 has been established in a former paper [3].

Regularity criteria for the three-dimensional Navier-Stokes equations

Indiana University Mathematics Journal, 2008

In this paper we consider the three-dimensional Navier-Stokes equations subject to periodic boundary conditions or in the whole space. We provide sufficient conditions, in terms of one component of the velocity field, or alternatively in terms of one component of the pressure gradient, for the regularity of strong solutions to the three-dimensional Navier-Stokes equations.

Uniform Regularity for the Navier–Stokes Equation with Navier Boundary Condition

Archive for Rational Mechanics and Analysis, 2012

We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier-Stokes equation with Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in L ∞ . This allows to get the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.