Dragoş Iftimie

Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions

We consider the Navier–Stokes equations with Navier friction boundary conditions and prove two results. First, in the case of a bounded domain we prove that weak Leray solutions converge (locally in time in dimension ≥3 and globally in time in dimension 2) as the viscosity goes to 0 to a strong solution of the Euler equations, provided that the initial data converge in L2 to a sufficiently smooth limit. Second, we consider the case of a half-space and anisotropic viscosities: we fix the horizontal viscosity, send the vertical viscosity to 0 and prove convergence to the expected limit system under a weaker hypothesis on the initial data.

A Uniqueness Result for the Navier--Stokes Equations with Vanishing Vertical Viscosity

Chemin et al. [M2AN Math. Model. Numer. Anal., 34 (2000), pp. 315--335.] considered the three-dimensional Navier--Stokes equations with vanishing vertical viscosity. Assuming that the initial velocity is square-integrable in the horizontal direction and Hs in the vertical direction, they prove existence of solutions for s>1/2 and uniqueness of solutions for s>3/2. Here, we close the gap between existence and uniqueness, proving uniqueness of solutions for s>1/2. Standard techniques are used.

Incompressible Flow Around a Small Obstacle and the Vanishing Viscosity Limit

In this article we consider viscous flow in the exterior of an obstacle satisfying the standard no-slip boundary condition at the surface of the obstacle. We seek conditions under which solutions of the Navier-Stokes system in the exterior domain converge to solutions of the Euler system in the full space when both viscosity and the size of the obstacle vanish. We prove that this convergence is true assuming two hypotheses: first, that the initial exterior domain velocity converges strongly in L2 to the full-space initial velocity and second, that the diameter of the obstacle is smaller than a suitable constant times viscosity, or, in other words, that the obstacle is sufficiently small. The convergence holds as long as the solution to the limit problem is known to exist and stays sufficiently smooth. This work complements the study of incompressible flow around small obstacles, which has been carried out in [4–6].

L p -Solutions of the Steady-State Navier–Stokes Equations with Rough External Forces

In this paper we address the existence, the asymptotic behavior and stability in L p and L p, ∞, , for solutions to the steady state 3D Navier–Stokes equations with possibly very singular external forces. We show that under certain smallness conditions of the forcing term there exists solutions to the stationary Navier–Stokes equations in L p spaces, and we prove the stability of these solutions. Namely, we prove that such small steady state solutions attract time dependent solutions with large initial velocity driven by the same forcing. We also give non-existence results of stationary solutions in L p , for .

Global existence and uniqueness of solutions for the equations of third grade fluids

Abstract We consider the equations governing the motion of third grade fluids in R n , n=2,3 . We show global existence of solutions without any smallness assumption, by assuming only that the initial velocity belongs to the Sobolev space H 2 . The uniqueness of such solutions is also proven in dimension two.

Remarks on the vanishing obstacle limit for a 3D viscous incompressible fluid

In [Math. Ann. 336 (2006), 449-489], the authors consider the two-dimensional Navier-Stokes equations in the exterior of an obstacle shrinking to a point and determine the limit velocity. Here we consider the same problem in the three-dimensional case, proving that the limit velocity is a solution of the Navier-Stokes equations in the full space.

On the large-time behavior of two-dimensional vortex dynamics

In this paper we prove two results regarding the large-time behavior of vortex dynamics in the full plane. In the first result we show that the total integral of vorticity is confined in a region of diameter growing at most like the square root of time. In the second result we show that if a dynamic rescaling of the absolute value of vorticity with spatial scale growing linearly with time converges weakly, then it must converge to a discrete sum of Dirac masses. This last result extends in scope a previous result by the authors, valid for nonnegative initial vorticity on a half-plane.

Asymptotics of solutions to the Navier–Stokes system in exterior domains

We consider the incompressible Navier-Stokes equations with the Dirichlet boundary condition in an exterior domain of

Large Time Behavior for Vortex Evolution in the Half-Plane

\mathbb{R}^n

Remarques sur la limite α→0 pour les fluides de grade 2

with