Interpolation, extrapolation, Morrey spaces and local energy control for the Navier--Stokes equations

Barker recently proved new weak-strong uniqueness results for the Navier-Stokes equations based on a criterion involving Besov spaces and a proof through interpolation between Besov-H{\"o}lder spaces and L 2. We improve slightly his results by considering Besov-Morrey spaces and interpolation between Besov-Morrey spaces and L 2 uloc. Let u 0 a divergence-free vector field on R 3. We shall consider weak solutions to the Cauchy initial value problem for the Navier-Stokes equations which satisfy energy estimates. The differential Navier-Stokes equations read as $\partial$ t u + u. $\nabla$ u = $\Delta$ u -- $\nabla$p div u = 0 u(0, .) = u 0

Real Interpolation method, Lorentz spaces and refined Sobolev inequalities

In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities

Local stability of energy estimates for the Navier--Stokes equations

We study the regularity of the weak limit of a sequence of dissipative solutions to the Navier--Stokes equations when no assumptions is made on the behavior of the pressures

The role of the pressure in the partial regularity theory for weak solutions of the Navier--Stokes equations

We study the role of the pressure in the partial regularity theory for weak solutions of the Navier--Stokes equations. By introducing the notion of dissipative solutions, due to Duchon \& Robert, we will provide a generalization of the Caffarelli, Kohn and Nirenberg theory. Our approach gives a new enlightenment of the role of the pressure in this theory in connection to Serrin's local regularity criterion

Frequency decay for Navier-Stokes stationary solutions

We consider stationary Navier-Stokes equations in R 3 with a regular external force and we prove exponential frequency decay of the solutions. Moreover, if the external force is small enough, we give a pointwise exponential frequency decay for such solutions according to the K41 theory. If a damping term is added to the equation, a pointwise decay is obtained without the smallness condition over the force

The Navier-Stokes equations in mixed-norm time-space parabolic Morrey spaces

We discuss the Navier-Stokes equations with forces in the mixed norm time-space parabolic Morrey spaces of Krylov

Weighted energy estimates for the incompressible Navier-Stokes equations and applications to axisymmetric solutions without swirl

We consider a family of weights which permit to generalize the Leray procedure to obtain weak suitable solutions of the 3D incom-pressible Navier-Stokes equations with initial data in weighted L 2 spaces. Our principal result concerns the existence of regular global solutions when the initial velocity is an axisymmetric vector field without swirl such that both the initial velocity and its vorticity belong to L 2 ((1 + r 2) -- $\gamma$ 2 dx), with r = x 2 1 + x 2 2 and $\gamma$ $\in$ (0, 2)