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

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

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

Entre la « main de l’œil » et l’ « œil digital », proximité et profondeur : la dimension haptique à l’horizon du Cinéma 3D et des acteurs virtuels

Si certaines œuvres du cinéma 3D vérifient le rôle privilégié de la « main de l’œil » dans l’haptique deleuzien d’autres, par leur irréductibilité à l’opposition entre espace haptique lisse et espace optique strié, nous conduisent à dépasser cette axiologie dualiste, non sans interroger la réversibilité même du tactile qui se manifeste par la rationalisation de l’ « œil digital » et l’illusion de maîtrise de la main démiurgique. Irréductible à ces écueils, la « profondeur haptique » tend non seulement à dépasser ce système duel mais à développer la potentialité empathique du toucher, que ce soit sous la forme de l’empathie kinesthésique développée à l’égard des acteurs virtuels ou par la mise en jeu d’un « regard haptique » ouvert au vertige de leur altérité radicale.If some 3D movies seem to testify the privileged role of “the eye’s hand” in Deleuze’s theory of haptic, other ones, through their irreducibility to the opposition between the smooth haptic space and the striate optical space, lead us beyond this dualistic axiology not without questioning the reversibility of the tactile which appears in the rationalization of the "digital eye" and the delusion of control by the demiurgic hand. The haptic "depth" tends not only to bypass this dual system but to develop the empathic potentiality of the sense of touch, either through the kinaesthetic empathy developed towards virtual actors or through the involvement of a "haptic outlook" open to the dizziness of their radical otherness