### A regularity criterion for the dissipative quasi-geostrophic equations

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In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross surface flow in the Euler equations (incompressible for the fluid and compressible for the gas). The obtained system is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of this system to finally construct a two-layer kinetic scheme...

In this paper we investigate the motion of a rigid ball in an incompressible perfect fluid occupying ${\mathbb{R}}^{2}$. We prove the global in time existence and the uniqueness of the classical solution for this fluid-structure problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid-structure problem obtained by incorporating some dissipation.

In this paper, we investigate the controllability of an underwater vehicle immersed in an infinite volume of an inviscid fluid whose flow is assumed to be irrotational. Taking as control input the flow of the fluid through a part of the boundary of the rigid body, we obtain a finite-dimensional system similar to Kirchhoff laws in which the control input appears through both linear terms (with time derivative) and bilinear terms. Applying Coron’s return method, we establish some local controllability...

In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done...

This note is devoted to the study of a bi-fluid generalization of the nonlinear shallow-water equations. It describes the evolution of the interface between two fluids of different densities. In the case of a two-dimensional interface, this systems contains unexpected nonlocal terms (that are of course not present in the usual one-fluid shallow water equations). We show here how to derive this systems from the two-fluid Euler equations and then show that it is locally well-posed.

Building upon the techniques introduced in [15], for any $\theta <\frac{1}{10}$ we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are Hölder-continuous with exponent $\theta $. A famous conjecture of Onsager states the existence of such dissipative solutions with any Hölder exponent $\theta <\frac{1}{3}$. Our theorem is the first result in this direction.

In this paper, we investigate the problem of fast rotating fluids between two infinite plates with Dirichlet boundary conditions and “turbulent viscosity” for general ${L}^{2}$ initial data. We use dispersive effect to prove strong convergence to the solution of the bimensionnal Navier-Stokes equations modified by the Ekman pumping term.

In this paper, we investigate the problem of fast rotating fluids between two infinite plates with Dirichlet boundary conditions and “turbulent viscosity” for general L2 initial data. We use dispersive effect to prove strong convergence to the solution of the bimensionnal Navier-Stokes equations modified by the Ekman pumping term.

On considère l’équation d’Euler incompressible dans le plan. Dans le cas où le tourbillon est positif et à support compact on montre que le support du tourbillon croît au plus comme $O{\left[(tlogt)\right]}^{1/4}$, améliorant la borne $O\left({t}^{1/3}\right)$ obtenue par C. Marchioro. Dans le cas où le tourbillon change de signe, on donne un exemple de tourbillon initial tel que la croissance du diamètre du support du tourbillon est exactement $O\left(t\right)$. Enfin, dans le cas du demi-plan et du tourbillon initial positif et à support compact, on montre que le...

We investigate the evolution of singularities in the boundary of a vortex patch for two-dimensional incompressible Euler equations. We are particularly interested in cusp-like singularities which, according to numerical simulations, are stable. In this paper, we first prove that, unlike the case of a corner-like singularity, the cusp-like singularity generates a lipschitzian velocity. We then state a global result of persistence of conormal regularity with respect to vector fields vanishing at a...

We study the convergence in the vanishing viscosity limit of the stationary incompressible Navier-Stokes equation towards the stationary Euler equation, in the presence of an arbitrary force term. This requires that the fluid is allowed to pass through some open part of the boundary.