On the global well-posedness for the Boussinesq system with horizontal dissipation

In this paper, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is an axisymmetric without swirl, we prove the global well-posedness for this system. In the absence of vertical dissipation, there is no smoothing effect on the vertical derivatives. To make up this shortcoming, we first establish a magic relationship between $\frac{u^{r}}{r}$ and $\frac{\omega_\theta}{r}$ by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis. This together with the structure of the coupling of \eqref{eq1.1} entails the desired regularity.Comment: 32page

On the global well-posedness of a class of Boussinesq- Navier-Stokes systems

In this paper we consider the following 2D Boussinesq-Navier-Stokes systems \partial_{t}u+u\cdot\nabla u+\nabla p+ |D|^{\alpha}u &= \theta e_{2} \partial_{t}\theta+u\cdot\nabla \theta+ |D|^{\beta}\theta &=0 \quad with $\textrm{div} u=0$ and $0<\beta<\alpha<1$. When $\frac{6-\sqrt{6}}{4}<\alpha< 1$, $1-\alpha<\beta\leq f(\alpha)$, where $f(\alpha)$ is an explicit function as a technical bound, we prove global well-posedness results for rough initial data.Comment: 23page

Interaction of vortices in viscous planar flows

We consider the inviscid limit for the two-dimensional incompressible Navier-Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter \nu, and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex system is well-posed on the interval [0,T]. Under these assumptions, we prove that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a superposition of Lamb-Oseen vortices whose centers evolve according to a viscous regularization of the point vortex system. Convergence holds uniformly in time, in a strong topology which allows to give an accurate description of the asymptotic profile of each individual vortex. In particular, we compute to leading order the deformations of the vortices due to mutual interactions. This allows to estimate the self-interactions, which play an important role in the convergence proof.Comment: 39 pages, 1 figur

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co

On the global well-posedness of the Euler-Boussinesq system with fractional dissipation

International audienc

Time-dependent delta-interactions for 1D Schrödinger Hamiltonians.

The non autonomous Cauchy problem i∂tu = − ∂ 2 xxu+α(t)δ0u with ut=0 = u0 is considered in L 2 (R). The regularity assumptions for α are accurately analyzed and show that the general results for non autonomous linear evolution equations in Banach spaces are far from being optimal. In the mean time, this article shows an unexpected application of paraproduct techniques, initiated by J.M. Bony for nonlinear partial differential equations, to a classical linear problem