A KAM Approach to the Inviscid Limit for the 2D Navier-Stokes Equations.

In this paper, we investigate the inviscid limit for time-quasi-periodic solutions of the incompressible Navier-Stokes equations on the two-dimensional torus , with a small time-quasi-periodic external force. More precisely, we construct solutions of the forced Navier-Stokes equation, bifurcating from a given time quasi-periodic solution of the incompressible Euler equations and admitting vanishing viscosity limit to the latter, uniformly for all times and independently of the size of the external perturbation. Our proof is based on the construction of an approximate solution, up to an error of order and on a fixed point argument starting with this new approximate solution.

Space Quasi-Periodic Steady Euler Flows Close to the Inviscid Couette Flow.

We prove the existence of steady stream functions, solutions for the Euler equation in a vorticity-stream function formulation in the two dimensional channel . These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash-Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions, slightly deforming the linear solutions and retaining the horizontal quasi-periodic structure.

Almost Global Existence for Some Hamiltonian PDEs with Small Cauchy Data on General Tori.

In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain's Lemma which provides a partition of the "resonant sites" of the Laplace operator on irrational tori.

On the Stability of Periodic Multi-Solitons of the KdV Equation.

In this paper we obtain the following stability result for periodic multi-solitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size , a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least . To the best of our knowledge, this is the first stability result of such type for periodic multi-solitons of large size of an integrable PDE.

Giant diverticulum of the sigmoid colon with perforation. Report of a case.

We present a case of perforated giant diverticulum of the sigmoid colon. This condition is extremely rare and only a few cases have so far been reported in the literature. Our case involved a 55-year old woman.