Global well-posedness for KdV in Sobolev Spaces of negative index

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H^s({\mathbb{R}), -3/10<s.Comment: 5 pages. Electronic Journal of Differential equations (submitted

Almost sure existence of global weak solutions for super-critical Navier-Stokes equations

In this paper we show that after suitable data randomization there exists a large set of super-critical periodic initial data, in $H^{-\alpha}({\mathbb T}^d)$ for some $\alpha(d) > 0$, for both 2d and 3d Navier-Stokes equations for which global energy bounds are proved. As a consequence, we obtain almost sure super-critical global weak solutions. We also show that in 2d these global weak solutions are unique.Comment: 22 pages, a revised argument in Section 5, the $d=3$ cas

On the nonlinear Dysthe equation

This work is dedicated to putting on a solid analytic ground the theory of local well-posedness for the two dimensional Dysthe equation. This equation can be derived from the incompressible Navier-Stokes equation after performing an asymptotic expansion of a wavetrain modulation to the fourth order. Recently, this equation has been used to numerically study rare phenomena on large water bodies such as rogue waves. In order to study well-posedness, we use Strichartz, and improved smoothing and maximal function estimates. We follow ideas from the pioneering work of Kenig, Ponce and Vega, but since the equation is highly anisotropic, several technical challenges had to be resolved. We conclude our work by also presenting an ill-posedness result.Comment: Some remarks and references adde

Near-linear dynamics in KdV with periodic boundary conditions

Near linear evolution in Korteweg de Vries (KdV) equation with periodic boundary conditions is established under the assumption of high frequency initial data. This result is obtained by the method of normal form reduction

Linear waves in sheared flows. Lower bound of the vorticity growth and propagation discontinuities in the parameters space

This study provides sufficient conditions for the temporal monotonic decay of enstrophy for two-dimensional perturbations traveling in the incompressible, viscous, plane Poiseuille and Couette flows. Extension of J. L. Synge's procedure (1938) to the initial-value problem allowed us to find the region of the wavenumber-Reynolds number map where the enstrophy of any initial disturbance cannot grow. This region is wider than the kinetic energy's one. We also show that the parameters space is split in two regions with clearly distinct propagation and dispersion properties