87 research outputs found
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 for some , 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 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
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