155 research outputs found
Sharp estimates for pseudodifferential operators with symbols of limited smoothness and commutators
We consider here pseudo-differential operators whose symbol
is not infinitely smooth with respect to . Decomposing such symbols into
four -sometimes five- components and using tools of paradifferential calculus,
we derive sharp estimates on the action of such pseudo-differential operators
on Sobolev spaces and give explicit expressions for their operator norm in
terms of the symbol . We also study commutator estimates
involving such operators, and generalize or improve the so-called Kato-Ponce
and Calderon-Coifman-Meyer estimates in various ways.Comment: Accepted for publication in Journal of Functional Analysi
Derivation of asymptotic two-dimensional time-dependent equations for ocean wave propagation
A general method for the derivation of asymptotic nonlinear shallow water and
deep water models is presented. Starting from a general dimensionless version
of the water-wave equations, we reduce the problem to a system of two equations
on the surface elevation and the velocity potential at the free surface. These
equations involve a Dirichlet-Neumann operator and we show that all the
asymptotic models can be recovered by a simple asymptotic expansion of this
operator, in function of the shallowness parameter (shallow water limit) or the
steepness parameter (deep water limit). Based on this method, a new
two-dimensional fully dispersive model for small wave steepness is also
derived, which extends to uneven bottom the approach developed by Matsuno
\cite{matsuno3} and Choi \cite{choi}. This model is still valid in shallow
water but with less precision than what can be achieved with Green-Naghdi
model, when fully nonlinear waves are considered. The combination, or the
coupling, of the new fully dispersive equations with the fully nonlinear
shallow water Green-Naghdi equations represents a relevant model for describing
ocean wave propagation from deep to shallow waters
Mathematics for 2d Interfaces
We present here a survey of recent results concerning the mathematical
analysis of instabilities of the interface between two incompressible, non
viscous, fluids of constant density and vorticity concentrated on the
interface. This configuration includes the so-called Kelvin-Helmholtz (the two
densities are equal), Rayleigh-Taylor (two different, nonzero, densities) and
the water waves (one of the densities is zero) problems. After a brief review
of results concerning strong and weak solutions of the Euler equation, we
derive interface equations (such as the Birkhoff-Rott equation) that describe
the motion of the interface. A linear analysis allows us to exhibit the main
features of these equations (such as ellipticity properties); the consequences
for the full, non linear, equations are then described. In particular, the
solutions of the Kelvin-Helmholtz and Rayleigh-Taylor problems are necessarily
analytic if they are above a certain threshold of regularity (a consequence is
the illposedness of the initial value problem in a non analytic framework). We
also say a few words on the phenomena that may occur below this regularity
threshold. Finally, special attention is given to the water waves problem,
which is much more stable than the Kelvin-Helmholtz and Rayleigh-Taylor
configurations. Most of the results presented here are in 2d (the interface has
dimension one), but we give a brief description of similarities and differences
in the 3d case.Comment: Survey. To appear in Panorama et Synth\`ese
A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations
We study the well-posedness of the initial value problem for a wide class of
singular evolution equations. We prove a general well-posedness theorem under
three assumptions easy to check: the first controls the singular part of the
equation, the second the behavior of the nonlinearities, and the third one
assumes that an energy estimate can be found for the linearized system. We
allow losses of derivatives in this energy estimate and therefore construct a
solution by a Nash-Moser iterative scheme. As an application to this general
theorem, we prove the well-posedness of the Serre and Green-Naghdi equation and
discuss the problem of their validity as asymptotic models for the water-waves
equations
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