35 research outputs found
The Euler Equations on Thin Domains
For the Euler equations in a thin domain Q_ε = Ω×(0, ε), Ω a rectangle in R^2, with initial data in (W^(2,q)(Qε))^3, q > 3, bounded uniformly in ε, the classical solution is shown to exist on a time interval (0, T(ε)), where T(є) → +∞ as є → 0. We compare this solution with that of a system of limiting equations on Ω
Generic transversality of heteroclinic and homoclinic orbits for scalar parabolic equations
In this paper, we consider the scalar reaction-diffusion equations on a bounded domain of class . We show that the heteroclinic and homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic orbits are transverse, generically with respect to f. One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka-Smale property, the generic hyperbolicity of periodic orbits remaining unproved
Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics
Geophysical fluids all exhibit a common feature: their aspect ratio (depth to horizontal width) is very small. This leads to an asymptotic model widely used in meteorology, oceanography, and limnology, namely the hydrostatic approximation of the time-dependent incompressible Navier–Stokes equations. It relies on the hypothesis that pressure increases linearly in the vertical direction. In the following, we prove a convergence and existence theorem for this model by means of anisotropic estimates and a new time-compactness criterium.Fonds Franco-Espagnol D.R.E.I.FMinisterio de Educación y Cienci
Generic Morse-Smale property for the parabolic equation on the circle
In this paper, we show that, for scalar reaction-diffusion equations
on the circle , the Morse-Smale property is
generic with respect to the non-linearity . In \cite{CR}, Czaja and Rocha
have proved that any connecting orbit, which connects two hyperbolic periodic
orbits, is transverse and that there does not exist any homoclinic orbit,
connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown
that, generically with respect to the non-linearity , all the equilibria and
periodic orbits are hyperbolic. Here we complete these results by showing that
any connecting orbit between two hyperbolic equilibria with distinct Morse
indices or between a hyperbolic equilibrium and a hyperbolic periodic orbit is
automatically transverse. We also show that, generically with respect to ,
there does not exist any connection between equilibria with the same Morse
index. The above properties, together with the existence of a compact global
attractor and the Poincar\'e-Bendixson property, allow us to deduce that,
generically with respect to , the non-wandering set consists in a finite
number of hyperbolic equilibria and periodic orbits . The main tools in the
proofs include the lap number property, exponential dichotomies and the
Sard-Smale theorem. The proofs also require a careful analysis of the
asymptotic behavior of solutions of the linearized equations along the
connecting orbits
A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
The purpose of this paper is to enhance a correspondence between the dynamics
of the differential equations on and those
of the parabolic equations on a bounded
domain . We give details on the similarities of these dynamics in the
cases , and and in the corresponding cases ,
and dim() respectively. In addition to
the beauty of such a correspondence, this could serve as a guideline for future
research on the dynamics of parabolic equations