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Geometric optics and instability for NLS and Davey-Stewartson models

Rémi Carles, Eric Dumas, Christof Sparber (2012)

Journal of the European Mathematical Society

We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schrödinger equations with Gauge invariant power-law nonlinearities and nonlocal perturbations. The model includes the Davey-Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variants. Our analysis reveals a new localization phenomenon for nonlocal perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev...

Geometric structure of magnetic walls

Myriam Lecumberry (2005)

Journées Équations aux dérivées partielles

After a short introduction on micromagnetism, we will focus on a scalar micromagnetic model. The problem, which is hyperbolic, can be viewed as a problem of Hamilton-Jacobi, and, similarly to conservation laws, it admits a kinetic formulation. We will use both points of view, together with tools from geometric measure theory, to prove the rectifiability of the singular set of micromagnetic configurations.

Geometrical methods in hydrodynamics

Adrian Constantin (2001)

Journées équations aux dérivées partielles

We describe some recent results on a specific nonlinear hydrodynamical problem where the geometric approach gives insight into a variety of aspects.

Geometry of fluid motion

Boris Khesin (2002/2003)

Séminaire Équations aux dérivées partielles

We survey two problems illustrating geometric-topological and Hamiltonian methods in fluid mechanics: energy relaxation of a magnetic field and conservation laws for ideal fluid motion. More details and results, as well as a guide to the literature on these topics can be found in [3].

Geometry of KDV (1): Addition and the unimodular spectral classes.

Henry P. McKean (1986)

Revista Matemática Iberoamericana

This is the first of three papers on the geometry of KDV. It presents what purports to be a foliation of an extensive function space into which all known invariant manifolds of KDV fit naturally as special leaves. The two main themes are addition (each leaf has its private one) and unimodal spectral classes (each leaf has a spectral interpretation).

Germes de configurations legendriennes stables et fonctions d'Airy-Weber généralisées

Nguyen Hu'u Du'c, Frédéric Pham (1991)

Annales de l'institut Fourier

On sait depuis Maslov, Arnold, etc... associer à presque tout germe de variété lagrangienne ou legendrienne lisse une classe de fonctions oscillantes qui sous des hypothèses génériques à la Thom fournissent des modèles universels pour le comportement d’une onde lumineuse au voisinage de la caustique.Le présent article étend cette construction à une classe de situations où la variété caractéristique est un germe singulier (union de composantes lisses), qui peut néanmoins être stable en ce sens que...

Global attractor for Navier-Stokes equations in cylindrical domains

Bernard Nowakowski, Wojciech M. Zajączkowski (2009)

Applicationes Mathematicae

Global and regular solutions of the Navier-Stokes system in cylindrical domains have already been obtained under the assumption of smallness of (1) the derivative of the velocity field with respect to the variable along the axis of cylinder, (2) the derivative of force field with respect to the variable along the axis of the cylinder and (3) the projection of the force field on the axis of the cylinder restricted to the part of the boundary perpendicular to the axis of the cylinder. With the same...

Global attractor for the Navier-Stokes equations in a cylindrical pipe

Piotr Kacprzyk (2010)

Annales Polonici Mathematici

Global existence of regular special solutions to the Navier-Stokes equations describing the motion of an incompressible viscous fluid in a cylindrical pipe has already been shown. In this paper we prove the existence of the global attractor for the Navier-Stokes equations and convergence of the solution to a stationary solution.

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