The aim of this note is to give a short review of our recent work (see [5]) with Miguel A. Alejo and Luis Vega, concerning the -stability, and asymptotic stability, of the -soliton of the Korteweg-de Vries (KdV) equation.
We consider the time-periodic Oseen flow around a rotating body in ℝ³. We prove a priori estimates in -spaces of weak solutions for the whole space problem under the assumption that the right-hand side has the divergence form. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional term -(ω ∧ x)·∇u + ω ∧ u in the equation of momentum where ω denotes the angular velocity. We prove the existence of generalized weak solutions in -space...
We investigate sufficient and possibly necessary conditions for the L2 stability of the upwind first order finite volume scheme for Maxwell equations, with metallic and absorbing boundary conditions. We yield a very general sufficient condition, valid for any finite volume partition in two and three space dimensions. We show this condition is necessary for a class of regular meshes in two space dimensions. However, numerical tests show it is not necessary in three space dimensions even on regular...
Au regard de la première partie de son œuvre, D’Alembert est reconnu aujourd’hui comme le fondateur de la théorie des équations aux dérivées partielles. La résolution de ces équations dans le cadre de problèmes physico-mathématiques dans ses neuf tomes d’Opuscules mathématiques (1761–1783) reste cependant peu étudiée par les historiens. Nous examinons ici cette question à la lumière de ses recherches sur les cordes vibrantes et l’écoulement des fluides dans ce corpus tardif. Celles-ci nous permettent...
The motion of a viscous incompressible fluid flow in bounded domains with a smooth boundary can be described by the nonlinear Navier-Stokes equations. This description corresponds to the so-called Eulerian approach. We develop a new approximation method for the Navier-Stokes equations in both the stationary and the non-stationary case by a suitable coupling of the Eulerian and the Lagrangian representation of the flow, where the latter is defined by the trajectories of the particles of the fluid....
In recent years, mirror symmetry for open strings has exhibited some new connections between symplectic and enumerative geometry (A-model) and complex algebraic geometry (B-model) that in a sense lie between classical and homological mirror symmetry. I review the rôle played in this story by matrix factorizations and the Calabi-Yau/Landau-Ginzburg correspondence.