We formulate sufficient conditions for regularity up to the boundary of a weak solution v in a subdomain Ω × (t₁,t₂) of the time-space cylinder Ω × (0,T) by means of requirements on one of the eigenvalues of the rate of deformation tensor. We assume that Ω is a cube.
In this proceedings article we shall survey a series of results on the stability of self-similar solutions of the vortex filament equation. This equation is a geometric flow for curves in and it is used as a model for the evolution of a vortex filament in fluid mechanics. The main theorem give, under suitable assumptions, the existence and description of solutions generated by curves with a corner, for positive and negative times. Its companion theorem describes the evolution of perturbations...
On considère l’équation d’Euler incompressible dans le plan. Dans le cas où le tourbillon est positif et à support compact on montre que le support du tourbillon croît au plus comme , améliorant la borne obtenue par C. Marchioro. Dans le cas où le tourbillon change de signe, on donne un exemple de tourbillon initial tel que la croissance du diamètre du support du tourbillon est exactement . Enfin, dans le cas du demi-plan et du tourbillon initial positif et à support compact, on montre que le...
We investigate the evolution of singularities in the boundary of a vortex patch for two-dimensional incompressible Euler equations. We are particularly interested in cusp-like singularities which, according to numerical simulations, are stable. In this paper, we first prove that, unlike the case of a corner-like singularity, the cusp-like singularity generates a lipschitzian velocity. We then state a global result of persistence of conormal regularity with respect to vector fields vanishing at a...
We consider a model of migrating population occupying a compact domain Ω in the plane. We assume the Malthusian growth of the population at each point x ∈ Ω and that the mobility of individuals depends on x ∈ Ω. The evolution of the probability density u(x,t) that a randomly chosen individual occupies x ∈ Ω at time t is described by the nonlocal linear equation , where φ(x) is a given function characterizing the mobility of individuals living at x. We show that the asymptotic behaviour of u(x,t)...
We prove the exact boundary controllability of the 3-D Euler equation of incompressible inviscid fluids on a regular connected bounded open set when the control operates on an open part of the boundary that meets any of the connected components of the boundary.
We study the boundary controllability of a nonlinear Korteweg–de Vries equation with the Dirichlet boundary condition on an interval with a critical length for which it has been shown by Rosier that the linearized control system around the origin is not controllable. We prove that the nonlinear term gives the local controllability around the origin.