Treibich-Verdier potentials and the stationary (m)KDV hierarchy.
Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation
We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates on the approximation of the solution by the particle system. Finally, a transportation inequality...
Triangulation formelle de certains systèmes de Pfaff complètement intégrables et application à l’étude des systèmes non linéaires
Trudinger–Moser inequality on the whole plane with the exact growth condition
Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modied versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function....
Truncated gradient flows of the van der Waals free energy.
Truncated spectral regularization for an ill-posed non-linear parabolic problem
It is known that the nonlinear nonhomogeneous backward Cauchy problem , with , where is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on and , that a solution of the above problem satisfies an integral equation involving the spectral representation of , which is also ill-posed. Spectral truncation is used...
Tunnel effect and symmetries for non-selfadjoint operators
We study low lying eigenvalues for non-selfadjoint semiclassical differential operators, where symmetries play an important role. In the case of the Kramers-Fokker-Planck operator, we show how the presence of certain supersymmetric and -symmetric structures leads to precise results concerning the reality and the size of the exponentially small eigenvalues in the semiclassical (here the low temperature) limit. This analysis also applies sometimes to chains of oscillators coupled to two heat baths,...
Tunnel effect for semiclassical random walk
In this note we describe recent results on semiclassical random walk associated to a probability density which may also concentrate as the semiclassical parameter goes to zero. The main result gives a spectral asymptotics of the close to eigenvalues. This problem was studied in [1] and relies on a general factorization result for pseudo-differential operators. In this note we just sketch the proof of this second theorem. At the end of the note, using the factorization, we give a new proof of the...
Tusk type conditions and thinness for the parabolic operator of order ...
Two blow-up regimes for supercritical nonlinear Schrödinger equations
We consider the focusing nonlinear Schrödinger equations . We prove the existence of two finite time blow up dynamics in the supercritical case and provide for each a qualitative description of the singularity formation near the blow up time.
Two component regularity for the Navier-Stokes equations.
Two conservative difference schemes for the generalized Rosenau equation.
Two constant sign solutions for a nonhomogeneous Neumann boundary value problem
We consider a nonlinear Neumann problem with a nonhomogeneous elliptic differential operator. With some natural conditions for its structure and some general assumptions on the growth of the reaction term we prove that the problem has two nontrivial solutions of constant sign. In the proof we use variational methods with truncation and minimization techniques.
Two dimensional incompressible ideal flow around a thin obstacle tending to a curve
Two Dimensional Variational Problems with Thin Obstacles.
Two endpoint bounds for generalized Radon transforms in the plane.
Two examples of an alternative approach to systems analogous to the heat polynomials
Two examples of fattening for the curvature flow with a driving force
We provide two examples of a regular curve evolving by curvature with a forcing term, which degenerates in a set having an interior part after a finite time.
Two Families of Mixed Finite Elements for Second Order Elliptic Problems.
Two functionals for which minimizers are also minimizers.