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In certain applications one is interested in the long-time behavior of systems described by a linear partial differential equation. For example, in kinetic equations one studies the decay to equilibrium of various linear and nonlinear systems. For the Kramers–Fokker–Planck equation, which will be studied here, exponential decay was shown in Talay (1999) and… (More)

- Frédéric Hérau
- 2002

We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weakCauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the… (More)

- Frédéric Hérau
- Asymptotic Analysis
- 2006

We consider an inhomogeneous linear Boltzmann equation, with an external confining potential. The collision operator is a simple relaxation toward a local Maxwellian, therefore without diffusion. We prove the exponential time decay toward the global Maxwellian, with an explicit rate of decay. The methods are based on hypoelliptic methods transposed here to… (More)

In the first part of this work, we consider second order supersymmetric differential operators in the semiclassical limit, including the Kramers-Fokker-Planck operator, such that the exponent of the associated Maxwellian φ is a Morse function with two local minima and one saddle point. Under suitable additional assumptions of dynamical nature, we establish… (More)

We consider operators of Kramers-Fokker-Planck type in the semi-classical limit such that the exponent of the associated Maxwellian is a Morse function with two local minima and a saddle point. Under suitable additional assumptions we establish the complete asymptotics of the exponentially small splitting between the first two eigenvalues. Résumé On… (More)

- Frédéric Hérau, KAREL PRAVDA-STAROV
- 2010

We establish global hypoelliptic estimates for linear Landau-type operators. Linear Landau-type equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to… (More)

- Virginie Bonnaillie-Noël, Marc Dambrine, Frédéric Hérau, Grégory Vial
- SIAM J. Math. Analysis
- 2010

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to wellposed variational problems under a sign condition of a coefficient. Nevertheless situations where this condition is violated appeared in several works. The wellposedness of such problems was still open. This… (More)

- Frédéric Hérau
- 2008

We consider the linear Fokker-Planck equation in a confining potential in space dimension d ≥ 3. Using spectral methods, we prove bounds on the derivatives of the solution for short and long time, and give some applications. Résumé: On considère l’équation de Fokker-Planck avec un potentiel confinant en dimension d ≥ 3. Avec des méthodes spectrales on donne… (More)

- Frédéric Hérau
- 2005

We consider the non-linear VPFP system with a coulombian repulsive interaction potential and a generic confining potential in space dimension d ≥ 3. Using spectral and kinetic methods we prove the existence and uniqueness of a mild solution with bounds uniform in time in weighted spaces, and for small total charge we find an explicit exponential rate of… (More)

We study a semiclassical random walk with respect to a probability measure with a finite number n0 of wells. We show that the associated operator has exactly n0 exponentially close to 1 eigenvalues (in the semiclassical sense), and that the other are O(h) away from 1. We also give an asymptotic of these small eigenvalues. The key ingredient in our approach… (More)