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Onétudie des estimations semiclassiques sur la résolvente d'opérateurs qui ne sont ni ellip-tiques ni autoadjoints, que l'on utilise pourétudier leprobì eme de Cauchy. En particulier on obtient une description précise du spectre pres de l'axe imaginaire, et des estimations de résolventè a l'intérieur du pseudo-spectre. On applique ensuite les résultatsà… (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
- 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 weak-Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the… (More)

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)

- 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 lineariza-tion of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to… (More)

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to wellposed varia-tional 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.… (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)

- 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'´ equation de Fokker-Planck avec un potentiel confi-nant en dimension d ≥ 3. Avec des méthodes spectrales on… (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)

- Frédéric Hérau, Wei-Xi Li
- 2011

In this paper we study a Landau-type operator with an external force. It is a linear model of the Landau equation near Maxwellian distributions. Making use of multiplier method, we get the global hypoelliptic estimate under suitable assumptions on the external potential.