113 research outputs found
Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation
The aim of the present paper is twofold:(1) We carry on with developing an
abstract method for deriving decay estimates on the semigroup associated to
non-symmetric operators in Banach spaces as introduced in [10]. We extend the
method so as to consider the shrinkage of the functional space. Roughly
speaking, we consider a class of operators writing as a dissipative part plus a
mild perturbation, and we prove that if the associated semigroup satisfies a
decay estimate in some reference space then it satisfies the same decay
estimate in another-smaller or larger-Banach space under the condition that a
certain iterate of the "mild perturba- tion" part of the operator combined with
the dissipative part of the semigroup maps the larger space to the smaller
space in a bounded way. The cornerstone of our approach is a factorization
argument, reminiscent of the Dyson series.(2) We apply this method to the
kinetic Fokker-Planck equation when the spatial domain is either the torus with
periodic boundary conditions, or the whole space with a confinement potential.
We then obtain spectral gap es- timates for the associated semigroup for
various metrics, including Lebesgue norms, negative Sobolev norms, and the
Monge-Kantorovich-Wasserstein distance W\_1.Comment: Some typos corrected, proof of Lemma 4.7 only sketched to shorten the
paper, 41 page
Lyapunov functionals for boundary-driven nonlinear drift-diffusions
We exhibit a large class of Lyapunov functionals for nonlinear
drift-diffusion equations with non-homogeneous Dirichlet boundary conditions.
These are generalizations of large deviation functionals for underlying
stochastic many-particle systems, the zero range process and the
Ginzburg-Landau dynamics, which we describe briefly. As an application, we
prove linear inequalities between such an entropy-like functional and its
entropy production functional for the boundary-driven porous medium equation in
a bounded domain with positive Dirichlet conditions: this implies exponential
rates of relaxation related to the first Dirichlet eigenvalue of the domain. We
also derive Lyapunov functions for systems of nonlinear diffusion equations,
and for nonlinear Markov processes with non-reversible stationary measures
Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus
For a general class of linear collisional kinetic models in the torus,
including in particular the linearized Boltzmann equation for hard spheres, the
linearized Landau equation with hard and moderately soft potentials and the
semi-classical linearized fermionic and bosonic relaxation models, we prove
explicit coercivity estimates on the associated integro-differential operator
for some modified Sobolev norms. We deduce existence of classical solutions
near equilibrium for the full non-linear models associated, with explicit
regularity bounds, and we obtain explicit estimates on the rate of exponential
convergence towards equilibrium in this perturbative setting. The proof are
based on a linear energy method which combines the coercivity property of the
collision operator in the velocity space with transport effects, in order to
deduce coercivity estimates in the whole phase space
About L-P estimates for the spatially homogeneous Boltzmann equation
For the homogeneous Boltzmann equation with (cutoff or non cutoff) hard
potentials, we prove estimates of propagation of Lp norms with a weight (, large enough), as well as
appearance of such weights. The proof is based on some new functional
inequalities for the collision operator, proven by elementary means
On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates
The spatially homogeneous Boltzmann equation with hard potentials is
considered for measure valued initial data having finite mass and energy. We
prove the existence of \emph{weak measure solutions}, with and without angular
cutoff on the collision kernel; the proof in particular makes use of an
approximation argument based on the Mehler transform. Moment production
estimates in the usual form and in the exponential form are obtained for these
solutions. Finally for the Grad angular cutoff, we also establish uniqueness
and strong stability estimate on these solutions
On the Mean Field and Classical Limits of Quantum Mechanics
The main result in this paper is a new inequality bearing on solutions of the
-body linear Schr\"{o}dinger equation and of the mean field Hartree
equation. This inequality implies that the mean field limit of the quantum
mechanics of identical particles is uniform in the classical limit and
provides a quantitative estimate of the quality of the approximation. This
result applies to the case of interaction potentials. The quantity
measuring the approximation of the -body quantum dynamics by its mean field
limit is analogous to the Monge-Kantorovich (or Wasserstein) distance with
exponent . The inequality satisfied by this quantity is reminiscent of the
work of Dobrushin on the mean field limit in classical mechanics [Func. Anal.
Appl. 13 (1979), 115-123]. Our approach of this problem is based on a direct
analysis of the -particle Liouville equation, and avoids using techniques
based on the BBGKY hierarchy or on second quantization
Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solutions and tail behavior
We consider the spatially homogeneous Boltzmann equation for inelastic hard
spheres, in the framework of so-called constant normal restitution
coefficients. We prove the existence of self-similar solutions, and we give
pointwise estimates on their tail. We also give general estimates on the tail
and the regularity of generic solutions. In particular we prove Haff 's law on
the rate of decay of temperature, as well as the algebraic decay of
singularities. The proofs are based on the regularity study of a rescaled
problem, with the help of the regularity properties of the gain part of the
Boltzmann collision integral, well-known in the elastic case, and which are
extended here in the context of granular gases.Comment: 41 page
Approach to equilibrium for the phonon Boltzmann equation
We study the asymptotics of solutions of the Boltzmann equation describing
the kinetic limit of a lattice of classical interacting anharmonic oscillators.
We prove that, if the initial condition is a small perturbation of an
equilibrium state, and vanishes at infinity, the dynamics tends diffusively to
equilibrium. The solution is the sum of a local equilibrium state, associated
to conserved quantities that diffuse to zero, and fast variables that are
slaved to the slow ones. This slaving implies the Fourier law, which relates
the induced currents to the gradients of the conserved quantities.Comment: 23 page
Quantitative lower bounds for the full Boltzmann equation, Part I: Periodic boundary conditions
We prove the appearance of an explicit lower bound on the solution to the
full Boltzmann equation in the torus for a broad family of collision kernels
including in particular long-range interaction models, under the assumption of
some uniform bounds on some hydrodynamic quantities. This lower bound is
independent of time and space. When the collision kernel satisfies Grad's
cutoff assumption, the lower bound is a global Maxwellian and its asymptotic
behavior in velocity is optimal, whereas for non-cutoff collision kernels the
lower bound we obtain decreases exponentially but faster than the Maxwellian.
Our results cover solutions constructed in a spatially homogeneous setting, as
well as small-time or close-to-equilibrium solutions to the full Boltzmann
equation in the torus. The constants are explicit and depend on the a priori
bounds on the solution.Comment: 37 page
Towards an -theorem for granular gases
The -theorem, originally derived at the level of Boltzmann non-linear
kinetic equation for a dilute gas undergoing elastic collisions, strongly
constrains the velocity distribution of the gas to evolve irreversibly towards
equilibrium. As such, the theorem could not be generalized to account for
dissipative systems: the conservative nature of collisions is an essential
ingredient in the standard derivation. For a dissipative gas of grains, we
construct here a simple functional related to the original ,
that can be qualified as a Lyapunov functional. It is positive, and results
backed by three independent simulation approaches (a deterministic spectral
method, the stochastic Direct Simulation Monte Carlo technique, and Molecular
Dynamics) indicate that it is also non-increasing. Both driven and unforced
cases are investigated
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