250 research outputs found
Regularity and mass conservation for discrete coagulation-fragmentation equations with diffusion
We present a new a-priori estimate for discrete coagulation-fragmentation
systems with size-dependent diffusion within a bounded, regular domain confined
by homogeneous Neumann boundary conditions. Following from a duality argument,
this a-priori estimate provides a global bound on the mass density and
was previously used, for instance, in the context of reaction-diffusion
equations.
In this paper we demonstrate two lines of applications for such an estimate:
On the one hand, it enables to simplify parts of the known existence theory and
allows to show existence of solutions for generalised models involving
collision-induced, quadratic fragmentation terms for which the previous
existence theory seems difficult to apply. On the other hand and most
prominently, it proves mass conservation (and thus the absence of gelation) for
almost all the coagulation coefficients for which mass conservation is known to
hold true in the space homogeneous case.Comment: 24 page
On the speed of approach to equilibrium for a collisionless gas
We investigate the speed of approach to Maxwellian equilibrium for a
collisionless gas enclosed in a vessel whose wall are kept at a uniform,
constant temperature, assuming diffuse reflection of gas molecules on the
vessel wall. We establish lower bounds for potential decay rates assuming
uniform bounds on the initial distribution function. We also obtain a
decay estimate in the spherically symmetric case. We discuss with particular
care the influence of low-speed particles on thermalization by the wall.Comment: 22 pages, 1 figure; submitted to Kinetic and Related Model
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
Global existence and full regularity of the Boltzmann equation without angular cutoff
We prove the global existence and uniqueness of classical solutions around an
equilibrium to the Boltzmann equation without angular cutoff in some Sobolev
spaces. In addition, the solutions thus obtained are shown to be non-negative
and in all variables for any positive time. In this paper, we study
the Maxwellian molecule type collision operator with mild singularity. One of
the key observations is the introduction of a new important norm related to the
singular behavior of the cross section in the collision operator. This norm
captures the essential properties of the singularity and yields precisely the
dissipation of the linearized collision operator through the celebrated
H-theorem
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
Celebrating Cercignani's conjecture for the Boltzmann equation
Cercignani's conjecture assumes a linear inequality between the entropy and
entropy production functionals for Boltzmann's nonlinear integral operator in
rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities
and spectral gap inequalities, this issue has been at the core of the renewal
of the mathematical theory of convergence to thermodynamical equilibrium for
rarefied gases over the past decade. In this review paper, we survey the
various positive and negative results which were obtained since the conjecture
was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani,
powerful mind and great scientist, one of the founders of the modern theory
of the Boltzmann equation. 24 pages. V2: correction of some typos and one
ref. adde
Regularizing effect and local existence for non-cutoff Boltzmann equation
The Boltzmann equation without Grad's angular cutoff assumption is believed
to have regularizing effect on the solution because of the non-integrable
angular singularity of the cross-section. However, even though so far this has
been justified satisfactorily for the spatially homogeneous Boltzmann equation,
it is still basically unsolved for the spatially inhomogeneous Boltzmann
equation. In this paper, by sharpening the coercivity and upper bound estimates
for the collision operator, establishing the hypo-ellipticity of the Boltzmann
operator based on a generalized version of the uncertainty principle, and
analyzing the commutators between the collision operator and some weighted
pseudo differential operators, we prove the regularizing effect in all (time,
space and velocity) variables on solutions when some mild regularity is imposed
on these solutions. For completeness, we also show that when the initial data
has this mild regularity and Maxwellian type decay in velocity variable, there
exists a unique local solution with the same regularity, so that this solution
enjoys the regularity for positive time
Modeling and Simulation of Thick Sprays through Coupling of a Finite Volume Euler Equation Solver and a Particle Method for a Disperse Phase
We present here the principles of the coupling between an efficient numerical method for hyperbolic systems, namely the FVCF scheme (that is, a finite volume scheme used in the context of non conservative equations arising in multiphase flows), on the one hand; and a particle method for the Vlasov-Boltzmann equation of PIC-DSMC type (that is, in which macroscopic quantities are computed in each cell by adding quantities attached to the particles, and where integrals are computed thanks to a random sampling), on the other hand. Numerical results illustrating this coupling are shown for a problem involving a spray (droplets inside an underlying gas) in a pipe which is modeled by a 1D fluid-kinetic system
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