16 research outputs found
On the discrete Safronov-Dubovskii coagulation equation: well-posedness, mass-conservation and asymptotic behaviour
The global existence of mass-conserving weak solutions to the
Safronov-Dubovskii coagulation equation is shown for the coagulation kernels
satisfying the at most linear growth for large sizes. In contrast to previous
works, the proof mainly relies on the de la Vallee-Poussin theorem [8, Theorem
7.1.6], which only requires the finiteness of the first moment of the initial
condition. By showing the necessary regularity of solutions, it is shown that
the weak solutions con-structed herein are indeed classical solutions. Under
additional restrictions on the initial data, the uniqueness of solutions is
also shown. Finally, the continuous dependence on the initial data and the
large-time behaviour of solutions are also addressed
Mathematical aspects of coagulation-fragmentation equations
We give an overview of the mathematical literature on the coagulation-like equations, from an analytic deterministic perspective. In Section 1 we present the coagulation type equations more commonly encountered in the scientific and mathematical literature and provide a brief historical overview of relevant works. In
Section 2 we present results about existence and uniqueness of solutions in some of those systems, namely the discrete Smoluchowski and coagulation-fragmentation: we start by a brief description of the functional spaces, and then review the results on existence of solutions with a brief description of the main ideas of the proofs. This part closes with the consideration of uniqueness results. In Sections 3 and 4 we are concerned with several aspects of the solutions behaviour.We pay special attention
to the long time convergence to equilibria, self-similar behaviour, and density conservation or lack thereof
Existence of self-similar profile for a kinetic annihilation model
We show the existence of a self-similar solution for a modified Boltzmann
equation describing probabilistic ballistic annihilation. Such a model
describes a system of hard-spheres such that, whenever two particles meet, they
either annihilate with probability or they undergo an
elastic collision with probability . For such a model, the number
of particles, the linear momentum and the kinetic energy are not conserved. We
show that, for smaller than some explicit threshold value ,
a self-similar solution exists.Comment: This new version supersedes and replaces the previous one. We found a
mistake in the previous (and published) version of the manuscript and
explained how to fix it in "Erratum to "Existence of self-similar profile for
a kinetic annihilation model" [J. Differential Equations 254 (7) (2013)
3023-3080]. J. Differential Equations 257 (2014), no. 8, 3071-3074." This
version provides a complete and corrected version of the previous manuscrip
An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach
We summarise the properties and the fundamental mathematical results
associated with basic models which describe
coagulation and fragmentation processes in a deterministic manner
and in which cluster size is a discrete quantity (an integer
multiple of some basic unit size).
In particular, we discuss Smoluchowski's equation for aggregation,
the Becker-Döring model of simultaneous aggregation and fragmentation,
and more general models involving coagulation and fragmentation
An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach
We summarise the properties and the fundamental mathematical resultsassociated with basic models which describecoagulation and fragmentation processes in a deterministic mannerand in which cluster size is a discrete quantity (an integermultiple of some basic unit size).In particular, we discuss Smoluchowski's equation for aggregation,the Becker-Döring model of simultaneous aggregation and fragmentation,and more general models involving coagulation and fragmentation
EQUILIBRIUM SOLUTION TO THE INELASTIC BOLTZMANN EQUATION DRIVEN BY A PARTICLES THERMAL BATH
International audienceWe show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host-medium with a fixed distribution. This is achieved by controlling the Lp-norms, the moments and the regularity of the solutions for the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states
Control of star formation by supersonic turbulence
Understanding the formation of stars in galaxies is central to much of modern
astrophysics. For several decades it has been thought that stellar birth is
primarily controlled by the interplay between gravity and magnetostatic
support, modulated by ambipolar diffusion. Recently, however, both
observational and numerical work has begun to suggest that support by
supersonic turbulence rather than magnetic fields controls star formation. In
this review we outline a new theory of star formation relying on the control by
turbulence. We demonstrate that although supersonic turbulence can provide
global support, it nevertheless produces density enhancements that allow local
collapse. Inefficient, isolated star formation is a hallmark of turbulent
support, while efficient, clustered star formation occurs in its absence. The
consequences of this theory are then explored for both local star formation and
galactic scale star formation. (ABSTRACT ABBREVIATED)Comment: Invited review for "Reviews of Modern Physics", 87 pages including 28
figures, in pres