3,274 research outputs found
Self-similar solutions for the LSW model with encounters
The LSW model with encounters has been suggested by Lifshitz and Slyozov as a
regularization of their classical mean-field model for domain coarsening to
obtain universal self-similar long-time behavior. We rigorously establish that
an exponentially decaying self-similar solution to this model exist, and show
that this solutions is isolated in a certain function space. Our proof relies
on setting up a suitable fixed-point problem in an appropriate function space
and careful asymptotic estimates of the solution to a corresponding homogeneous
problem.Comment: 22 page
Kramers' formula for chemical reactions in the context of Wasserstein gradient flows
We derive Kramers' formula as singular limit of the Fokker-Planck equation
with double-well potential. The convergence proof is based on the Rayleigh
principle of the underlying Wasserstein gradient structure and complements a
recent result by Peletier, Savar\'e and Veneroni.Comment: revised proofs, 12 pages, 1 figur
Non-self-similar behavior in the LSW theory of Ostwald ripening
The classical Lifshitz-Slyozov-Wagner theory of domain coarsening predicts
asymptotically self-similar behavior for the size distribution of a dilute
system of particles that evolve by diffusional mass transfer with a common mean
field. Here we consider the long-time behavior of measure-valued solutions for
systems in which particle size is uniformly bounded, i.e., for initial measures
of compact support.
We prove that the long-time behavior of the size distribution depends
sensitively on the initial distribution of the largest particles in the system.
Convergence to the classically predicted smooth similarity solution is
impossible if the initial distribution function is comparable to any finite
power of distance to the end of the support. We give a necessary criterion for
convergence to other self-similar solutions, and conditional stability theorems
for some such solutions. For a dense set of initial data, convergence to any
self-similar solution is impossible.Comment: 31 pages, LaTeX2e; Revised version, to appear in J. Stat. Phy
Optimal bounds for self-similar solutions to coagulation equations with product kernel
We consider mass-conserving self-similar solutions of Smoluchowski's
coagulation equation with multiplicative kernel of homogeneity . We establish rigorously that such solutions exhibit a singular behavior
of the form as . This property had been
conjectured, but only weaker results had been available up to now
Self-similar gelling solutions for the coagulation equation with diagonal kernel
We consider Smoluchowski's coagulation equation in the case of the diagonal
kernel with homogeneity . In this case the phenomenon of gelation
occurs and solutions lose mass at some finite time. The problem of the
existence of self-similar solutions involves a free parameter , and one
expects that a physically relevant solution (i.e. nonnegative and with
sufficiently fast decay at infinity) exists for a single value of ,
depending on the homogeneity . We prove this picture rigorously for
large values of . In the general case, we discuss in detail the
behaviour of solutions to the self-similar equation as the parameter
changes
On a thermodynamically consistent modification of the Becker-Doering equations
Recently, Dreyer and Duderstadt have proposed a modification of the
Becker--Doering cluster equations which now have a nonconvex Lyapunov function.
We start with existence and uniqueness results for the modified equations. Next
we derive an explicit criterion for the existence of equilibrium states and
solve the minimization problem for the Lyapunov function. Finally, we discuss
the long time behavior in the case that equilibrium solutions do exist
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