89 research outputs found
Poincar\'e and log-Sobolev inequalities for mixtures
This work studies mixtures of probability measures on and
gives bounds on the Poincar\'e and the log-Sobolev constant of two-component
mixtures provided that each component satisfies the functional inequality, and
both components are close in the -distance. The estimation of those
constants for a mixture can be far more subtle than it is for its parts. Even
mixing Gaussian measures may produce a measure with a Hamiltonian potential
possessing multiple wells leading to metastability and large constants in
Sobolev type inequalities. In particular, the Poincar\'e constant stays bounded
in the mixture parameter whereas the log-Sobolev may blow up as the mixture
ratio goes to or . This observation generalizes the one by Chafa\"i and
Malrieu to the multidimensional case. The behavior is shown for a class of
examples to be not only a mere artifact of the method.Comment: 13 page
Macroscopic limit of the Becker-D\"oring equation via gradient flows
This work considers gradient structures for the Becker-D\"oring equation and
its macroscopic limits. The result of Niethammer [17] is extended to prove the
convergence not only for solutions of the Becker-D\"oring equation towards the
Lifshitz-Slyozov-Wagner equation of coarsening, but also the convergence of the
associated gradient structures. We establish the gradient structure of the
nonlocal coarsening equation rigorously and show continuous dependence on the
initial data within this framework. Further, on the considered time scale the
small cluster distribution of the Becker--D\"oring equation follows a
quasistationary distribution dictated by the monomer concentration
Analysis of the implicit upwind finite volume scheme with rough coefficients
We study the implicit upwind finite volume scheme for numerically
approximating the linear continuity equation in the low regularity
DiPerna-Lions setting. That is, we are concerned with advecting velocity fields
that are spatially Sobolev regular and data that are merely integrable. We
prove that on unstructured regular meshes the rate of convergence of
approximate solutions generated by the upwind scheme towards the unique
distributional solution of the continuous model is at least 1/2. The numerical
error is estimated in terms of logarithmic Kantorovich-Rubinstein distances and
provides thus a bound on the rate of weak convergence.Comment: 27 pages. To appear in Numerische Mathemati
A non-local problem for the Fokker-Planck equation related to the Becker-D\"{o}ring model
This paper concerns a Fokker-Planck equation on the positive real line
modeling nucleation and growth of clusters. The main feature of the equation is
the dependence of the driving vector field and boundary condition on a
non-local order parameter related to the excess mass of the system.
The first main result concerns the well-posedness and regularity of the
Cauchy problem. The well-posedness is based on a fixed point argument, and the
regularity on Schauder estimates. The first a priori estimates yield H\"older
regularity of the non-local order parameter, which is improved by an iteration
argument.
The asymptotic behavior of solutions depends on some order parameter
depending on the initial data. The system shows different behavior depending on
a value , determined from the potentials and diffusion coefficient.
For , there exists an equilibrium solution
. If the solution converges strongly to
, while if the solution converges
weakly to . The excess gets lost due
to the formation of larger and larger clusters. In this regard, the model
behaves similarly to the classical Becker-D\"oring equation.
The system possesses a free energy, strictly decreasing along the evolution,
which establishes the long time behavior. In the subcritical case
the entropy method, based on suitable weighted logarithmic Sobolev inequalities
and interpolation estimates, is used to obtain explicit convergence rates to
the equilibrium solution.
The close connection of the presented model and the Becker-D\"oring model is
outlined by a family of discrete Fokker-Planck type equations interpolating
between both of them. This family of models possesses a gradient flow
structure, emphasizing their commonality.Comment: Minor revised version accepted for publication in Discrete &
Continuous Dynamical Systems -
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