535 research outputs found
Geometric properties of cones with applications on the Hellinger-Kantorovich space, and a new distance on the space of probability measures
We study general geometric properties of cone spaces, and we apply them on
the Hellinger--Kantorovich space We exploit a two-parameter scaling property of the
Hellinger-Kantorovich metric
and we prove the existence of a distance on the space of Probability measures that
turns the Hellinger--Kantorovich space
into a cone
space over the space of probabilities measures
We provide a two parameter rescaling of geodesics in
and for
we obtain a full characterization of the geodesics. We
finally prove finer geometric properties, including local-angle condition and
partial -semiconcavity of the squared distances, that will be used in a
future paper to prove existence of gradient flows on both spaces
BV solutions and viscosity approximations of rate-independent systems
In the nonconvex case solutions of rate-independent systems may develop jumps
as a function of time. To model such jumps, we adopt the philosophy that rate
independence should be considered as limit of systems with smaller and smaller
viscosity. For the finite-dimensional case we study the vanishing-viscosity
limit of doubly nonlinear equations given in terms of a differentiable energy
functional and a dissipation potential which is a viscous regularization of a
given rate-independent dissipation potential. The resulting definition of 'BV
solutions' involves, in a nontrivial way, both the rate-independent and the
viscous dissipation potential, which play a crucial role in the description of
the associated jump trajectories. We shall prove a general convergence result
for the time-continuous and for the time-discretized viscous approximations and
establish various properties of the limiting BV solutions. In particular, we
shall provide a careful description of the jumps and compare the new notion of
solutions with the related concepts of energetic and local solutions to
rate-independent systems
A gradient system with a wiggly energy and relaxed EDP-convergence
If gradient systems depend on a microstructure, we want to derive a
macroscopic gradient structure describing the effective behavior of the
microscopic effects. We introduce a notion of evolutionary Gamma-convergence
that relates the microscopic energy and the microscopic dissipation potential
with their macroscopic limits via Gamma-convergence. This new notion
generalizes the concept of EDP-convergence, which was introduced in
arXiv:1507.06322, and is called "relaxed EDP-convergence". Both notions are
based on De Giorgi's energy-dissipation principle, however the special
structure of the dissipation functional in terms of the primal and dual
dissipation potential is, in general, not preserved under Gamma-convergence. By
investigating the kinetic relation directly and using general forcings we still
derive a unique macroscopic dissipation potential.
The wiggly-energy model of James et al serves as a prototypical example where
this nontrivial limit passage can be fully analyzed.Comment: 43 pages, 8 figure
Interaction of modulated pulses in the nonlinear Schroedinger equation with periodic potential
We consider a cubic nonlinear Schroedinger equation with periodic potential.
In a semiclassical scaling the nonlinear interaction of modulated pulses
concentrated in one or several Bloch bands is studied. The notion of closed
mode systems is introduced which allows for the rigorous derivation of a finite
system of amplitude equations describing the macroscopic dynamics of these
pulses
Optimal transport in competition with reaction: the Hellinger-Kantorovich distance and geodesic curves
We discuss a new notion of distance on the space of finite and nonnegative
measures which can be seen as a generalization of the well-known
Kantorovich-Wasserstein distance. The new distance is based on a dynamical
formulation given by an Onsager operator that is the sum of a Wasserstein
diffusion part and an additional reaction part describing the generation and
absorption of mass.
We present a full characterization of the distance and its properties. In
fact the distance can be equivalently described by an optimal transport problem
on the cone space over the underlying metric space. We give a construction of
geodesic curves and discuss their properties
A rate-independent model for the isothermal quasi-static evolution of shape-memory materials
This note addresses a three-dimensional model for isothermal stress-induced
transformation in shape-memory polycrystalline materials. We treat the problem
within the framework of the energetic formulation of rate-independent processes
and investigate existence and continuous dependence issues at both the
constitutive relation and quasi-static evolution level. Moreover, we focus on
time and space approximation as well as on regularization and parameter
asymptotics.Comment: 33 pages, 3 figure
High-frequency averaging in semi-classical Hartree-type equations
We investigate the asymptotic behavior of solutions to semi-classical
Schroedinger equations with nonlinearities of Hartree type. For a weakly
nonlinear scaling, we show the validity of an asymptotic superposition
principle for slowly modulated highly oscillatory pulses. The result is based
on a high-frequency averaging effect due to the nonlocal nature of the Hartree
potential, which inhibits the creation of new resonant waves. In the proof we
make use of the framework of Wiener algebras.Comment: 13 pages; Version 2: Added Remark 2.
Recommended from our members
Uniform exponential decay for reaction-diffusion systems with complex-balanced mass-action kinetics : dedicated to Bernold Fiedler on the occasion of his sixtieth birthday
We consider reaction-diffusion systems on a bounded domain with no-flux
boundary conditions. All reactions are given by the mass-action law and are
assumed to satisfy the complex-balance condition. In the case of a diagonal
diffusion matrix, the relative entropy is a Liapunov functional. We give an
elementary proof for the Liapunov property as well a few explicit examples
for the condition of complex or detailed balancing. We discuss three methods
to obtain energy-dissipation estimates, which guarantee exponential decay of
the relative entropy, all of which rely on the log-Sobolev estimate and
suitable handling of the reaction terms as well as the mass-conservation
relations. The three methods are (i) a convexification argument based on the
authors joint work with Haskovec and Markowich, (ii) a series of analytical
estimates derived by Desvillettes, Fellner, and Tang, and (iii) a compactness
argument of developed by Glitzky and Hünlich
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