8,812 research outputs found
Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations
We establish a connection between Optimal Transport Theory and classical
Convection Theory for geophysical flows. Our starting point is the model
designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal
Transport problems. This model can be seen as a generalization of the
Darcy-Boussinesq equations, which is a degenerate version of the
Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate
different variants of the NSB equations (in particular what we call the
generalized Hydrostatic-Boussinesq equations) to various models involving
Optimal Transport (and the related Monge-Ampere equation. This includes the 2D
semi-geostrophic equations and some fully non-linear versions of the so-called
high-field limit of the Vlasov-Poisson system and of the Keller-Segel for
Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model
can be related to the magnetic relaxation model studied by Arnold and Moffatt
to obtain stationary solutions of the Euler equations with prescribed topology
Shape optimisation for a class of semilinear variational inequalities with applications to damage models
The present contribution investigates shape optimisation problems for a class
of semilinear elliptic variational inequalities with Neumann boundary
conditions. Sensitivity estimates and material derivatives are firstly derived
in an abstract operator setting where the operators are defined on polyhedral
subsets of reflexive Banach spaces. The results are then refined for
variational inequalities arising from minimisation problems for certain convex
energy functionals considered over upper obstacle sets in . One
particularity is that we allow for dynamic obstacle functions which may arise
from another optimisation problems. We prove a strong convergence property for
the material derivative and establish state-shape derivatives under regularity
assumptions. Finally, as a concrete application from continuum mechanics, we
show how the dynamic obstacle case can be used to treat shape optimisation
problems for time-discretised brittle damage models for elastic solids. We
derive a necessary optimality system for optimal shapes whose state variables
approximate desired damage patterns and/or displacement fields
The F model on dynamical quadrangulations
The dynamically triangulated random surface (DTRS) approach to Euclidean
quantum gravity in two dimensions is considered for the case of the elemental
building blocks being quadrangles instead of the usually used triangles. The
well-known algorithmic tools for treating dynamical triangulations in a Monte
Carlo simulation are adapted to the problem of these dynamical
quadrangulations. The thus defined ensemble of 4-valent graphs is appropriate
for coupling to it the 6- and 8-vertex models of statistical mechanics. Using a
series of extensive Monte Carlo simulations and accompanying finite-size
scaling analyses, we investigate the critical behaviour of the 6-vertex F model
coupled to the ensemble of dynamical quadrangulations and determine the matter
related as well as the graph related critical exponents of the model.Comment: LaTeX, 43 pages, 10 figures, 7 tables; substantially shortened and
revised version as published, for more details refer to V1, to be found at
http://arxiv.org/abs/hep-lat/0409028v
Out of equilibrium dynamics of classical and quantum complex systems
Equilibrium is a rather ideal situation, the exception rather than the rule
in Nature. Whenever the external or internal parameters of a physical system
are varied its subsequent relaxation to equilibrium may be either impossible or
take very long times. From the point of view of fundamental physics no generic
principle such as the ones of thermodynamics allows us to fully understand
their behaviour. The alternative is to treat each case separately. It is
illusionary to attempt to give, at least at this stage, a complete description
of all non-equilibrium situations. Still, one can try to identify and
characterise some concrete but still general features of a class of out of
equilibrium problems - yet to be identified - and search for a unified
description of these. In this report I briefly describe the behaviour and
theory of a set of non-equilibrium systems and I try to highlight common
features and some general laws that have emerged in recent years.Comment: 36 pages, to be published in Compte Rendus de l'Academie de Sciences,
T. Giamarchi e
Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities
We consider a diffuse interface model for an incompressible isothermal
mixture of two viscous Newtonian fluids with different densities in a bounded
domain in two or three space dimensions. The model is the nonlocal version of
the one recently derived by Abels, Garcke and Gr\"{u}n and consists in a
Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard
equation. The density of the mixture depends on an order parameter. For this
nonlocal system we prove existence of global dissipative weak solutions for the
case of singular double-well potentials and non degenerate mobilities. To this
goal we devise an approach which is completely independent of the one employed
by Abels, Depner and Garcke to establish existence of weak solutions for the
local Abels et al. model.Comment: 43 page
Beginner's guide to Aggregation-Diffusion Equations
The aim of this survey is to serve as an introduction to the different
techniques available in the broad field of Aggregation-Diffusion Equations. We
aim to provide historical context, key literature, and main ideas in the field.
We start by discussing the modelling and famous particular cases: Heat
equation, Fokker-Plank, Porous medium, Keller-Segel,
Chapman-Rubinstein-Schatzman, Newtonian vortex, Caffarelli-V\'azquez,
McKean-Vlasov, Kuramoto, and one-layer neural networks. In Section 4 we present
the well-posedness frameworks given as PDEs in Sobolev spaces, and
gradient-flow in Wasserstein. Then we discuss the asymptotic behaviour in time,
for which we need to understand minimisers of a free energy. We then present
some numerical methods which have been developed. We conclude the paper
mentioning some related problems
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