190 research outputs found
Homogenization Theory: Periodic and Beyond (online meeting)
The objective of the workshop has been to review the latest developments in homogenization theory for a large category of equations and settings arising in the modeling of solid, fluids, wave propagation, heterogeneous media, etc. The topics approached have covered periodic and nonperiodic deterministic homogenization, stochastic homogenization, regularity theory, derivation of wall laws and detailed study of boundary layers,..
Homogenization of nonlinear stochastic partial differential equations in a general ergodic environment
In this paper, we show that the concept of sigma-convergence associated to
stochastic processes can tackle the homogenization of stochastic partial
differential equations. In this regard, the homogenization problem for a
stochastic nonlinear partial differential equation is studied. Using some deep
compactness results such as the Prokhorov and Skorokhod theorems, we prove that
the sequence of solutions of this problem converges in probability towards the
solution of an equation of the same type. To proceed with, we use a suitable
version of sigma-convergence method, the sigma-convergence for stochastic
processes, which takes into account both the deterministic and random
behaviours of the solutions of the problem. We apply the homogenization result
to some concrete physical situations such as the periodicity, the almost
periodicity, the weak almost periodicity, and others.Comment: To appear in: Stochastic Analysis and Application
Reduction Methods in Climate Dynamics -- A Brief Review
We review a range of reduction methods that have been, or may be useful for
connecting models of the Earth's climate system of differing complexity. We
particularly focus on methods where rigorous reduction is possible. We aim to
highlight the main mathematical ideas of each reduction method and also provide
several benchmark examples from climate modelling
On operator estimates in homogenization of nonlocal operators of convolution type
The paper studies a bounded symmetric operator in
with here is a small
positive parameter. It is assumed that is a non-negative
function such that and the moments , , are finite. It is also assumed
that is -periodic both in and function such
that and . Our
goal is to study the limit behaviour of the resolvent
, as . We show that, as
, the operator
converges in the operator norm in to the resolvent
of the effective operator being a
second order elliptic differential operator with constant coefficients of the
form . We then obtain sharp in
order estimates of the rate of convergence
A two-scale model of two-phase ow in porous media ranging from porespace to the macro scale
We will derive two-scale models for two-phase flow in porous media, with the microscale
given by the porescale. The resulting system will account for balance of mass, momentum
and energy. To this aim, we will combine a generalization of Rajagopal’s and Srinivasa’s assumption
of maximum rate of entropy production [39, 20, 21] with formal asymptotic expansion. The
microscopic model will be based on phase fields, in particular to the full Cahn-Hilliard-Navier-
Stokes-Fourier model derived in [23] with the boundary conditions from [20]. Using a generalized
notion of characteristic functions, we will show that the solutions to the two-scale model macroscopically
behave like classical solutions to a system of porous media flow equations. Relative
permeabilities and capillary pressure relations are outcomes of the theory and exist only for special
cases. Therefore, the two-scale model can be considered as a true generalization of classical
models providing more information on the microscale thereby making the introduction of hysteresis
superfluous
Interplay of Theory and Numerics for Deterministic and Stochastic Homogenization
The workshop has brought together experts in the broad field of partial differential equations with highly heterogeneous coefficients. Analysts and computational and applied mathematicians have shared results and ideas on a topic of considerable interest both from the theoretical and applied viewpoints. A characteristic feature of the workshop has been to encourage discussions on the theoretical as well as numerical challenges in the field, both from the point of view of deterministic as well as stochastic modeling of the heterogeneities
Recent Advances in Industrial and Applied Mathematics
This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress
Recent Advances in Industrial and Applied Mathematics
This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress
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