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Title: Variational Principle for General Diffusion Problems

Abstract

We employ the Monge-Kantorovich mass transfer theory to study the existence of solutions for a large class of parabolic partial differential equations. We deal with nonhomogeneous nonlinear diffusion problems(of Fokker-Planck type) with time-dependent coefficients. This work greatly extends the applicability of known techniques based on constructing weak solutions by approximation with time-interpolants of minimizers arising from Wasserstein-type implicit schemes. It also generalizes previous results of the authors, where proofs of convergence in the case of a right-hand side in the equation is given by these methods. To prove the existence of weak solutions we establish an interesting maximum principle for such equations. This involves comparison with the solution for the corresponding homogeneous, time-independent equation.

Authors:
;  [1]
  1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213 (United States)
Publication Date:
OSTI Identifier:
21064222
Resource Type:
Journal Article
Journal Name:
Applied Mathematics and Optimization
Additional Journal Information:
Journal Volume: 50; Journal Issue: 3; Other Information: DOI: 10.1007/s00245-004-0801-2; Copyright (c) 2004 Springer; www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0095-4616
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; COMPARATIVE EVALUATIONS; CONVERGENCE; DIFFUSION; FOKKER-PLANCK EQUATION; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; TIME DEPENDENCE; VARIATIONAL METHODS

Citation Formats

Petrelli, Luca, and Tudorascu, Adrian. Variational Principle for General Diffusion Problems. United States: N. p., 2004. Web. doi:10.1007/S00245-004-0801-2.
Petrelli, Luca, & Tudorascu, Adrian. Variational Principle for General Diffusion Problems. United States. https://doi.org/10.1007/S00245-004-0801-2
Petrelli, Luca, and Tudorascu, Adrian. 2004. "Variational Principle for General Diffusion Problems". United States. https://doi.org/10.1007/S00245-004-0801-2.
@article{osti_21064222,
title = {Variational Principle for General Diffusion Problems},
author = {Petrelli, Luca and Tudorascu, Adrian},
abstractNote = {We employ the Monge-Kantorovich mass transfer theory to study the existence of solutions for a large class of parabolic partial differential equations. We deal with nonhomogeneous nonlinear diffusion problems(of Fokker-Planck type) with time-dependent coefficients. This work greatly extends the applicability of known techniques based on constructing weak solutions by approximation with time-interpolants of minimizers arising from Wasserstein-type implicit schemes. It also generalizes previous results of the authors, where proofs of convergence in the case of a right-hand side in the equation is given by these methods. To prove the existence of weak solutions we establish an interesting maximum principle for such equations. This involves comparison with the solution for the corresponding homogeneous, time-independent equation.},
doi = {10.1007/S00245-004-0801-2},
url = {https://www.osti.gov/biblio/21064222}, journal = {Applied Mathematics and Optimization},
issn = {0095-4616},
number = 3,
volume = 50,
place = {United States},
year = {2004},
month = {10}
}