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Optimal transport methods in partial differential equation

Grant number: 15/20962-7
Support type:Scholarships abroad - Research Internship - Post-doctor
Effective date (Start): February 22, 2016
Effective date (End): March 21, 2016
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Lucas Catão de Freitas Ferreira
Grantee:Matheus Correia dos Santos
Supervisor abroad: Jose Antonio Carrillo de La Plata
Home Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Local de pesquisa : Imperial College London, England  
Associated to the scholarship:14/23326-1 - Optimal mass transport and nonlocal PDEs, BP.PD

Abstract

This project is intended to the study of recent developments in the theory of optimal transport and its applications to nonlinear nonlocal partial differential equations. These equations arise from systems where there exists conservation of mass and where the individuals of the system maintain an interaction among themselves. In some cases, the model can be identified as a gradient flow of the associated entropy on a infinite dimensional Riemannian manifold with respect to the Wasserstein metric. More specifically, we are interested in the analysis of the existence, uniqueness, well-posedness, asymptotic behaviour and steady-state properties of kinetic models like, the recent fractional versions of the porous medium equation and semidiscrete and discrete versions of the nonlinear Fokker-Plank equation where the solution can be identified as a gradient flow for finite Markov chains). We will analyse how the new non-local metrics defined on the space of probability measures can give us qualitative results for equations of the type $\partial_t\rho + (-\Delta)^s\varphi(\rho)=0$, like the classical Wasserstein metric gives on problems of the type $\partial\rho + \nabla\cdot(\rho\nabla U'(\rho))=0$. Furthermore, discrete versions of these metrics have shown to be very fruitful to generate functional inequalities like log-Sobolev and Talagran for semidiscrete problems as the linear Fokker-Planck case. We will study the non-linear case and see whether is possible to find an associated Lyapunov functinal which allow us to obtain information about the long-time asymptotic behaviour of solutions to the semidiscrete version.

Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
CARRILLO, JOSE A.; JUENGEL, ANSGAR; SANTOS, MATHEUS C. Displacement convexity for the entropy in semi-discrete non-linear Fokker-Planck equations. EUROPEAN JOURNAL OF APPLIED MATHEMATICS, v. 30, n. 6, SI, p. 1103-1122, DEC 2019. Web of Science Citations: 1.

Please report errors in scientific publications list by writing to: cdi@fapesp.br.