Continuity equations with non smooth velocity: quantitative estimates and applications to nonlinear problems
Third-party funded project
Project title Continuity equations with non smooth velocity: quantitative estimates and applications to nonlinear problems
Principal Investigator(s) Crippa, Gianluca
Project Members Bohun, Anna
Ligabue, Silvia
Organisation / Research unit Departement Mathematik und Informatik / Analysis (Crippa)
Project start 01.10.2014
Probable end 30.09.2018
Status Completed
Abstract

 

We investigate well-posedness and further properties for the continuity equation and for the ordinary differential equation out of the classical smooth (Lipschitz regular) framework.
The leading theme is the search for quantitative estimates: stability, compactness and regularity statements in which we give an explicit control of the quantities under analysis in terms of natural bounds on the data.
This quantitative analysis allows us to study several applications to nonlinear partial differential equations: we address various questions about existence, continuity with respect to initial data, convergence of singular approximations and convergence of nonlocal approximations for the Euler equation, the Vlasov-Poisson equation, and the Burgers' equation.

We investigate well-posedness and further properties for the continuity equation and for the ordinary differential equation out of the classical smooth (Lipschitz regular) framework.

The leading theme is the search for quantitative estimates: stability, compactness and regularity statements in which we give an explicit control of the quantities under analysis in terms of natural bounds on the data.

This quantitative analysis allows us to study several applications to nonlinear partial differential equations: we address various questions about existence, continuity with respect to initial data, convergence of singular approximations and convergence of nonlocal approximations for the Euler equation, the Vlasov-Poisson equation, and the Burgers' equation.

 

Keywords continuity equation, ordinary differential equation, regular Lagrangian flow, geometric measure theory, nonlocal conservation laws, two-dimensional Euler equation
Financed by Swiss National Science Foundation (SNSF)
Follow-up project of 1121993 Continuity equations with non smooth velocity: fluid dynamics and further applications
   

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20/10/2018