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Continuity equations with non smooth velocity: fluid dynamics and further applications
Third-party funded project
Project title Continuity equations with non smooth velocity: fluid dynamics and further applications
Principal Investigator(s) Crippa, Gianluca
Project Members Bohun, Anna
Organisation / Research unit Departement Mathematik und Informatik / Analysis (Crippa)
Project start 01.04.2012
Probable end 30.09.2014
Status Completed
Abstract

 

We will address various open problems related to the behaviour of the continuity equation and of the associated ordinary differential equation when the vector field governing the transport process lacks the usual (Lipschitz) regularity properties. The motivations come from the applications of such results to nonlinear problems, originating in fluid dynamics or in the theory of conservation laws.
Besides exploiting hyperbolic PDEs techniques, the analysis requires new tools from geometric measure theory, properly adapted in order to describe and control the irregular behaviours under consideration.
One first line of work regards a precise understanding of further suitable weak settings in which the continuity equation and the ordinary differential equation are well-posed and enjoy additional properties (compactness or regularity of solutions, for instance). A second line will address some questions on two-dimensional incompressible nonviscous fluids, mainly in the framework of measure-valued vorticity.

We will address various open problems related to the behaviour of the continuity equation and of the associated ordinary differential equation when the vector field governing the transport process lacks the usual (Lipschitz) regularity properties. The motivations come from the applications of such results to nonlinear problems, originating in fluid dynamics or in the theory of conservation laws. 

Besides exploiting hyperbolic PDEs techniques, the analysis requires new tools from geometric measure theory, properly adapted in order to describe and control the irregular behaviours under consideration. 

One first line of work regards a precise understanding of further suitable weak settings in which the continuity equation and the ordinary differential equation are well-posed and enjoy additional properties (compactness or regularity of solutions, for instance). A second line will address some questions on two-dimensional incompressible nonviscous fluids, mainly in the framework of measure-valued vorticity.

 

Financed by Swiss National Science Foundation (SNSF)
   

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