Doktorkolloquium: Anna Bohun
Several physical phenomena arising in fluid dynamics and kinetic equations can be modeled by nonlinear transport PDE. Such quantities are the vorticity of a fluid, or the density of a collection of particles advected by a velocity field which is highly irregular. The theory of characteristics provides a link between this PDE and the ODE dX/dt=b(t,X(t,x)), where $b$ is the velocity field. Given a vector field with Sobolev or BV regularity and bounded divergence, the theory of DiPerna-Lions and Ambrosio gives a good notion of solution to the ordinary differential equation using the concept of regular Lagrangian flow. I will discuss the recent works with Crippa-Bouchut regarding Lagrangian flows associated to velocity fields with anisotropic regularity: those with gradient given by the singular integral of an L1 function in some directions, and the singular integral of a measure in others. This answers positively the question of existence of Lagrangian solutions to the Vlasov Poisson and Euler equations with L1 data.
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