Doktorkolloquium: Anna Bohun

Flows of singular vector fields and applications to fluid and kinetic equations

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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