Construction of nonunique solutions of the transport and continuity equation for Sobolev vector fields in DiPerna–Lions’ theory

The HADES seminar on Tuesday, October 3rd will be at 3:30pm in Room 740.

Speaker: Anuj Kumar

Abstract: In this talk, we are concerned with DiPerna–Lions’ theory for the transport equation. In the first part of the talk, I will discuss a few results regarding the nonuniqueness of trajectories of the associated ODE. Alberti ’12 asked the following question: are there continuous Sobolev vector fields with bounded divergence such that the set of initial conditions for which the trajectories are not unique is of full measure? We construct an explicit example of divergence-free H\”older continuous Sobolev vector field for which trajectories are not unique on a set of full measure, which then answers the question of Alberti. The construction is based on building an appropriate Cantor set and a “blob flow” vector field to translate cubes in space. The vector field constructed also implies nonuniqueness in the class of measure solutions. The second part to talk is a more recent work jointly with E. Bruè and M. Colombo. We construct nonunique solutions of the continuity equation in the class L^\infty in time and L^r in space. We prove nonuniqueness in the range of exponents beyond what is available using the method of convex integration and sharply match with the range of uniqueness of solutions from Bruè, Colombo, De Lellis’ 21.

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