Transport by Sobolev Vector Fields: Regularity and Asymptotic Mixing

In 1989, DiPerna and Lions developed a well-posedness theory for bounded solutions of transport equations driven by Sobolev vector fields. A related conjecture on mixing efficiency in incompressible flows was formulated by Bressan in 2003 and remains open. First, I will review the existing well-posedness theory and discuss recent developments in regularity theory, including a new result. Then, I will survey progress on Bressan’s mixing conjecture and present a new result establishing its asymptotic validity for time-periodic vector fields. This result is obtained by adapting tools from smooth ergodic theory, such as Lyapunov exponents, to the non-smooth setting of regular Lagrangian flows. The talk is based on joint work with Elia Brué, Maria Colombo and Guido De Philippis.