Quantitative Regularity Estimates for Compressible Transport Equations

These notes aim at presenting some recent estimates for transport equations with rough, i.e., non-smooth, velocity fields. Our final goal is to use those estimates to obtain new results on complex systems where the transport equation is coupled to other PDE’s: A driving example being the compressible Navier–Stokes system. But for simplicity, we work in the linear setting where the velocity field is given and only briefly sketch at the end of the notes how to use the new theory for nonlinear estimates. After reviewing some of the classical results, we focus on /quantitative/ estimates, in the absence of any bounds on the divergence of the velocity fields (or any corresponding bound on the Jacobian of the Lagrangian flow) for which a new approach is needed.