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In these lectures we will discuss some estimates for transport equations with non-smooth velocity fields. The overarching goal is to apply this theory to nonlinear systems, where the transport equation is coupled to a nonlinear PDE for the velocity field (e.g, the compressible Navier-Stokes equations). With this motivation in mind, even though we will mostly focus on the linear case, we will derive estimates which are: (1) quantitative estimates of regularity; (2) avoid bounds on the divergence of the velocity field and lower/upper bounds on the transported density.
The lecture is organised according to the following schedule:
1. Short review of the Cauchy-Lipschitz theory
2. Lagrangian estimates for flows with Sobolev velocity
3. Eulerian approaches, renormalised solutions
4. Quantitative estimates for compressible transport
5. (Outline of) Applications to nonlinear coupled systems
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