Seminar Analysis and Mathematical Physics: Thérèse Moerschell (EPFL)

Examples of non-uniqueness for the advection-diffusion equation

The advection-diffusion equation is known to have unique solutions for any vector field that is L^2 in time and in space. But what happens when we have slightly less than square integrability? In this talk we will explore two examples of vector fields in L^p(0,T;L^q(\T^d)) made of shear flows that prove the non-uniqueness of solutions whenever we have p<2 or q<2. We will first show that they give different solutions to the advection equation and then use the Feynman-Kac formula to show that diffusion has little effect if our parameters are well-tuned.
This is part of my Master's thesis, supervised by Massimo Sorella and Maria Colombo.

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