Seminar Analysis: Paolo Bonicatto (SISSA Trieste)

Uniqueness of weak solutions to transport equation with two-dimensional nearly incompressible BV vector field

Given a bounded, autonomous vector field b: R² → R², we study the uniqueness of bounded solutions to the initial value problem for the related transport equation

(1)  ∂tu + b · ∇u = 0. 

We prove that uniqueness of weak solutions holds under the assumptions that b is of class BV and it is nearly incompressible. Our proof is based on a splitting technique (introduced previously by Alberti, Bianchini and Crippa) that allows to reduce (1) to a family of 1-dimensional equations which can be solved explicitly, thus yielding uniqueness for the original problem.

In order to perform this program, we use Disintegration Theorem and known results on the structure of level sets of Lipschitz maps: this is done after a suitable localization of the problem, in which we exploit also Ambrosio’s superposition principle. 

This is joint work with S. Bianchini and N. A. Gusev. 

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