Loss of regularity for the continuity equation with non-Lipschitz velocity field

28 Mar 2019 Alberti Giovanni Crippa Gianluca Mazzucato Anna L.

We consider the Cauchy problem for the continuity equation in space dimension ${d \geq 2}$. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces $W^{1,p}$, for $1 \leq p<\infty$, and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time... (read more)

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