Existence and uniqueness for the transport of currents by Lipschitz vector fields

Abstract

This work establishes the existence and uniqueness of solutions to the initialvalue problem for the geometric transport equation d dt Tt + L b Tt = 0 in the class of k-dimensional integral or normal currents Tt (t being the time variable) under the natural assumption of Lipschitz regularity of the driving vector field b. Our argument relies crucially on the notion of decomposability bundle introduced recently by Alberti and Marchese. In the particular case of 0-currents, this also yields a new proof of the uniqueness for the continuity equation in the class of signed measures.

On the Transport of Currents

“…t ψ(t, x) = α (t)β(Φ −t (x)) + α(t)∇β(Φ −t (x)) • d dt Φ −t (x) = α (t)β(Φ −t (x)) − α(t)∇β(Φ −t (x)) • b(Φ −t (x)),where we used the defining property of Φ. On the other hand, by an elementary computation on the directional derivative of the flow (see[11, Lemma 2.3]) we haveb(x) • ∇ x ψ(t, x) = α(t)b(x) • ∇β(Φ −t (x))DΦ −t (x) = α(t)∇β(Φ −t (x)) • DΦ −t (x)[b(x)] = α(t)∇β(Φ −t (x)) • b(Φ −t (x)).…”