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SUMMARY:Mixing Estimates for Sobolev and BV Vector Fields - Lucas Huysmans
  (MPI Leipzig)
DTSTART:20260223T140000Z
DTEND:20260223T150000Z
UID:TALK242167@talks.cam.ac.uk
CONTACT:48139
DESCRIPTION:I will discuss recent work on quantitative estimates for the p
 assive scalar transport equation when the advecting velocity field is belo
 w the Lipschitz regularity class. In this setting\, the classical Cauchy-L
 ipschitz theory fails to give control on the rate at which the solution ma
 y grow\, usually measured by Sobolev norms. Despite this\, well-posedness 
 of the transport equation has been known when the vector field lies in W^{
 1\,1} since 1989\, and for vector fields in BV since 2004\, under the addi
 tional assumption that the divergence remains bounded. It was conjectured 
 by Bressan in 2003 that the same quantitative "mixing estimates" given by 
 the Cauchy-Lipschitz theory should hold also in this setting. Since then\,
  much progress has been made in recovering these estimates when the vector
  field lies in W^{1\,p} with p>1\, however the endpoint case p=1\, and in 
 particular BV\, has remained a challenging open problem. In this talk I wi
 ll discuss the first quantitative mixing and stability estimates for a pas
 sive scalar transported by W^{1\,1} and BV autonomous vector fields with z
 ero divergence. The proof involves quantitative weak harmonic estimates in
  BV\, and a double pigeonhole principle to quantify the weak-* compactness
  of the solution space.
LOCATION:MR13
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