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SUMMARY:On the sticky particle solutions to the pressureless Euler system 
 in general dimension - Sara Daneri (Gran Sasso Science Institute\, L'Aquil
 a)
DTSTART:20220214T133000Z
DTEND:20220214T143000Z
UID:TALK167324@talks.cam.ac.uk
DESCRIPTION:In this talk we consider the pressureless Euler system in dime
 nsion greater than or equal to two. Several works have been devoted to the
  search of solutions which satisfy the following adhesion or sticky partic
 le principle: if two particles of the fluid do not interact\, then they mo
 ve freely keeping constant velocity\, otherwise they join with velocity gi
 ven by the balance of momentum. For initial data given by a finite number 
 of particles pointing each in a given direction\, in general dimension\, i
 t is easy to show that a global sticky particle solution always exists and
  is unique. In dimension one\, sticky particle solutions have been proved 
 to exist and be unique.&nbsp\; In dimension greater or equal than two\, it
  was shown that as soon as the initial data is not concentrated on a finit
 e number of particles\, it might lead to non-existence or non-uniqueness o
 f sticky particle solutions.\nIn collaboration with S. Bianchini\,&nbsp\; 
 we show that&nbsp\; even though the sticky particle solutions are not well
 -posed for every measure-type initial data\, there exists a comeager set o
 f initial data in the weak topology giving rise to a unique sticky particl
 e solution. Moreover\, for any of these initial data the sticky particle &
 nbsp\;solution is unique also in the larger class of dissipative solutions
  (where trajectories are allowed to cross) and is given by a trivial free 
 flow concentrated on trajectories which do not intersect. In particular fo
 r such initial data there is only one dissipative solution and its dissipa
 tion is equal to zero. Thus\, for a comeager set of initial data the probl
 em of finding sticky particle solutions is well-posed\, but the dynamics t
 hat one &nbsp\;sees is trivial. Our notion of dissipative solution is lagr
 angian and therefore general enough to include weak and measure-valued sol
 utions.
LOCATION:Seminar Room 1\, Newton Institute
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