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SUMMARY:Rough solutions of the 3-D compressible Euler equations - Qian Wan
 g (Oxford)
DTSTART:20220131T140000Z
DTEND:20220131T150000Z
UID:TALK168788@talks.cam.ac.uk
CONTACT:Dr Greg Taujanskas
DESCRIPTION:I will talk about my work on the compressible Euler equations.
  We prove the local-in-time existence the solution of the compressible Eul
 er equations in 3-D\, for the Cauchy data of the velocity\, density and vo
 rticity (v\,\\varrho\, \\omega) in H^s^ x H^s^ x H^s'^\, for s' in (2\,s).
  The result extends the  sharp  result of Smith-Tataru and  Wang\,  establ
 ished in the irrotational case\, i.e \\omega=0\, which  is  known to be op
 timal for s greater than 2. At the opposite extreme\, in the incompressibl
 e case\, i.e. with a constant density\,  the result is known to  hold for\
 n \\omega\\in H^s^\, s greater than 3/2 and fails for s less than or equal
  to 3/2\, see the work of Bourgain-Li.  It is thus natural to conjecture t
 hat the optimal result should be  (v\,\\varrho\, \\omega) in H^s^ x H^s^ x
  H^s'^\, s greater than 2\, s' greater than 3/2. We view our work here as 
 an important step in proving the  conjecture. The  main difficulty in esta
 blishing sharp well-posedness results for general compressible Euler flow 
 is  due to the highly nontrivial interaction between  the  sound waves\, g
 overned by quasilinear wave equations\, and vorticity which is transported
  by the flow. To overcome this difficulty\, we separate the dispersive par
 t of sound wave from the transported part\, and gain regularity significan
 tly by exploiting the nonlinear structure of the system and the geometric 
 structures of the acoustic spacetime.
LOCATION:CMS\, MR13
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