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CATEGORIES:Partial Differential Equations seminar
SUMMARY:Rough solutions of the 3-D compressible Euler equa
tions - Qian Wang (Oxford)
DTSTART;TZID=Europe/London:20220131T140000
DTEND;TZID=Europe/London:20220131T150000
UID:TALK168788AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/168788
DESCRIPTION:I will talk about my work on the compressible Eule
r equations. We prove the local-in-time existence
the solution of the compressible Euler equations i
n 3-D\, for the Cauchy data of the velocity\, dens
ity and vorticity (v\,\\varrho\, \\omega) in H^s^
x H^s^ x H^s'^\, for s' in (2\,s). The result exte
nds the sharp result of Smith-Tataru and Wang\,
established in the irrotational case\, i.e \\ome
ga=0\, which is known to be optimal for s greate
r than 2. At the opposite extreme\, in the incompr
essible 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 e
qual to 3/2\, see the work of Bourgain-Li. It is
thus natural to conjecture that 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 pro
ving the conjecture. The main difficulty in esta
blishing sharp well-posedness results for general
compressible Euler flow is due to the highly nont
rivial interaction between the sound waves\, gov
erned by quasilinear wave equations\, and vorticit
y which is transported by the flow. To overcome th
is difficulty\, we separate the dispersive part of
sound wave from the transported part\, and gain r
egularity significantly by exploiting the nonlinea
r structure of the system and the geometric struct
ures of the acoustic spacetime.
LOCATION:CMS\, MR13
CONTACT:Dr Greg Taujanskas
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