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CATEGORIES:Partial Differential Equations seminar
SUMMARY:Advances in the theory of multi-dimensional shock
waves - Jared Speck (Vanderbilt University)
DTSTART;TZID=Europe/London:20221114T140000
DTEND;TZID=Europe/London:20221114T150000
UID:TALK192236AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/192236
DESCRIPTION:A shock singularity in a quasilinear hyperbolic PD
E solution is a mild singularity such that one of
the solutionâ€™s derivatives blows up\, though the s
olution itself remains bounded. Importantly\, the
mild nature of the singularity opens the door to t
he possibility that the solution might be continue
d uniquely as a weak solution past the singularity
\, under suitable selection criteria. While the ri
gorous 1D theory is in a mature stage due to the a
vailability of well-posedness results for BV initi
al data\, multi-dimensional hyperbolic PDEs are ty
pically ill-posed in BV. Consequently\, the theory
of multi-dimensional shocks is permeated with fun
damental open problems\, many with deep ties to ge
ometry. Despite the challenges in higher dimension
s\, for specific systems\, including the compressi
ble Euler equations and relativistic Euler equatio
ns in 3D\, there has been dramatic progress in the
last 15 years\, starting with Christodoulouâ€™s 200
7 monograph on shock formation in irrotational sol
utions. In this talk\, after providing an introduc
tion to the 1D problem\, I will give a non-technic
al description of recent advances in multi-dimensi
ons\, with a focus on the multi-dimensional compre
ssible Euler equations with vorticity and entropy.
Many recent results are based on a new formulatio
n of compressible Euler flow exhibiting miraculous
geo-analytic structures and regularity properties
\, and the analysis fundamentally relies on nonlin
ear geometric optics. In particular\, I will descr
ibe my recent series of works on the 3D compressib
le Euler equations with vorticity and entropy\, wh
ich\, for open sets of initial data\, reveal the f
ull structure of the maximal classical development
\, including the full structure of the singular se
t as well the emergence of a Cauchy horizon from t
he singularity. Finally\, time permitting\, I will
discuss some of the many open problems in the fie
ld. Various aspects of this program are joint with
L. Abbrescia\, J. Luk\, and M. Disconzi.
LOCATION:CMS\, MR13
CONTACT:Zexing Li
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