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CATEGORIES:Analysis Seminar
SUMMARY:Long-time behaviour of solutions to the equivarian
t Harmonic Map Heat Flow and Landau-Lifschitz equa
tions - Dimitrios Roxanas\, University of Edinburg
h
DTSTART:20171031T160000Z
DTEND:20171031T170000Z
UID:TALK2812AT
URL:/talk/index/2812
DESCRIPTION:We study the asymptotic behaviour of corotational
solutions to the Harmonic Map Heat Flow from $\\ma
thbb{R}^2$ to $\\mathbb{S}^2.$ In particular\, we
give criteria in terms of the initial data: we com
pare their energy to the energy of the static solu
tions\, which are setting the natural threshold be
tween\, on one hand\, global existence and decay\,
and on the other\, blow-up or lack of decay. \n\n
We first recover Struwe's global existence result
for a below threshold scenario\; we also show deca
y to zero as time goes to infinity. The proof in t
his case follows the "concentration-compactness pl
us rigidity" approach of Kenig and Merle\, origina
lly developed for dispersive equations. \n\nWe the
n proceed to extend results of Gustafson\, Nakanis
hi and Tsai for above threshold maps. Employing a
characterization of blowing-up solutions and a ne
w stability result\, based on modulation arguments
and spectral considerations\, we show that soluti
ons in an above threshold class (but not close to
the harmonic maps) exist globally and asymptotical
ly relax to a rescaled static solution.\n\nIf time
permits\, we will briefly sketch how these ideas
can be used in the study of above-threshold soluti
ons to the Landau-Lifschitz equation.\n
LOCATION:Lecture Theater C\, Watson Building
CONTACT:Yuzhao Wang
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