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SUMMARY:Hamilton-Jacobi equation with random initial condition
DTSTART;VALUE=DATE-TIME:20230327T070000Z
DTEND;VALUE=DATE-TIME:20230331T090000Z
DTSTAMP;VALUE=DATE-TIME:20240624T172500Z
UID:indico-event-475@indico.gssi.it
DESCRIPTION:Speakers: Fraydoun Rezakhanlou (University of California)\n\nL
ecturer: Fraydoun Rezakhanlou (University of California)Abstract: Hamilton
–Jacobi equation (HJE) is one of the most popular and studied PDE which
enjoys vast applications in numerous areas of science. Originally HJEs wer
e formulated in connection with the completely integrable Hamiltonian ODEs
of celestial mechanics. They have also been used to study the evolution o
f the value functions in control and differential game theory. HJE associa
ted with space-time stationary Hamiltonian functions are used to study tur
bulence in hydrodynamics. Several growth models in physics and biology are
described by such HJEs and their viscous variants. In these models\, a ra
ndom interface separates regions associated with different phases and the
interface can be locally approximated by the graph of a solution to a HJE.
Naturally we would like to understand how the randomness affects the solu
tions and how the statistics of solutions are propagated with time. Lagran
gian techniques in Aubry-Mather theory for action-minimizing trajectories\
, PDE techniques of weak KAM theory\, and probabilistic methods related to
first/last passage percolation problems have been employed to study long-
time behavior of solutions. Most notably\, a unique invariant measure has
been constructed for any prescribed average velocity for some important ex
amples of Hamiltonian functions. In these lectures I will give an overview
of some of the existing results for the statistics of random solutions to
HJEs. In particular\, I will discuss a systematic approach for constructi
ng Gibbsian solutions to Hamilton-Jacobi PDEs by exploring the Eularian de
scription of the shock dynamics. Such Gibbsian solutions depend on kernels
satisfying kinetic-like equations reminiscent of the Smoluchowski model f
or coagulating and fragmenting particles.References: Updated notes of the
lecture will be made available at https://math.berkeley.edu/~rezakhan/KHJE
_l'Aquila.pdf Information. This minicourse is part of the school "Scali
ng Limits and Generalized Hydrodynamics"\, see https://indico.gssi.it/e/Sc
alingLimits.\n\nhttps://indico.gssi.it/event/475/
URL:https://indico.gssi.it/event/475/
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