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CATEGORIES:Partial Differential Equations seminar
SUMMARY:On weak and measure-valued solutions to compressib
le Euler and similar systems - Agnieszka Swierczew
ska-Gwiazda\, University of Warsaw
DTSTART;TZID=Europe/London:20150527T150000
DTEND;TZID=Europe/London:20150527T160000
UID:TALK59349AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/59349
DESCRIPTION:The theory for gravity driven avalanche flows is q
ualitatively similar to that of compressible fluid
dynamics. I will present one of the models descr
ibing flow of granular avalanches - the Savage-Hut
ter model. The derivation of considered continuum
flow models essentially bases on the fact that the
characteristic length in the flowing direction is
in general much larger than the thickness of an a
valanche. Such an approach resulted in depth-avera
ged equation governed by generalized system of sha
llow water equations (Saint-Venant equations). The
evolution of granular avalanches along an incline
d slope is described by the mass conservation law
and momentum balance law.\nOriginally the model wa
s derived in one-dimensional setting. Our interest
is mostly directed to two-dimensional extension.
As the solutions of the Savage-Hutter system deve
lop shock waves and other singularities characteri
stic for hyperbolic system of conservation laws. A
ccordingly\, any mathematical theory based on the
classical concept of smooth solutions fails as soo
n as we are interested in global-in-time solutions
to the system.\nI will start with presenting the
concept of measure-valued solutions (generalizatio
n by DiPerna and Majda). Then I will show how the
method of convex integration\, recently adapted to
the incompressible Euler system by De Lellis and
Szekelyhidi\, can be applied to show that the Sava
ge-Hutter system is always solvable but not well p
osed in the class of weak solutions.\nThe talk is
based on the following results:\n\n[1] P. Gwiazda
On measure-valued solutions to a two-dimensional
gravity-driven avalanche flow model. Math. Methods
Appl. Sci. 28 (2005)\, no. 18\, 2201-2223.\n\n[2]
E. Feireisl\, P. Gwiazda\, and A. Swierczewska Gw
iazda. On weak solutions to the 2d Savage-Hutter m
odel of the motion of a gravity driven avalanche\n
flow\, arXiv:1502.06223.\n\n[3] P. Gwiazda\, A. S
wierczewska-Gwiazda\, and E. Wiedemann. Weak-stron
g uniqueness for measure-valued solutions of the S
avage-Hutter equations\, arXiv:1503.05246
LOCATION:CMS\, MR14
CONTACT:Amit Einav
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