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SUMMARY:On weak and measure-valued solutions to compressible Euler and sim
 ilar systems - Agnieszka Swierczewska-Gwiazda\, University of Warsaw
DTSTART:20150527T140000Z
DTEND:20150527T150000Z
UID:TALK59349@talks.cam.ac.uk
CONTACT:Amit Einav
DESCRIPTION:The theory for gravity driven avalanche flows is qualitatively
  similar to that of compressible fluid dynamics. I  will present one of th
 e models describing flow of granular avalanches - the Savage-Hutter model.
  The derivation of considered continuum flow models essentially bases on t
 he fact that the characteristic length in the flowing direction is in gene
 ral much larger than the thickness of an avalanche. Such an approach resul
 ted in depth-averaged equation governed by generalized system of shallow w
 ater equations (Saint-Venant equations). The evolution of granular avalanc
 hes along an inclined slope is described by the mass conservation law and 
 momentum balance law.\nOriginally the model was derived in one-dimensional
  setting. Our interest is mostly directed to two-dimensional extension.  A
 s the solutions of the Savage-Hutter system develop shock waves and other 
 singularities characteristic for hyperbolic system of conservation laws. A
 ccordingly\, any mathematical theory based on the classical concept of smo
 oth solutions fails as soon as we are interested in global-in-time solutio
 ns to the system.\nI will start with presenting the concept of measure-val
 ued solutions (generalization 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 th
 e Savage-Hutter system is always solvable but not well posed 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 a
 valanche flow model. Math. Methods Appl. Sci. 28 (2005)\, no. 18\, 2201-22
 23.\n\n[2] E. Feireisl\, P. Gwiazda\, and A. Swierczewska Gwiazda. On weak
  solutions to the 2d Savage-Hutter model of the motion of a gravity driven
  avalanche\nflow\,  arXiv:1502.06223.\n\n[3] P. Gwiazda\, A. Swierczewska-
 Gwiazda\, and E. Wiedemann. Weak-strong uniqueness for measure-valued solu
 tions of the Savage-Hutter equations\,  arXiv:1503.05246
LOCATION:CMS\, MR14
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