Seminar on Partial Differential Equations
18.10.16 09:00
Matteo Caggio
Regularity criteria for the Navier-Stokes equations based on one component of velocity more
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Intenational Summer School on Evolution Equations
Date: July 11 - 15, 2016
Place: Faculty of Mathematics and Physics, Charles University in Prague, Malostranské náměstí 25, Prague
Lectures will be given by: Bernard Dacorogna (EPFL, Switzerland), Anne-Laure Dalibard (École Normale Supérieur, France), Igor Kukavica (USC Dornsife, USA), Franck Merle (Université de Cergy-Pontoise, France), Gigliola Staffilani (MIT, USA) and Andrew Stuart (University of Warwick, Great Britain).
Organizers: Eduard Feireisl (IM CAS), Pavel Krejčí (IM CAS), Josef Málek (Charles University in Prague), Ondřej Kreml (IM CAS), Rudolf Kinc (AMCA Agency)
The Equadiff conferences are a series of international meetings devoted to the field of differential equations in the broadest sense.
On July 6-10, 2015, hosted by Université Claude Bernard Lyon 1 at the heart of the second largest urban area of France, Equadiff 2015 continues the tradition of the Equadiff series
in Prague (2013), Loughborough (2011), Brno (2009), Vienna (2007), Bratislava (2005), and Hasselt (2003).
With 15 special guest speakers and 26 selected minisymposia, the conference aims at bringing together world-wide experts.
International Mathematical Summer School coordinated by E. Rocca (Universita di Milano and WIAS Berlin) and E.Feireisl (IM Praha).
Courses given by: J. Ball (Oxford), C.M.Dafermos (Brown University), E.Feireisl (IM Praha), F.Otto (Max Planck Institute, Leipzig)
The main goal of the present research proposal is to build up a general mathematical theory describing the motion of a compressible, viscous, and heat conductive fluid. Our approach is based on the concept of generalized (weak) solutions satisfying the basic physical principles of balance of mass, momentum, and energy. The energy balance is expressed in terms of a variant of entropy inequality supplemented with an integral identity for the total energy balance.
We propose to identify a class of suitable weak solutions, where admissibility is based on a direct application of the principle of maximal entropy production compatible with Second law of thermodynamics. Stability of the solution family will be investigated by the method of relative entropies constructed on the basis of certain thermodynamic potentials as ballistic free energy.
The new solution framework will be applied to multiscale problems, where several characteristic scales become small or extremely large. We focus on mutual interaction of scales during this process and identify the asymptotic behavior of the quantities that are filtered out in the singular limits. We also propose to study the influence of the geometry of the underlying physical space that may change in the course of the limit process. In particular, problems arising in homogenization and optimal shape design in combination with various singular limits are taken into account.
The abstract approximate scheme used in the existence theory will be adapted in order to develop adequate numerical methods. We study stability and convergence of these methods using the tools developed in the abstract part, in particular, the relative entropies.