Date: 18.02  20.02.2013 
Place: INF 294, SR 214 
Organizers: Anna MarciniakCzochra, Piotr Gwiazda, Filip Klawe 
Abstract: For most of equations of mathematical physics, it is impossible to show the existence of classical solutions. On the other hand, the concept of distribution solutions (or weak solutions) does not allow us to pose certain problems correctly. As an example, we can point out the lack of uniqueness in the class of distribution solutions of the simplest Burger's equation. To remedy such problems, the notion of entropy or renormalization has been introduced. It is postulated that, in addition to a weak formulation of the problem, the equation satisfies certain additional condition in entropic or renormalized sense (for a sufficiently rich family of entropies/renormalizations). Another application of entropy is the method of generalized relative entropy used to study the long time behavior of solutions to partial differential equations. The seminar will be devoted to introducing this approach and its applications. In particular we will focus on: Kruzkov theory of entropy solutions for scalar hyperbolic equations [2], Di PernaLions theory of renormalized solutions to the transport equation [36], theory of renormalized and entropy solutions to elliptic and parabolic equations [78], method of relative entropy in dynamics of biological populations [910].

Participation in the seminar requires a background in partial differential equations and functional analysis. Bibliography 

Program 
Monday 18.02.2013, 14.0017.00 
Aneta WróblewskaKamińska i Piotr Orliński: "Theory of renormalized solutions of elliptic
equations with given data in L1" (Abstract) 
Tuesday 19.02.2013, 9.0012.00 
Paweł Subko: "Renormalization theory for transport equation" (Abstract) 
Filip Klawe: "Renormalised solutions of nonlinear parabolic
problems with L1 data" (Abstract) 
Wednesday 20.02.2013, 9.0012.00 
Ewelina Zatorska: "An application of the renormalized solutions of
the steady continuity equation in the
compressible NavierStokes equations" (Abstract) 
Piotr Minakowski: "Introduction to the Boltzman equation" (Abstract) 