The aim of the semester Beyond hydrodynamic equations is to create an environment to deepen the understanding of evolutionary problems modeled by partial differential equations by developing their mathematical analysis and to to address new challenges in the study of nonlinear systems of partial differential equations. Many fascinating physical life, or biological phenomena require extremely nontrivial mathematical descriptions.

Introducing advanced types of PDEs in mathematical biology, physics, social sciences  brought a new quality of modeling, however in many cases did not leave a hope for well-posedness of classical solutions. That is why the methods of functional analysis have developed so widely in order to search for weak solutions, and sometimes even solutions in a weaker sense – measure-valued solutions.

However, the mere existence of solutions is not fully satisfactory. To understand well the process one asks further questions on properties of solutions, like uniqueness, asymptotic behaviour and relations among different systems. There are various methods that are particularly powerful like compactness methods or relative entropy method. The semester will give particularly opportunity to  exploit them.

As the semester will focus on key thematic areas such as modeling, existence, regularity of solutions, asymptotics, multiscale analysis, numerical analysis, which require well understanding of already established knowledge and the development of new analytical methods, advanced functional analysis and PDEs techniques.

During the semester very diverse phenomena are going to be considered (among others): thermodynamic models, polymeric models, kinetic models, collective models arising from social sciences and life sciences.

We will concentrate on showing that there are definite advantages of identifying different processes in the language of hydrodynamics what allows for the presentation of a rich variety of behaviors.

Simons Scientific Leaders

• Young-Pil Choi (Department of Mathematics, Yonsei University, Seoul, Republic of Korea)

• Euard Feireisl (Institute of Mathematics, Academy of Science of Czech Rep., Prague, Czech Rep.)

• Jakub Skrzeczkowski (Mathematical Institute, University of Oxford, Oxford, United Kingdom)

Junior Simons Fellows

• Sougata Mahanta (Institute of Mathematics, Czech Academy of Science)

• Jiří Púček (Institute of Mathematics Czech Academy of Sciences)

• Simon Schneider (Johannes Gutenberg University Mainz, Germany)

Organizers of the semester

• Piotr Gwiazda (IMPAS, Poland)

• Aneta Wróblewska-Kamińska (IMPAS, Poland)