Models arising in engineering and natural sciences are usually described by partial differential equations (PDEs) and their natural modifications. From that point of view, PDEs form an indispensable element of applied mathematics and sometimes a bridge in cross-subject studies. Our PDEs group investigates analytical (Gwiazda, Mucha, Rybka, Zajaczkowski) and numerical (Dryja, Wozniakowski) aspects of this branch of mathematics. The core topics are systems arising in fluid mechanics, however a number of important directions are derived from models of chemical/thermodynamical reactions. The key points in the research plan for the next 5 years are located in transport-kinetic theory, singular/degenerated parabolic problems, the development of finite element methods, and classical questions of mathematical fluid mechanics.
Particular attention will be paid to several directions concerned with physical systems of PDEs, from the viewpoint of numerical analysis and regularity. The group is composed of young leaders, post-docs, and more than 10 PhD students. The youth of the team is one of its great advantages.
Transport-kinetic theory. The needs of applications (like cancer cell models) require considering system with measures as solutions. The basic problems rely on redefining the meaning of a solution, questions of stability (distance/topology), and on development of nonlinear analysis (suitable notions of convergence of measures). Years of development of pure mathematical tools permit finally the construction of new theories for difficult models arising in natural sciences. Thanks to international collaboration, the group has obtained break-through results in these areas.
Singular/degenerate systems. Models of crystal growth and image processing often deliver systems of PDEs which are not found in existing theories. Such systems, with the total variation flow involving non-linearity, will be examined from the point of view of qualitative properties of their solutions, with particular interests in such phenomena like facets (flat region), edges, or segmentations. It is a new field in PDEs and the competition is rapidly increasing. This is largely due to the fact that each important step in the theory may result in a new and improved tools for image processing. The interdisciplinary character of this study requires no explanation.
Finite element methods. As an important example of the progress in this theory can serve the local discontinuous Galerkin method (LDG). This method permits analysing directly systems with discontinuous coefficients. As a result, one is able to gain not only a better ”precision”, but also improved efficiency of the constructed code. This technique seems to be very promising. Of course it is just one example of many important directions of research in this area. One should stress that, although only the last point derives from numerical analysis, all of the three directions that have been pointed out are connected to numerical simulations. Mathematical results will allow one to construct new codes, an indispensable input in an interdisciplinary collaboration.