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IMA Annual Program Year Workshop

Dynamical Systems in Studies of Partial Differential Equations

Dynamical Systems in Studies of Partial Differential Equations

September 24-28, 2012

University of Minnesota, Twin Cities | |

Boston University | |

Georgia Institute of Technology |

The theory of partial differential equations (PDEs) is a broad research field, rapidly growing in close connections with other mathematical disciplines and applied sciences. In this workshop, connections between the theories of dynamical systems and PDEs will be explored from several points of view.
Infinite-dimensional dynamical systems generated by evolutionary PDEs provide the most immediate examples of interplay between the two theories. Extensions of well-established results and techniques from finite-dimensional dynamical systems (invariant manifolds, bifurcations, KAM theory) have proved very useful in qualitative studies of PDEs. On the other hand, specific questions for PDEs brought about stimulating problems in the theory of dynamical systems, such as the existence of finite-dimensional attractors and their behavior under (regular or singular) perturbations. Entire (or eternal) solutions, which emerged as key objects in these problems, have long served as organizing centers for qualitative investigations of dissipative evolutionary PDEs and they continue to play an important role in other modern approaches to PDEs (for example, entire solutions of spatially extended systems have been examined in connection with traveling waves, and classification of entire solutions for specific classes of PDEs have been shown to have important implications on the structure of singularities of solutions).
In quite a different way, dynamical systems have been used for the investigation of solutions of PDEs, which are not originally set up as models of evolution phenomena. The spatial dynamics of elliptic equations on unbounded cylinders is an example of such an approach. The key underlying idea that interesting solutions can be found by studying ODEs on manifolds in the state space has been successfully applied in such problems and in many other PDEs.
As it often happens in studies of evolutionary PDEs, in particular those on unbounded spatial domains, applications of standard results from dynamical systems may be hindered by obstacles, such as the presence of the essential spectrum of the linearized problem. Yet, even when standard results do not apply, the conclusions they would lead to can often be proved by other methods, like the renormalization (rescaling) techniques. Examples of such conclusions can be found in studies of blow up of solutions of parabolic equations. In these results, the role of dynamical systems is in the guideline they provide for the investigation.
Rigorous computational approaches to PDEs will also form an important part of the workshop.
We believe that a workshop focusing on these and related applications of dynamical systems in the theory of PDEs will provide a platform for an exciting exchange of ideas between specialists with different backgrounds in PDEs, dynamical systems, and their applications. It is our intention to have different backgrounds and approaches represented in the lectures and informal discussions, and we expect the workshop to stimulate new collaborations.