Evolution equations of physics, fluids, and geometry: asymptotics and singularities (12w5137)
James Colliander (University of British Columbia)
Stephen Gustafson (University of British Columbia)
Slim Ibrahim (University of Victoria)
Nader Masmoudi (Courant Institute)
Kenji Nakanishi (Kyoto University)
Tai-Peng Tsai (University of British Columbia)
The last decade or so has seen a flowering in the study of nonlinear dispersive PDE, and the field seems to have advanced to the point where our level of understanding of these (comparatively subtle) equations is in some cases approaching that of their (longer-studied) parabolic counterparts. Thus one goal of this workshop is the usual one: an exchange of the latest advances in this fast-moving area - with a particular focus on studies of singularity formation and asymptotics - to keep participants aware of the state-of-the-art, to facilitate new collaborations, and to spur new innovations.
At the same time, however, the perspective is beginning to broaden so that, on one hand, some approaches which might more traditionally be thought of as "parabolic" are being brought to bear on dispersive problems (a great example is the highly influential recent work of C. Kenig and F. Merle, building on ideas of J. Bourgain and others, which develops concentration-compactness-type ideas in the dispersive setting), while on the other hand, experience gained in confronting the subtleties of dispersive problems has been productively applied to parabolic problems, including various geometric flows, and even the formidable Navier-Stokes equations (one example here is intriguing recent work of C. Kenig and G. Koch). Thus, another goal of the proposed workshop is to highlight this seeming 'convergence' in the hope of promoting a significant cross-fertilization of ideas, approaches, and tools. To this end, the program will include discussion of work on some parabolic equations (for example Navier-Stokes, some geometric flows, and reaction-diffusion systems), again focusing on the key issues of singularities and asymptotics, in which the tools have significant overlap with those typically used in dispersive problems.
In parallel with all the recent theoretical progress, there have been significant advances in the numerical computation of solutions of nonlinear evolution problems (especially dispersive ones), and in particular in the numerical study of blow-up phenomena. A third goal of the proposed workshop is to exploit this new numerical insight in order to raise new questions for study, and to stimulate new theoretical work. To this end, we will invite several of the leading experts in such computations.
A final goal of the proposed workshop is to promote the development of junior talent in the field, and facilitate the integration of young scientists into the community. To that end, we anticipate allotting about 10 workshop places for graduate students, postdoctoral fellows, and new faculty (these are not included on the list below, but would be decided upon before invitations are made).