The 71st Midwest Partial Differential Equations Seminar May 11-12, 2013 Room TBA East Hall University of Michigan Ann Arbor, MI   Abstract Detail   Title: Lower bound for the rate of blow-up of singular solutions of the three-dimensional Zakharov system We consider the Zakharov system which describes the propagation of Langmuir waves in a non-magnetized plasma. It was derived by Zakharov (1972) in the form of a coupled system governing the electric field complex amplitude and the density fluctuations of ions. Heuristic arguments and numerical simulations show that solutions may blow-up in a finite time both in two and three dimensions. In two dimensions, there exist exact self-similar blowing-up solutions. In addition, Merle (1996) established a lower bound for the rate of blow-up of singular solutions in the energy space. This rate is optimal. In three dimensions, there are no known explicit blowing-up solutions. Self-similar solutions exist only asymptotically close to collapse. In the present work, we assume that the solution blows up in a finite time and we establish a lower bound for the rate of blow-up of certain Sobolev norms of the solution. The main ingredients are a local well-posedness result (in the spirit of Ginibre-Tsutsumi-Velo 1997) and a contradiction argument adapted from Cazenave and Weissler (1990). This is a joint work with Jim Colliander and Magda Czubak. Catherine Sulem Universtity of Toronto Email:  sulem@math.toronto.edu Phone: 416-9784378   BACK to Speaker Listing BACK to Conference Home   Sponsors National Science Foundation, Department of Mathematics at the University of Michigan