10/5/2009

Triston Roy
IAS

In this talk I will discuss the global existence of H^{2} solutions of a barely energy supercritical equation, namely a loglog defocusing quitic wave equation. The first step is to control the $L_{t}^{4} L_{x}^{12}$ norm of the solution on an arbitrary long-time interval [0,T] by an a priori bound of its $L_{t}^{\infty} H^{2}([0,T])$ norm. To do that we divide the interval into subintervals on which the $L_{t}^{4} L_{x}^{12}$ norm of the solution is substantial. We prove that the length of these subintervals is also substantial. Then, by using a variant of a Morawetz-type estimate from Shatah-Struwe (the variant we use is due to Bahouri-Gerard), we can control the length of large sequences of these subintervals. Eventually we can estimate their number. The second estimate is to estimate a posteriori the $L_{t}^{\infty} H^{2}([0,T])$ norm of the solution. This is done by a connectedness-in-time argument, using the Strichartz estimates and the control of the $L_{t}^{4} L_{x}^{12}$ norm.