Stable small-data shock formation for wave equations in 3D

Wednesday, October 23, 2013 -
2:30pm to 3:30pm
PLEASE NOTE SPECIAL DAY AND TIME.  I will present some preliminary results obtained in collaboration with G. Holzegel, S. Klainerman, and W. Wong. Our main result is a proof of stable shock formation in solutions to a class of nonlinear wave equations in three spatial dimensions. The data are small and belong to a standard Sobolev space. Our work provides a detailed geometric description of the dynamics from $t=0$ until the first shock. The main precursor to our work is the remarkable result of D. Christodoulou, who in 2007 proved an analogous shock formation result for the irrotational region of small-data solutions to the relativistic Euler equations. The result was later extended by Christodoulou-Miao to apply to the non-relativistic Euler equations. Our work recovers both of these results as special cases, and it also applies to a larger class of equations. Our work also generalizes previous small-data singularity formation results of F. John, S. Alinhac, and many others.   I will highlight some of the main ideas behind our proof. Our analysis is based on a carefully chosen mix of i) purely geometric analysis in the spirit of Christodoulou's approach and ii) estimates for the components of tensorfields relative to rectangular coordinates, with an emphasis on the latter. The net effect is that our approach is shorter than that of Christodoulou, less precise in some ways, and more precise in others.
Speaker: 
Jared Speck
MIT
Event Location: 
Fine Hall 314