Blow-up of QNLW with Small Initial Data à la Christodoulou

A Geometric Perspective on Shock Formation

These slides are our first foray into understanding the work of Demetrius Christodoulou (and also Fritz John and Serge Alinhac) concerning shock formation for quasilinear wave equations in the small initial data regime. For nonlinear wave equations in higher spatial dimensions ($d \geq 2$) there are two competing effects trying to drive the long-term evolution of the solution, when the initial data is small. First is the dispersive decay: waves tend to spread out, and conservation of energy forces the amplitude to decay. Second is nonlinear resonance: the nonlinearity for the wave equation can drive a positive feedback that tries to grow the amplitude. The long-term behavior of a wave is (to first approximation) determined by the battle between these two effects.

When the spatial dimension $d \geq 4$, it is known that the dispersive decay always wins, provided we start with small initial data. This is because the higher dimension gives "more room" for the wave to spread out, and the amplitude decay is stronger. Determining under what circumstances when $d = 1, 2, 3$ can the dispersive decay win is an active field of research started since the mid 1980s by my PhD advisor Sergiu Klainerman (with many contributors since then). A sufficient structural property for dispersive decay to win is now generally called a "null condition".

Generally for quasilinear equations the regime where the nonlinear resonance wins is hard to understand, in contrast to the semilinear equations. For the semilinear equations the main part of the evolution is determined by the linear operator which is unchanged, so the dispersive decay effects are the same for both small and large solutions. For quasilinear equations the main part of the evolution is now effected also by the solution itself, which can lead to drastic corrections to the dispersive decay property, rendering the picture much harder to understand.

The shock regime is an exception. A shock singularity is a higher order singularity: so the geometry of the main part of the quasilinear equation looks, to lowest order, like the standard linear evolution. This allows us to accurately compare the dispersive decay effect with the nonlinear resonance effect even near the onset of shock. Explaining how this works is the goal of the attached PDF.

Willie WY Wong
Associate Professor

My research interests include partial differential equations, geometric analysis, fluid dynamics, and general relativity.