Exploring global dynamics and blowup in some nonlinear PDEs

CDSNS Colloquium
Friday, February 24, 2023 - 3:30pm for 1 hour (actually 50 minutes)
Skiles 006 and Online
Jonathan Jaquette – Brown University – jaquette@bu.edu
Jorge Gonzalez


Conservation laws and Lyapunov functions are powerful tools for proving the global existence or stability of solutions to PDEs, but for most complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative.

In a recent series of papers, we have shown (using computer-assisted-proofs) that this equation exhibits rich dynamical behavior existing globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup.