Mathematics Colloquium: Uncovering unstable blowup in PDEs through an exploration of invariant manifolds

Speaker:  Jonathan Jaquette, New Jersey Institute of Technology

Title: Uncovering unstable blowup in PDEs through an exploration of invariant manifolds

Abstract: When a PDE that generates an analytic semiflow blows up, its solutions may be continued in complex time around the singularity potentially producing a branched Riemann surface. The work Cho et al. [\emph{Jpn. J. Ind. Appl. Math.} 33 (2016): 145-166] investigated this phenomena for the quadratic heat equation $ u_t = u_{xx} + u^2$. When solutions are continued for purely imaginary time, a nonlinear Schr\"odinger equation (NLS) $i u_t =  u_{xx} + u^2$ for $ x \in \mathbb{T}$ is obtained, and the authors conjectured that this NLS is globally well-posed for real initial data. Using a mix of analytical and computer-assisted techniques, we have shown that this equation exhibits rich dynamical structure punctuated by (presumably unstable) blowup solutions. It is also of note that the nonlinearity here, a complex quadratic, is essentially the same nonlinearity as in the Constantin-Lax-Majda equation, a 1D model for incompressible fluids.

In recent work we have identified real initial data whose numerical solution blows up, in contradiction of the conjecture by Cho et al. Furthermore, the set of real initial data which blows up under the NLS dynamics appears to occur on a codimension-1 manifold, and we conjecture that it is precisely the stable manifold of the zero equilibrium for the nonlinear heat equation.  We apply the parameterization method to study the internal dynamics of this manifold, offering a heuristic argument in support of our conjecture.  The solution exhibits self-similar blowup and potentially nontrivial self-similar dynamics, however the proper scaling ansatz remains elusive. 

Time:  Friday, April 11, 3:30pm – 4:20pm

Place:  Exploratory Hall, room 4106

Zoom and In-person