Two topics in self-similar singularity formation
Early Career Math Colloquium
Two topics in self-similar singularity formation
Series: Early Career Math Colloquium
Location: Online
Presenter: Federico Pasqualotto, Duke University
Self-similarity is a powerful tool to investigate the "universal" behavior of certain nonlinear PDEs in which coherent structures (such as singularities) appear. In this talk, I will describe two results in which self-similarity plays a crucial role.
The first result, joint with Sung-Jin Oh, concerns singularity formation for a wide class of one-dimensional models. We construct approximately self-similar "shock forming" solutions to a class of dispersive and dissipative perturbations of the classical Burgers equation. This class includes the Whitham equation arising in water waves and the fractional KdV equation with dispersive term of order $\alpha \in [0,1)$. Our result appears to be the first construction of gradient blow-up for fractional KdV in the range $\alpha \in [2/3,1)$.
In the second result, joint with Tarek Elgindi, we focus on the axisymmetric 3D Euler equations in the whole space $\mathbb{R}^3$ (and its 2D approximate model, the Boussinesq system) . We provide a novel mechanism for self-similar singularity formation starting from $C^{1,\alpha}$ initial data with non-trivial swirl. Unlike previous constructions, the blow-up happens away from the symmetry axis.