Seminar on Analysis, Geometry and PDEs, "Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation", Federico Pasqualotto (Duke University)
The talk is given in the Hörmander auditorium, but can also be followed on Zoom: https://lu-se.zoom.us/j/63015319260. Contact Erik Wahlén for the password.
Abstract: In this talk, I will describe a singularity formation scenario for a general class of dispersive and dissipative perturbations of the classical Burgers equation. This class includes the Whitham equation in water waves, the fractional KdV equation with dispersive term of order $\alpha \in [0,1)$, and the fractal Burgers equation with dissipative term of order $\beta \in [0,1)$.
We show the existence of solutions whose gradient blows up in finite time, starting from smooth initial data ("wave breaking"). Our theorem appears to be the first construction of gradient blow-up for fractional KdV in the range $\alpha \in [2/3,1)$. We follow a self-similar approach, treating the dispersive term perturbatively. We show that the blow-up is stable when $\alpha < 2/3$. On the other hand, for $\alpha \geq 2/3$, the solution is constructed by perturbing an underlying unstable self-similar Burgers profile. In the final part of the talk, I will indicate how these observations can be used to address certain blow-up problems in higher space dimensions.
This is joint work with Sung-Jin Oh (UC Berkeley).
Tid: 2022-11-22 13:30 till 14:30
erik [dot] wahlen [at] math [dot] lu [dot] se