Department of Mathematics

 In this talk, I will discuss a construction of “shock forming” solutions to a class of dispersive and dissipative perturbations of the Burgers equation. This class includes the fractional KdV equation with dispersive term of order α in [0,1), the Whitham equation arising in water waves, and the fractal Burgers equation with dissipation term of order β in [0,1

Our result seems to be the first construction of gradient blow-up for fractional KdV in the range α in [2/3,1). We construct blow-up solutions by a self-similar approach, treating the dispersive term as perturbative.

The blow-up is stable for α < 2/3. However, for α ≥ 2/3, the solution is constructed by perturbing an underlying unstable self-similar Burgers profile. The construction is carried out by means of a weighted L² approach, which may be of independent interest.

This is joint work with Sung-Jin Oh.