Federico Pasqualotto, UC Berkeley

Abstract: 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∈[0,1), the Whitham equation arising in water waves, and the fractal Burgers equation with dissipation term of order β∈[0,1). Our result seems to be the first construction of a shock forming solution for fractional KdV in the range α∈[2/3,1). We construct blow-up solutions by a self-similar approach, treating the dispersive term as perturbative. The blow up constructed is stable for α<2/3. However, for α≥2/3, the solution is constructed by perturbing an underlying unstable self-similar Burgers profile. This is joint work with Sung-Jin Oh (UC Berkeley).

 

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