Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation

Understand

We consider a class of dispersive and dissipative perturbations of the inviscid Burgers equation, which includes the fractional KdV equation of order $\alpha$, and the fractal Burgers equation of order $\beta$, where $\alpha, \beta \in [0,1)$, and the Whitham equation.

• For all $\alpha, \beta \in [0,1)$, we construct solutions whose gradient blows up at a point, and whose amplitude stays bounded, which therefore display a "shock-like" singularity.
• We moreover provide an asymptotic description of the blow-up.
• To our knowledge, this constitutes the first proof of gradient blow-up for the fKdV equation in the range $\alpha \in [2/3, 1)$, as well as the first description of explicit blow-up dynamics for the fractal Burgers equation in the range $\beta \in [2/3, 1)$.