Finite-time gradient-catastrophe conjecture for non-cavitating-violating initial data in one dimension

Consider the one-dimensional Amick-Schonbek system with initial data that violate the non-cavitation condition at one point. A gradient catastrophe means the formation of an unbounded spatial gradient in finite time.

Gradient-catastrophe conjecture. Initial data violating the non-cavitation condition in one point in 1D lead to a gradient catastrophe in finite time.

This conjecture is based on numerical one-dimensional studies, extending earlier numerical work cited by the authors. The analogous behavior appears to hold in two dimensions, but the paper reports insufficient numerical resolution to provide equally convincing evidence there.