5 problems

Let $\mathcal{C}$ denote the class of two-dimensional shallow water equations, let $G^\sim$ be its equivalence group, and call an admissible transformation independent when it is i…

Dispersion conjecture. Localised smooth initial data satisfying the non-cavitation condition do not lead to stable structures localised in both spatial dimension. The solutions exi…

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

Let $k=1$, let $u_0\in H^s(\mathbb{R})$ with $s>3/2$, and let $u$ be the corresponding solution of equation (8.0.1), with lifespan $T$. Finite-time wave breaking conjecture. The so…

Consider the class of two-dimensional shallow water equations with variable bottom topography, denoted by $\mathcal L$, and its spaces of zeroth-order conservation laws. Two cases…