We will consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows to define global solutions, even in the presence of blow-ups of the freezing rate. We will provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Holder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root behavior of the growth process pointed out by Stefan in his seminal paper [Ste89] from 1889 for the ordinary Stefan problem. In our second main theorem, we will establish the uniqueness of the global solutions, a first result of this kind in the context of growth processes with singular self-excitation when blow-ups are present. Based on joint work with Francois Delarue and Sergey Nadtochiy.[-]

We discuss properties of the viscous Hamilton-Jacobi equation$$\begin{cases}u_{t}-\Delta u=|D u|^{p} & \text { in }(0, \infty) \times \Omega, \\ u=0 & \text { in }(0, \infty) \times \partial \Omega, \\ u(0)=u_{0} & \text { in } \Omega,\end{cases}$$in the super-quadratic case $p>2$. Here $\Omega$ is a bounded domain in $\mathbf{R}^{\mathbf{N}}$. In the super-quadratic regime, solutions may be continuous but with a gradient blow up; in this case the second order equation exhibits very peculiar phenomena. Some properties are similar to first order problems, such as loss of boundary conditions and appearance of singularities, but the presence of diffusion let singularities appear and disappear, in a very unusual way. In the talk I will present results obtained in collaboration with Philippe Souplet which describe the qualitative behavior of the solution, starting from smooth initial data. This includes the analysis of blow-up rates, blow-up profiles, life after blow-up, loss and recovery of boundary conditions.[-]