11 problems

Let the preceding method produce a quadrature domain $\Om$ with three nodes, and let $T_0>0$ be a time. Consider a Hele–Shaw flow that starts with no liquid at time $T=0$ and has t…

Hedenmalm–Perdomo sharpness conjecture. The number $0.52$ can be replaced by $\frac12+\varepsilon$ for any positive $\varepsilon$ in the curvature condition, while retaining the co…

Electric-field insufficiency conjecture. When $cmu/(aQ)>cpi^2/2$, the electric field is too weak to prevent breakthrough caused by the hydrodynamic dipole. The claim concerns the a…

Let $\sigma>0$ be the ultraviolet cutoff in the Hele–Shaw problem, let $\kappa(\gamma)$ denote the boundary curvature at the boundary parameter $\gamma$, and consider the limit…

Interface-erosion scaling-limit conjecture. As $N\to\infty$, the processes $(D_{N^2t}^{i,N},gamma_{N^2t}^{i,N})$ converge in probability towards a solution $(D^i_t,gamma^i_t)$ to t…

Global existence and convergence conjecture. If $Sigma=hat{\mathbb{C}}$, assume $n>1$. Then there exists a unique solution $(D_t^i,gamma_t^i)$, $t\in[0,\infty)$, to the competitive…

Discrete-to-continuum conjecture. Realizations of the discrete random dynamics converge towards a corresponding solution to the competitive Hele-Shaw problem as the mesh size tends…

Let the initial free surface lie in a strip of width $2C$, and let $\kappa$ be the geometric constant such that the corresponding Hele-Shaw semi-flow becomes analytic after time…

Consider the porous medium equation … with $p=P_m(\rho):=\frac{m}{m-1}\rho^{m-1}$, $G'<0$, and $G(p_M)=0$ for some $p_M>0$. In the stiff pressure limit $m\to\infty$, let…

Laplacian growth is interface evolution driven by Laplacian fields, including Hele-Shaw flow, and the domain may have arbitrary connectivity. Surface tension is assumed to be small…

A Hele-Shaw flow describes viscous fluid motion in a narrow cell, and its connectivity is the connectivity of the fluid domain. Surface tension is neglected, so the interface evolv…