Formation of viscous fingers in regularized Laplacian growth

A systematic analytic treatment of local fluctuations in the regularized Laplacian growth problem is given. The interface dynamics is stabilized by a short-distance cutoff ℏ preventing the cusps production in a finite time. The regularization mechanism results in the violation of the incompressibility condition of the viscous fluid on a microscale in the vicinity of the moving interface, thus producing local fluctuations of pressure. Dissipation of fluctuations with time is described by universal Dyson Brownian motion, which reduces to the complex viscous Burgers equation in the hydrodynamic approximation. Because of the intrinsic instability of the interface dynamics, tiny fluctuations of pressure generate universal complex patterns with well developed fjords and fingers in a long time asymptotic.