Bi-Laplacian Growth Patterns in Disordered Media

Experiments in quasi 2-dimensional geometry (Hele Shaw cells) in which a fluid is injected into a visco-elastic medium (foam, clay or associating-polymers) show patterns akin to fracture in brittle materials, very different from standard Laplacian growth patterns of viscous fingering. An analytic theory is lacking since a pre-requisite to describing the fracture of elastic material is the solution of the bi-Laplace rather than the Laplace equation. In this Letter we close this gap, offering a theory of bi-Laplacian growth patterns based on the method of iterated conformal maps.Comment: Submitted to PRL. For further information see http://www.weizmann.ac.il/chemphys/ander

Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections

We consider a stochastic partial differential equation with two logarithmic nonlinearities, with two reflections at 1 and -1 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche, Gouden\`ege and Zambotti, we obtain existence and uniqueness of solution for initial conditions in the interval $(-1,1)$. Finally, we prove that the unique invariant measure is ergodic, and we give a result of exponential mixing

Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection

We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche and Zambotti, we use a method based on infinite dimensional equations, approximation by regular equations and convergence of the approximated semi-group. We obtain existence and uniqueness of solution for nonnegative intial conditions, results on the invariant measures, and on the reflection measures

Odometer of long-range sandpiles in the torus: mean behaviour and scaling limits

In \cite{Cipriani2016}, the authors proved that with the appropriate rescaling, the odometer of the (nearest neighbours) Divisible Sandpile in the unit torus converges to the bi-Laplacian field. Here, we study $\alpha$-long-range divisible sandpiles similar to those introduced in \cite{Frometa2018}. We obtain that for $\alpha \in (0,2)$, the limiting field is a fractional Gaussian field on the torus. However, for $\alpha \in [2,\infty)$, we recover the bi-Laplacian field. The central tool for our results is a careful study of the spectrum of the fractional Laplacian in the discrete torus. More specifically, we need the rate of divergence of such eigenvalues as we let the side length of the discrete torus goes to infinity. As a side result, we construct the fractional Laplacian built from a long-range random walk. Furthermore, we determine the order of the expected value of the odometer on the finite grid. \end{abstract}Comment: 35 pages, 4 figure

Extremal basic frequency of non-homogeneous plates

In this paper we propose two numerical algorithms to derive the extremal principal eigenvalue of the bi-Laplacian operator under Navier boundary conditions or Dirichlet boundary conditions. Consider a non-homogeneous hinged or clamped plate $\Omega$, the algorithms converge to the density functions on $\Omega$ which they yield the maximum or minimum basic frequency of the plate

Singular limits for the bi-laplacian operator with exponential nonlinearity in R4\R^4

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^{4}$ such that for some integer $d\geq1$ its $d$-th singular cohomology group with coefficients in some field is not zero, then problem {\Delta^{2}u-\rho^{4}k(x)e^{u}=0 & \hbox{in}\Omega, u=\Delta u=0 & \hbox{on}\partial\Omega, has a solution blowing-up, as $\rho\to0$, at $m$ points of $\Omega$, for any given number $m$.Comment: 30 pages, to appear in Ann. IHP Non Linear Analysi