Stochastic porous media equations and self-organized criticality: convergence to the critical state in all dimensions

If $X=X(t,\xi)$ is the solution to the stochastic porous media equation in $\cal O\subset\mathbb{R}^d$, $1\le d\le 3,$ modelling the self-organized criticaity and $X_c$ is the critical state, then it is proved that \int^\9_0m(\cal O\setminus\cal O^t_0)dt<\9, $\mathbb{P}{-a.s.}$ and \lim_{t\to\9}\int_{\cal O}|X(t)-X_c|d\xi=\ell<\9,\ \mathbb{P}{-a.s.} Here, $m$ is the Lebesgue measure and $\cal O^t_c$ is the critical region $\{\xi\in\cal O;$ $X(t,\xi)=X_c(\xi)\}$ and $X_c(\xi)\le X(0,\xi)$ a.e. $\xi\in\cal O$. If the stochastic Gaussian perturbation has only finitely many modes (but is still function-valued), \lim_{t\to\9}\int_K|X(t)-X_c|d\xi=0 exponentially fast for all compact $K\subset\cal O$ with probability one, if the noise is sufficiently strong. We also recover that in the deterministic case $\ell=0$

Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case

We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. The main idea consists in approximating the equation by equations with monotone non-degenerate coefficients and deriving some new analytical properties of the solution

Boundary Controllability and Observability of a Viscoelastic String

In this paper we consider an integrodifferential system, which governs the vibration of a viscoelastic one-dimensional object. We assume that we can act on the system at the boundary and we prove that it is possible to control both the position and the velocity at every point of the body and at a certain time $T$, large enough. We shall prove this result using moment theory and we shall prove that the solution of this problem leads to identify a Riesz sequence which solves controllability and observability. So, the result as presented here are constructive and can lead to simple numerical algorithms

Fenchel-Rockafellar type duality for a non-convex non-differential optimization problem

AbstractA Fenchel-Rockafellar type duality theorem is obtained for a non-convex and non-differentiable maximization problem by embedding the original problem in a family of perturbed problems. The recent results of Ivan Singer are developed in this more general framework. A relationship is also established between the solutions and optimal values of the primal and dual problems using the theory of subdifferential calculus

A numerical framework for modelling tire mechanics accounting for composite materials, large strains and frictional contact

Presentation delivered by Alejandro Cornejo from CIMNE during the 17th International Conference on Computational Plasticity, Fundamentals and Applications (COMPLAS) taking place from 5 – 7 of September in Barcelona, Spain.The Fatigue4Light project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 10100684

Weak formulation for singular diffusion equation with dynamic boundary condition

In this paper, we propose a weak formulation of the singular diffusion equation subject to the dynamic boundary condition. The weak formulation is based on a reformulation method by an evolution equation including the subdifferential of a governing convex energy. Under suitable assumptions, the principal results of this study are stated in forms of Main Theorems A and B, which are respectively to verify: the adequacy of the weak formulation; the common property between the weak solutions and those in regular problems of standard PDEs.Comment: 23 page

Approximating optimal control problems governed by variational inequalities

It is proposed an approximating method for optimal control problems governed by elliptic variational inequalities. Some applications and numerical examples are treated