Agenda de l’IDP

Séminaire Orléans

Pinning and depinning of interfaces in random media
Patrick Dondl (Bonn)
Thursday 17 March 2011 14:00 -  Orléans -  Salle de Séminaire

Résumé :
We consider a parabolic model for the evolution of an interface in random medium. The local velocity of the interface is governed by line tension and a competition between a constant external driving force $F>0$ and a heterogeneous random field $f(x,y,\omega)$, which describes the interaction of the interface with its environment. To be precise, let $(\Omega,\F,\P)$ be a probability space, $\omega \in \Omega$. We consider the evolution equation $$ \partial_t u(x,t,\omega) &= u_{xx}(x,t,\omega) - f(x,u(x,t,\omega),\omega) + F $$ with zero initial condition. The random field $f>0$ has the form of localized smooth obstacles of random strength. In particular, we are interested in the macroscopic, homogenized behavior of solutions to the evolution equation and their dependence on $F$. We prove that, under some assumptions on $f$, we have existence of a non-negative stationary solution for $F$ small enough. This means that all solutions to the evolution equation become stuck if the driving force is not sufficiently large. The proof relies on a percolation argument. Given stronger assumptions on $f$, but still without a uniform bound on the obstacle strength, we also show that for large enough $F$ the interface will propagate with a finite velocity. The two results combined show the emergence of a rate-independent hysteresis in systems subject to a viscous microscopic evolution law through the interaction with a random environment. Joint work with N. Dirr (Bath University) and M. Scheutzow (TU Berlin).

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