%0 Report
%D 2015
%T The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data
%A Gianni Dal Maso
%A Ilaria Lucardesi
%X Given a bounded open set $\Omega \subset \mathbb R^d$ with Lipschitz boundary and an increasing family $\Gamma_t$, $t\in [0,T]$, of closed subsets of $\Omega$, we analyze the scalar wave equation $\ddot{u} - div (A \nabla u) = f$ in the time varying cracked domains $\Omega\setminus\Gamma_t$. Here we assume that the sets $\Gamma_t$ are contained into a prescribed $(d-1)$-manifold of class $C^2$. Our approach relies on a change of variables: recasting the problem on the reference configuration $\Omega\setminus \Gamma_0$, we are led to consider a hyperbolic problem of the form $\ddot{v} - div (B\nabla v) + a \cdot \nabla v - 2 b \cdot \nabla \dot{v} = g$ in $\Omega \setminus \Gamma_0$. Under suitable assumptions on the regularity of the change of variables that transforms $\Omega\setminus \Gamma_t$ into $\Omega\setminus \Gamma_0$, we prove existence and uniqueness of weak solutions for both formulations. Moreover, we provide an energy equality, which gives, as a by-product, the continuous dependence of the solutions with respect to the cracks.
%G en
%U http://urania.sissa.it/xmlui/handle/1963/34629
%1 34832
%2 Mathematics
%4 1
%# MAT/05
%$ Submitted by lucardes@sissa.it (lucardes@sissa.it) on 2015-10-12T09:08:37Z
No. of bitstreams: 1
DM-Luc-15-SISSA.pdf: 376913 bytes, checksum: bfdcf0975c25c9366acb4195e0f08e79 (MD5)