SISSA Open Science

# The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data

 dc.contributor.author Dal Maso, Gianni dc.contributor.author Lucardesi, Ilaria dc.date.accessioned 2015-10-13T07:52:08Z dc.date.available 2015-10-13T07:52:08Z dc.date.issued 2015-10 dc.identifier.uri http://preprints.sissa.it/xmlui/handle/1963/34629 dc.description.abstract 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 $C^2$. en_US 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. dc.description.sponsorship MIUR Project Calculus of Variations" (PRIN 2010-11), en_US ERC Grant No. 290888 Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture''. dc.language.iso en en_US dc.relation.ispartofseries SISSA;47/2015/MATE dc.subject wave equation, second order linear hyperbolic equations, dynamic fracture mechanics, cracking domains. en_US dc.title The wave equation on domains with cracks growing on a prescribed path: existence, uniqueness, and continuous dependence on the data en_US dc.type Preprint en_US dc.subject.miur MAT/05 en_US dc.miur.area 1 en_US dc.contributor.area Mathematics en_US
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