Given a bounded open set $\Omega \subset \re^d$ 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 \textit{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.

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

Dal Maso, Gianni;
2017-01-01

Abstract

Given a bounded open set $\Omega \subset \re^d$ 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 \textit{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.
2017
2017
1
184
241
https://academic.oup.com/amrx/article/2017/1/184/2452787/The-Wave-Equation-on-Domains-with-Cracks-Growing
http://preprints.sissa.it/xmlui/handle/1963/34629
http://cvgmt.sns.it/paper/2817/
Dal Maso, Gianni; Lucardesi, I.
File in questo prodotto:
File Dimensione Formato  
DalMaso.pdf

non disponibili

Tipologia: Versione Editoriale (PDF)
Licenza: Non specificato
Dimensione 1.59 MB
Formato Adobe PDF
1.59 MB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/15978
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 21
  • ???jsp.display-item.citation.isi??? 18
social impact