We consider the hyperbolic system ü- div (A∇ u) = f in the time varying cracked domain Ω \ Γt, where the set Ω ⊂ Rdis open, bounded, and with Lipschitz boundary, the cracks Γt, t∈ [ 0 , T] , are closed subsets of Ω ¯ , increasing with respect to inclusion, and u(t) : Ω \ Γt→ Rdfor every t∈ [ 0 , T]. We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system v̈- div (B∇ v) + a∇ v- 2 ∇ v˙ b= g on the fixed domain Ω \ Γ0. Under these assumptions, we obtain existence and uniqueness of weak solutions for these two problems. Moreover, we show an energy equality for the functions v, which allows us to prove a continuous dependence result for both systems. The same study has already been carried out in [3, 7] in the scalar case.

Linear Hyperbolic Systems in Domains with Growing Cracks / Caponi, Maicol. - In: MILAN JOURNAL OF MATHEMATICS. - ISSN 1424-9286. - 85:1(2017), pp. 149-185. [10.1007/s00032-017-0268-7]

Linear Hyperbolic Systems in Domains with Growing Cracks

Caponi, Maicol
2017

Abstract

We consider the hyperbolic system ü- div (A∇ u) = f in the time varying cracked domain Ω \ Γt, where the set Ω ⊂ Rdis open, bounded, and with Lipschitz boundary, the cracks Γt, t∈ [ 0 , T] , are closed subsets of Ω ¯ , increasing with respect to inclusion, and u(t) : Ω \ Γt→ Rdfor every t∈ [ 0 , T]. We assume the existence of suitable regular changes of variables, which reduce our problem to the transformed system v̈- div (B∇ v) + a∇ v- 2 ∇ v˙ b= g on the fixed domain Ω \ Γ0. Under these assumptions, we obtain existence and uniqueness of weak solutions for these two problems. Moreover, we show an energy equality for the functions v, which allows us to prove a continuous dependence result for both systems. The same study has already been carried out in [3, 7] in the scalar case.
85
1
149
185
https://doi.org/10.1007/s00032-017-0268-7
https://link.springer.com/content/pdf/10.1007%2Fs00032-017-0268-7.pdf
http://cvgmt.sns.it/paper/3319/
Caponi, Maicol
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/85606
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