Weighted Inertia-Dissipation-Energy approach to doubly nonlinear wave equations
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909075622592512 |
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| author | Akagi, Goro Bögelein, Verena Marveggio, Alice Stefanelli, Ulisse |
| author_facet | Akagi, Goro Bögelein, Verena Marveggio, Alice Stefanelli, Ulisse |
| contents | We discuss a variational approach to doubly nonlinear wave equations of the form $ρu_{tt} + g (u_t) - Δu + f (u)=0$. This approach hinges on the minimization of a parameter-dependent family of uniformly convex functionals over entire trajectories, namely the so-called Weighted Inertia-Dissipation-Energy (WIDE) functionals. We prove that the WIDE functionals admit minimizers and that the corresponding Euler-Lagrange system is solvable in the strong sense. Moreover, we check that the parameter-dependent minimizers converge, up to subsequences, to a solution of the target doubly nonlinear wave equation as the parameter goes to $0$. The analysis relies on specific estimates on the WIDE minimizers, on the decomposition of the subdifferential of the WIDE functional, and on the identification of the nonlinearities in the limit. Eventually, we investigate the viscous limit $ρ\to 0$, both at the functional level and on that of the equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08856 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted Inertia-Dissipation-Energy approach to doubly nonlinear wave equations Akagi, Goro Bögelein, Verena Marveggio, Alice Stefanelli, Ulisse Analysis of PDEs We discuss a variational approach to doubly nonlinear wave equations of the form $ρu_{tt} + g (u_t) - Δu + f (u)=0$. This approach hinges on the minimization of a parameter-dependent family of uniformly convex functionals over entire trajectories, namely the so-called Weighted Inertia-Dissipation-Energy (WIDE) functionals. We prove that the WIDE functionals admit minimizers and that the corresponding Euler-Lagrange system is solvable in the strong sense. Moreover, we check that the parameter-dependent minimizers converge, up to subsequences, to a solution of the target doubly nonlinear wave equation as the parameter goes to $0$. The analysis relies on specific estimates on the WIDE minimizers, on the decomposition of the subdifferential of the WIDE functional, and on the identification of the nonlinearities in the limit. Eventually, we investigate the viscous limit $ρ\to 0$, both at the functional level and on that of the equation. |
| title | Weighted Inertia-Dissipation-Energy approach to doubly nonlinear wave equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.08856 |