Fredholmness and Smooth Dependence for Linear Time-Periodic Hyperbolic System
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2010
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| _version_ | 1866917134428274688 |
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| author | Kmit, Irina Recke, Lutz |
| author_facet | Kmit, Irina Recke, Lutz |
| contents | This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type $$ \partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,...,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the data $a_j$ and $b_{jk}$ and the reflection coefficients such that the system is Fredholm solvable. Moreover, we state conditions on the data such that for any right hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution-map is smooth with respect to appropriate function space norms. In particular, those conditions imply that no small denominator effects occur.
We show that perturbations of the coefficients $a_j$ lead to essentially different results than perturbations of the coefficients $b_{jk}$, in general. Our results cover cases of non-strictly hyperbolic systems as well as systems with discontinuous coefficients $a_j$ and $b_{jk}$, but they are new even in the case of strict hyperbolicity and of smooth coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1005_0689 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | Fredholmness and Smooth Dependence for Linear Time-Periodic Hyperbolic System Kmit, Irina Recke, Lutz Analysis of PDEs 35B10, 35L50, 47A53 This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type $$ \partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,...,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the data $a_j$ and $b_{jk}$ and the reflection coefficients such that the system is Fredholm solvable. Moreover, we state conditions on the data such that for any right hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution-map is smooth with respect to appropriate function space norms. In particular, those conditions imply that no small denominator effects occur. We show that perturbations of the coefficients $a_j$ lead to essentially different results than perturbations of the coefficients $b_{jk}$, in general. Our results cover cases of non-strictly hyperbolic systems as well as systems with discontinuous coefficients $a_j$ and $b_{jk}$, but they are new even in the case of strict hyperbolicity and of smooth coefficients. |
| title | Fredholmness and Smooth Dependence for Linear Time-Periodic Hyperbolic System |
| topic | Analysis of PDEs 35B10, 35L50, 47A53 |
| url | https://arxiv.org/abs/1005.0689 |