Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2011
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911309654654976 |
|---|---|
| author | Kmit, I. Recke, L. |
| author_facet | Kmit, I. Recke, L. |
| contents | This paper concerns linear first-order hyperbolic systems in one space dimension of the type $$ \partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state a non-resonance condition (depending on the coefficients $a_j$ and $b_{jj}$ and the boundary reflection coefficients), which implies Fredholm solvability of the problem in the space of continuous functions. Further, we state one more non-resonance condition (depending also on $\partial_ta_j$), which implies $C^1$-solution regularity. Moreover, we give examples showing that both non-resonance conditions cannot be dropped, in general. Those conditions are robust under small perturbations of the problem data. Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1108_2882 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems Kmit, I. Recke, L. Analysis of PDEs 35B10, 35L40, 47A53 This paper concerns linear first-order hyperbolic systems in one space dimension of the type $$ \partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state a non-resonance condition (depending on the coefficients $a_j$ and $b_{jj}$ and the boundary reflection coefficients), which implies Fredholm solvability of the problem in the space of continuous functions. Further, we state one more non-resonance condition (depending also on $\partial_ta_j$), which implies $C^1$-solution regularity. Moreover, we give examples showing that both non-resonance conditions cannot be dropped, in general. Those conditions are robust under small perturbations of the problem data. Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity. |
| title | Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems |
| topic | Analysis of PDEs 35B10, 35L40, 47A53 |
| url | https://arxiv.org/abs/1108.2882 |