Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems

Please always quote using this URN:urn:nbn:de:0296-matheon-11971
  • 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 kind of dissipativity condition (depending on the coefficients $a_j$ and $b_{jj}$ and the boundary reflection coefficients), which implies Fredholm solvability of the problem, i.e., either there is a nontrivial solution to the homogeneous problem (in this case the space of such solutions has finite dimension) or the nonhomogeneous problem is uniquely solvable for any right-hand side (in this case the solution depends continuously on the right-hand side). In particular, under those conditions no small denominator effects occur. Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity. Finally, in the case that all coefficients $a_j$ are $t$-independent, we show that the solutions are $C^\infty$-smooth if the data are $C^\infty$-smooth.

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Author:Irina Kmit, Lutz Recke
Referee:Alexander Mielke
Document Type:Preprint, Research Center Matheon
Date of first Publication:2013/02/08
Release Date:2013/02/08
Tag:Fredholm solvability; first-order hyperbolic systems; no small denominators; reflection boundary conditions; time-periodic solutions
Institute:Humboldt-Universität zu Berlin
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Bxx Qualitative properties of solutions / 35B10 Periodic solutions
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Lxx Hyperbolic equations and systems [See also 58J45] / 35L50 Initial-boundary value problems for first-order hyperbolic systems
47-XX OPERATOR THEORY / 47Axx General theory of linear operators / 47A53 (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]
Preprint Number:999
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