Prescribing analytic singularities for solutions of a class of vector fields on the torus
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- by Adalberto P. Bergamasco and Sérgio Luís Zani PDF
- Trans. Amer. Math. Soc. 357 (2005), 4159-4174 Request permission
Abstract:
We consider the operator $L=\partial _t+(a(t)+ib(t))\partial _x$ acting on distributions on the two-torus $\mathbb T^2,$ where $a$ and $b$ are real-valued, real analytic functions defined on the unit circle $\mathbb T^1.$ We prove, among other things, that when $b$ changes sign, given any subset $\Sigma$ of the set of the local extrema of the local primitives of $b,$ there exists a singular solution of $L$ such that the $t-$projection of its analytic singular support is $\Sigma ;$ furthermore, for any $\tau \in \Sigma$ and any closed subset $F$ of $\mathbb T^1_x$ there exists $u\in \mathcal D’(\mathbb T^2)$ such that $Lu\in C^\omega (\mathbb T^2)$ and $\operatorname {sing supp_A}(u)=\{\tau \}\times F.$ We also provide a microlocal result concerning the trace of $u$ at $t=\tau .$References
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Additional Information
- Adalberto P. Bergamasco
- Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação - USP, Caixa Postal 668, São Carlos, SP, 13560-970 Brasil
- Email: apbergam@icmc.usp.br
- Sérgio Luís Zani
- Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação - USP, Caixa Postal 668, São Carlos, SP, 13560-970 Brasil
- Email: szani@icmc.usp.br
- Received by editor(s): December 9, 2003
- Published electronically: May 20, 2005
- Additional Notes: The first author was partially supported by CNPq. Both authors were partially supported by FAPESP
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 357 (2005), 4159-4174
- MSC (2000): Primary 35A20, 35H10
- DOI: https://doi.org/10.1090/S0002-9947-05-03905-X
- MathSciNet review: 2159704