Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation
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- by Joseph Thirouin PDF
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Abstract:
In this paper, we study a quadratic equation on the one-dimensional torus \[ i \partial _t u = 2J\Pi (|u|^2)+\bar {J}u^2, \quad u(0, \cdot )=u_0,\] where $J=\int _\mathbb {T}|u|^2u \in \mathbb {C}$ has constant modulus, and $\Pi$ is the Szegő projector onto functions with nonnegative frequencies. Thanks to a Lax pair structure, we construct a flow on $BMO(\mathbb {T})\cap \mathrm {Im}\Pi$ which propagates $H^s$ regularity for any $s>0$, whereas the energy level corresponds to $s=1/2$. Then, for each $s>1/2$, we exhibit solutions whose $H^s$ norm goes to $+\infty$ exponentially fast, and we show that this growth is optimal.References
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Additional Information
- Joseph Thirouin
- Affiliation: Département de mathématiques et applications, École normale supérieure, CNRS, PSL Research University, 75005 Paris, France
- MR Author ID: 1199507
- Email: joseph.thirouin@ens.fr
- Received by editor(s): October 5, 2017
- Received by editor(s) in revised form: January 22, 2018
- Published electronically: September 24, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 3673-3690
- MSC (2010): Primary 37K10; Secondary 37K40, 35B45
- DOI: https://doi.org/10.1090/tran/7535
- MathSciNet review: 3896126