On two-sided estimates for the nonlinear Fourier transform of KdV

Jan-Cornelius  Molnar

doi:10.3934/dcds.2016.36.3339

Article Contents

2016, Volume 36, Issue 6: 3339-3356. Doi: 10.3934/dcds.2016.36.3339

This issue Previous Article Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy Next Article On elliptic systems with Sobolev critical exponent

On two-sided estimates for the nonlinear Fourier transform of KdV

Jan-Cornelius Molnar^1,

1.
Winterthrerstrasse 190, 8057 Zurich

Received: March 31, 2015

Revised: September 30, 2015

Published: May 2017

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Abstract

The KdV-equation $u_t = -u_{xxx} + 6uu_x$ on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearizing the KdV flow. The regularity properties of $u$ are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper we obtain two-sided polynomial estimates of all integer Sobolev norms $||u||_m$, $m\ge 0$, in terms of the weighted norms of the nonlinear Fourier transformed, which are linear in the highest order.

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Mathematics Subject Classification: Primary: 37K15; Secondary: 35Q53, 37K10.

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