Well-posedness for the non-integrable periodic fifth order KdV in Bourgain spaces

Ryan McConnell

doi:10.3934/dcds.2024068

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2024, Volume 44, Issue 11: 3507-3552. Doi: 10.3934/dcds.2024068

This issue Previous Article Stability of P-cyclic closed characteristics on P-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $ Next Article Probabilistic well-posedness for the nonlinear wave equation on $ B_2\times\mathbb{T} $

Well-posedness for the non-integrable periodic fifth order KdV in Bourgain spaces

Ryan McConnell

Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A.

Received: January 1, 2024

Early access: May 15, 2024

Published online: May 15, 2024

The author was partially supported by NSF grant DMS-2154031.

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Abstract

We study well-posedness for a non-integrable generalization of the fifth order KdV, the second member in the KdV heirarchy. In particular, we use differentiation-by-parts to establish well-posedness for $ s> 35/64 $ in low modulation restricted norm spaces, as well as non-linear smoothing of order $ \varepsilon < \min(2(s-35/64), 1) $. As corollaries, we obtain unconditional well-posedness for the non-integrable fifth order KdV for $ s > 1 $ and global well-posedness for the integrable fifth order KdV for $ s\geq 1 $. We also show local well-posedness for the non-integrable fifth order KdV for $ s > 1/2 $, contingent upon the conjectured $ L^8 $ Strichartz estimate. As an application of the nonlinear smoothing we obtain non-trivial upper bounds on the upper Minkowski dimension of the solution to the non-integrable fifth order KdV.

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Mathematics Subject Classification: Primary: 35A01, 37K10, 37L50.

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