Electron. J. Differential Equations, Vol. 2024 (2024), No. 05, pp. 1-25.

Global well-posedness for Cauchy problems of Zakharov-Kuznetsov equations on cylindrical spaces

Satoshi Osawa, Hideo Takaoka

Abstract:
We study the global well-posedness of the Zakharov-Kuznetsov equation on cylindrical spaces. Our goal is to establish the existence of global-in-time solutions below the energy class. To prove the results, we adapt the I-method to extend the local solutions globally in time. The main tool in our argument is multilinear estimates in the content of Bourgain's spaces. Using modified energies induced by the I-method, we obtain polynomial bounds on the \(H^s\) growth of global solutions.

Submitted January 3, 2024. Published January 22, 2024.
Math Subject Classifications: 35Q53, 42B37.
Key Words: Zakharov-Kuznetsov equation; low regularity; global well-posedness; bilinear estimate
DOI: 10.58997/ejde.2023405

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  Satoshi Osawa
Department of Mathematics
Kobe University
Kobe, 657-8501, Japan
email: sohsawa@math.kobe-u.ac.jp
Hideo Takaoka
Department of Mathematics
Kobe University
Kobe, 657-8501, Japan
email: takaoka@math.kobe-u.ac.jp

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