Discrete Schrödinger equation and ill-posedness for the Euler equation

In-Jee Jeong; Benoit Pausader

doi:10.3934/dcds.2017012

Article Contents

2017, Volume 37, Issue 1: 281-293. Doi: 10.3934/dcds.2017012

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Discrete Schrödinger equation and ill-posedness for the Euler equation

In-Jee Jeong^1, , and
Benoit Pausader^2,

1.
304 Fine Hall, Princeton, NJ 08544, USA
2.
Kassar House, 151 Thayer street, Providence, RI 02912, USA

* Corresponding author: In-Jee Jeong
* Corresponding author: In-Jee Jeong

Received: November 2015

Revised: September 2016

Published: September 2017

The second author is partly funded by the Sloan foundation and by the NSF grant DMS-1362940.

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Abstract

We consider the 2D Euler equation with periodic boundary conditions in a family of Banach spaces based on the Fourier coefficients, and show that it is ill-posed in the sense that 'norm inflation' occurs. The proof is based on the observation that the evolution of certain perturbations of the 'Kolmogorov flow' given in velocity by

$U(x,y) = \left( {\begin{array}{*{20}{c}}{\cos \;y}\\0\end{array}} \right)$

can be well approximated by the linear Schrödinger equation, at least for a short period of time.

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Mathematics Subject Classification: Primary:35Q31, 76D03;Secondary:37B55.

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References

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