Discrete Schrödinger equation and ill-posedness for the Euler equation

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January 2017, 37(1): 281-293. doi: 10.3934/dcds.2017012

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

Received November 2015 Revised September 2016 Published November 2016

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

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.

Keywords: The Euler equation, ill-posedness, norm inflation, discrete Schrödinger equation, Kolmogorov flow.

Mathematics Subject Classification: Primary:35Q31, 76D03;Secondary:37B5.

Citation: In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012

References:

[1]	J. Bourgain and D. Li, Strong Ill-posedness of the incompressible Euler equation inborderline Sobolev spaces, Invent. Math., 20 (2015), 97-157. doi: 10.1007/s00222-014-0548-6.
[2]	J. Bourgain and D. Li, Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces, Geom. Funct. Anal., 25 (2015), 1-86. doi: 10.1007/s00039-015-0311-1.
[3]	A. Cheskidov and R. Shvydkoy, Ill-posedness of the basic equations of fluid dynamics in Besov spaces, Proceedings of AMS, 138 (2010), 1059-1067. doi: 10.1090/S0002-9939-09-10141-7.
[4]	D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92 (1970), 102-163. doi: 10.2307/1970699.
[5]	T. Elgindi and N. Masmoudi, L^∞ Ill-posedness for a class of equations arising in hydrodynamics, preprint, arXiv: 1405.2478v2.
[6]	A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM Journal on Mathematical Analysis, 39 (2008), 1890-1920. doi: 10.1137/070689139.
[7]	T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbb{R}^3$, J. Func. Anal., 9 (1972), 296-305. doi: 10.1016/0022-1236(72)90003-1.
[8]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.
[9]	J. Mattingly and Ya. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations, Commun. Contemp. Math., 1 (1999), 497-516. doi: 10.1142/S0219199799000183.

show all references

References:

[1]	J. Bourgain and D. Li, Strong Ill-posedness of the incompressible Euler equation inborderline Sobolev spaces, Invent. Math., 20 (2015), 97-157. doi: 10.1007/s00222-014-0548-6.
[2]	J. Bourgain and D. Li, Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces, Geom. Funct. Anal., 25 (2015), 1-86. doi: 10.1007/s00039-015-0311-1.
[3]	A. Cheskidov and R. Shvydkoy, Ill-posedness of the basic equations of fluid dynamics in Besov spaces, Proceedings of AMS, 138 (2010), 1059-1067. doi: 10.1090/S0002-9939-09-10141-7.
[4]	D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., 92 (1970), 102-163. doi: 10.2307/1970699.
[5]	T. Elgindi and N. Masmoudi, L^∞ Ill-posedness for a class of equations arising in hydrodynamics, preprint, arXiv: 1405.2478v2.
[6]	A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM Journal on Mathematical Analysis, 39 (2008), 1890-1920. doi: 10.1137/070689139.
[7]	T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbb{R}^3$, J. Func. Anal., 9 (1972), 296-305. doi: 10.1016/0022-1236(72)90003-1.
[8]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.
[9]	J. Mattingly and Ya. Sinai, An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations, Commun. Contemp. Math., 1 (1999), 497-516. doi: 10.1142/S0219199799000183.

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