Discrete Schrödinger equation and ill-posedness for the Euler equation

Previous Article
Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations
DCDS Home
This Issue
Next Article
Liouville theorems for stable solutions of the weighted Lane-Emden system

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.

[1]	Tsukasa Iwabuchi, Kota Uriya. Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $\|u\|^2$. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1395-1405. doi: 10.3934/cpaa.2015.14.1395
[2]	Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253
[3]	Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565
[4]	Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations and Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002
[5]	Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863
[6]	Nobu Kishimoto. A remark on norm inflation for nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1375-1402. doi: 10.3934/cpaa.2019067
[7]	G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327
[8]	Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure and Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727
[9]	Adán J. Corcho. Ill-Posedness for the Benney system. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 965-972. doi: 10.3934/dcds.2006.15.965
[10]	Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic and Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019
[11]	Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082
[12]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao. Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm. Communications on Pure and Applied Analysis, 2003, 2 (1) : 33-50. doi: 10.3934/cpaa.2003.2.33
[13]	Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971
[14]	Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382
[15]	Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5449-5463. doi: 10.3934/dcdsb.2020353
[16]	Zehan Lin, Chongbin Xu, Caidi Zhao, Chujin Li. Statistical solution and Kolmogorov entropy for the impulsive discrete Klein-Gordon-Schrödinger-type equations. Discrete and Continuous Dynamical Systems - B, 2023, 28 (1) : 20-49. doi: 10.3934/dcdsb.2022065
[17]	Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027
[18]	Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure and Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261
[19]	Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072
[20]	Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
In-Jee Jeong Benoit Pausader
on Google Scholar
In-Jee Jeong Benoit Pausader

2021 Impact Factor: 1.588

Tools

Article outline

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
In-Jee Jeong Benoit Pausader
on Google Scholar
In-Jee Jeong Benoit Pausader

[Back to Top]