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May  2016, 15(3): 727-760. doi: 10.3934/cpaa.2016.15.727

Well-posedness and ill-posedness results for the Novikov-Veselov equation

Yannis Angelopoulos 1,

1. 

Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George Street, Toronto, On M5S 2E4, Canada

Received  September 2014 Revised  December 2015 Published  February 2016

In this paper we study the Novikov-Veselov equation and the related modified Novikov-Veselov equation in certain Sobolev spaces. We prove local well-posedness in $H^s (\mathbb{R}^2)$ for $s > \frac{1}{2}$ for the Novikov-Veselov equation, and local well-posedness in $H^s (\mathbb{R}^2)$ for $s > 1$ for the modified Novikov-Veselov equation. Finally we point out some ill-posedness issues for the Novikov-Veselov equation in the supercritical regime.
Keywords: Novikov-Veselov equation, Fourier restriction norm method., multilinear estimates, dispersive equation.
Mathematics Subject Classification: 35Q5.
Citation: Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure & Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727
References:
[1]

L. V. Bogdanov, The Veselov-Novikov equation as a natural generalization of the Korteweg de Vries equation,, \emph{Teoret. Mat. Fiz.}, 70 (1987), 309.   Google Scholar

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II: The KdV equation,, \emph{GAFA}, 3 (1993), 107.  doi: 10.1007/BF01896020.  Google Scholar

[3]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation,, \emph{GAFA}, 3 (1993), 315.  doi: 10.1007/BF01896259.  Google Scholar

[4]

J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data,, \emph{Sel. Math. New. Ser.}, 3 (1997), 115.  doi: 10.1007/s000290050008.  Google Scholar

[5]

N. Burq, Éstimations de Strichartz pour des parturbations à longue portée de l'opérateur de Schrödinger,, \emph{S\'eminaire \'E.D.P.}, (): 2001.   Google Scholar

[6]

A. Carbery, C. E. Kenig and S. Ziesler, Restriction for homogeneous polynomial surfaces in $\mathbbR^3$,, arXiv:1108.4123, ().  doi: 10.1090/S0002-9947-2012-05685-6.  Google Scholar

[7]

M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations,, arXiv:math/0311048, ().   Google Scholar

[8]

M. Christ and A. Kiselev, Maximal functions associated to filtrations,, \emph{J. Funct. Anal.}, 179 (2001), 409.  doi: 10.1006/jfan.2000.3687.  Google Scholar

[9]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation,, \emph{Differential Equations}, 31 (1995), 1002.   Google Scholar

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P. G. Grinevich, The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy,, \emph{Uspekhi Mat. Nauk}, 55 (2000), 3.  doi: 10.1070/rm2000v055n06ABEH000333.  Google Scholar

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P. G. Grinevich and S. V. Manakov, Inverse scattering problem for the two-dimensional Schrödinger operator, the $\bar{\partial}$-method and nonlinear equations,, \emph{Funktsional. Anal. i Prilozhen}, 20 (1986), 14.   Google Scholar

[12]

A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation,, \emph{Discrete Contin. Dyn. Syst. - A.}, 5 (2014), 2061.  doi: 10.3934/dcds.2014.34.2061.  Google Scholar

[13]

Z. Hani, A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds,, \emph{Analysis & PDE}, 5 (2012), 339.  doi: 10.2140/apde.2012.5.339.  Google Scholar

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A. Kazeykina, Solitons and Large Time Asymptotics of Solutions for the Novikov-Veselov Equation,, PhD thesis at École Polytechnique, (2012).   Google Scholar

[15]

A. Kazeykina and R. G. Novikov, Large time asymptotics for the Grinevich-Zakharov potentials,, \emph{Bulletin des Sciences Math\'ematiques}, 135 (2011), 374.  doi: 10.1016/j.bulsci.2011.02.003.  Google Scholar

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A. Kazeykina and R. G. Novikov, Absence of exponentially localized solitons for the Novikov-Veselov equation at negative energy,, \emph{Nonlinearity}, 24 (2011), 1821.  doi: 10.1088/0951-7715/24/6/007.  Google Scholar

[17]

A. Kazeykina and R. G. Novikov, A large time asymptotics for the solution of the Cauchy problem for the Novikov-Veselov equation at negative energy with nonsingular scattering data,, \emph{Inverse Problems}, 28 (2012).  doi: 10.1088/0266-5611/28/5/055017.  Google Scholar

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M. Lassas, J. L. Mueller and S. Siltanen, Mapping properties of the nonlinear Fourier transform in dimension two,, \emph{Comm. Partial Differential Equations}, 32 (2007), 591.  doi: 10.1080/03605300500530412.  Google Scholar

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M. Lassas, J. L. Mueller, S. Siltanen and A. Stahel, The Novikov-Veselov equation and the inverse scattering method, Part I,, \emph{Analysis, 241 (2012), 1322.  doi: 10.1016/j.physd.2012.04.010.  Google Scholar

[20]

F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation,, \emph{SIAM J. Math. Anal.}, 41 (2009), 1323.  doi: 10.1137/080739173.  Google Scholar

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F. Linares, A. Pastor and J.-C. Saut, Well-posedness for the ZK equation in a cylinder and on the background of a KdV soliton,, \emph{Communications PDE}, 35 (2010), 1674.  doi: 10.1080/03605302.2010.494195.  Google Scholar

[22]

F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation,, \emph{Discrete Contin. Dyn. Syst.}, 24 (2009), 547.  doi: 10.3934/dcds.2009.24.547.  Google Scholar

[23]

L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 32 (2015), 347.  doi: 10.1016/j.anihpc.2013.12.003.  Google Scholar

[24]

L. Molinet, J.-C. Saut and N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation,, \emph{Duke Mathematical Journal}, 115 (2002), 353.  doi: 10.1215/S0012-7094-02-11525-7.  Google Scholar

[25]

L. Molinet, J.-C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations,, \emph{SIAM J. Math. Anal.}, 33 (2001), 982.  doi: 10.1137/S0036141001385307.  Google Scholar

[26]

L. Molinet, J.-C. Saut and N. Tzvetkov, Global well-posedness for the KPII equation on the background of non localized solution,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 28 (2011), 653.  doi: 10.1016/j.anihpc.2011.04.004.  Google Scholar

[27]

C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis,, Vol. 1, (2013).   Google Scholar

[28]

R. G. Novikov, Absence of exponentially localized solitons for the Novikov-Veselov equation at positive energy,, \emph{Phys. Lett. A}, 375 (2011), 1233.  doi: 10.1016/j.physleta.2011.01.052.  Google Scholar

[29]

S. P. Novikov and A. P. Veselov, Finite-gap two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations,, \emph{Dokl. Akad. Nauk SSSR}, 279 (1984), 20.   Google Scholar

[30]

S. P. Novikov and A. P. Veselov, Two-dimensional Schrödinger operator: inverse scattering transform and evolutional equations. Solitons and coherent structures,, \emph{Phys. D}, 18 (1986), 267.  doi: 10.1016/0167-2789(86)90187-9.  Google Scholar

[31]

P. Perry, Miura maps and inverse scattering for the Novikov-Veselov equation,, \emph{Analysis & PDE}, ().  doi: 10.2140/apde.2014.7.311.  Google Scholar

[32]

L. Robbiano and C. Zuily, Strichartz estimates for Schrödinger equations with variable coefficients,, \emph{Soci\'et\'e Math\'ematique de France, (2005), 101.   Google Scholar

[33]

G. Staffilani and D. Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients,, \emph{Comm. Partial Differential Equations}, 27 (2002), 1337.  doi: 10.1081/PDE-120005841.  Google Scholar

[34]

T. Tao, Nonlinear Dispersive Equations: Local And Global Analysis,, Regional Conference Series In Mathematics, (2006).   Google Scholar

[35]

N. Tzvetkov, Remark on the local ill-posedness for KdV equation,, \emph{C.R. Acad. Sci. Paris S\'er. I Math.}, 329 (1999), 1043.  doi: 10.1016/S0764-4442(00)88471-2.  Google Scholar

show all references

References:
[1]

L. V. Bogdanov, The Veselov-Novikov equation as a natural generalization of the Korteweg de Vries equation,, \emph{Teoret. Mat. Fiz.}, 70 (1987), 309.   Google Scholar

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II: The KdV equation,, \emph{GAFA}, 3 (1993), 107.  doi: 10.1007/BF01896020.  Google Scholar

[3]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation,, \emph{GAFA}, 3 (1993), 315.  doi: 10.1007/BF01896259.  Google Scholar

[4]

J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data,, \emph{Sel. Math. New. Ser.}, 3 (1997), 115.  doi: 10.1007/s000290050008.  Google Scholar

[5]

N. Burq, Éstimations de Strichartz pour des parturbations à longue portée de l'opérateur de Schrödinger,, \emph{S\'eminaire \'E.D.P.}, (): 2001.   Google Scholar

[6]

A. Carbery, C. E. Kenig and S. Ziesler, Restriction for homogeneous polynomial surfaces in $\mathbbR^3$,, arXiv:1108.4123, ().  doi: 10.1090/S0002-9947-2012-05685-6.  Google Scholar

[7]

M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations,, arXiv:math/0311048, ().   Google Scholar

[8]

M. Christ and A. Kiselev, Maximal functions associated to filtrations,, \emph{J. Funct. Anal.}, 179 (2001), 409.  doi: 10.1006/jfan.2000.3687.  Google Scholar

[9]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation,, \emph{Differential Equations}, 31 (1995), 1002.   Google Scholar

[10]

P. G. Grinevich, The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy,, \emph{Uspekhi Mat. Nauk}, 55 (2000), 3.  doi: 10.1070/rm2000v055n06ABEH000333.  Google Scholar

[11]

P. G. Grinevich and S. V. Manakov, Inverse scattering problem for the two-dimensional Schrödinger operator, the $\bar{\partial}$-method and nonlinear equations,, \emph{Funktsional. Anal. i Prilozhen}, 20 (1986), 14.   Google Scholar

[12]

A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation,, \emph{Discrete Contin. Dyn. Syst. - A.}, 5 (2014), 2061.  doi: 10.3934/dcds.2014.34.2061.  Google Scholar

[13]

Z. Hani, A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds,, \emph{Analysis & PDE}, 5 (2012), 339.  doi: 10.2140/apde.2012.5.339.  Google Scholar

[14]

A. Kazeykina, Solitons and Large Time Asymptotics of Solutions for the Novikov-Veselov Equation,, PhD thesis at École Polytechnique, (2012).   Google Scholar

[15]

A. Kazeykina and R. G. Novikov, Large time asymptotics for the Grinevich-Zakharov potentials,, \emph{Bulletin des Sciences Math\'ematiques}, 135 (2011), 374.  doi: 10.1016/j.bulsci.2011.02.003.  Google Scholar

[16]

A. Kazeykina and R. G. Novikov, Absence of exponentially localized solitons for the Novikov-Veselov equation at negative energy,, \emph{Nonlinearity}, 24 (2011), 1821.  doi: 10.1088/0951-7715/24/6/007.  Google Scholar

[17]

A. Kazeykina and R. G. Novikov, A large time asymptotics for the solution of the Cauchy problem for the Novikov-Veselov equation at negative energy with nonsingular scattering data,, \emph{Inverse Problems}, 28 (2012).  doi: 10.1088/0266-5611/28/5/055017.  Google Scholar

[18]

M. Lassas, J. L. Mueller and S. Siltanen, Mapping properties of the nonlinear Fourier transform in dimension two,, \emph{Comm. Partial Differential Equations}, 32 (2007), 591.  doi: 10.1080/03605300500530412.  Google Scholar

[19]

M. Lassas, J. L. Mueller, S. Siltanen and A. Stahel, The Novikov-Veselov equation and the inverse scattering method, Part I,, \emph{Analysis, 241 (2012), 1322.  doi: 10.1016/j.physd.2012.04.010.  Google Scholar

[20]

F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation,, \emph{SIAM J. Math. Anal.}, 41 (2009), 1323.  doi: 10.1137/080739173.  Google Scholar

[21]

F. Linares, A. Pastor and J.-C. Saut, Well-posedness for the ZK equation in a cylinder and on the background of a KdV soliton,, \emph{Communications PDE}, 35 (2010), 1674.  doi: 10.1080/03605302.2010.494195.  Google Scholar

[22]

F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation,, \emph{Discrete Contin. Dyn. Syst.}, 24 (2009), 547.  doi: 10.3934/dcds.2009.24.547.  Google Scholar

[23]

L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 32 (2015), 347.  doi: 10.1016/j.anihpc.2013.12.003.  Google Scholar

[24]

L. Molinet, J.-C. Saut and N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation,, \emph{Duke Mathematical Journal}, 115 (2002), 353.  doi: 10.1215/S0012-7094-02-11525-7.  Google Scholar

[25]

L. Molinet, J.-C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations,, \emph{SIAM J. Math. Anal.}, 33 (2001), 982.  doi: 10.1137/S0036141001385307.  Google Scholar

[26]

L. Molinet, J.-C. Saut and N. Tzvetkov, Global well-posedness for the KPII equation on the background of non localized solution,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 28 (2011), 653.  doi: 10.1016/j.anihpc.2011.04.004.  Google Scholar

[27]

C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis,, Vol. 1, (2013).   Google Scholar

[28]

R. G. Novikov, Absence of exponentially localized solitons for the Novikov-Veselov equation at positive energy,, \emph{Phys. Lett. A}, 375 (2011), 1233.  doi: 10.1016/j.physleta.2011.01.052.  Google Scholar

[29]

S. P. Novikov and A. P. Veselov, Finite-gap two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations,, \emph{Dokl. Akad. Nauk SSSR}, 279 (1984), 20.   Google Scholar

[30]

S. P. Novikov and A. P. Veselov, Two-dimensional Schrödinger operator: inverse scattering transform and evolutional equations. Solitons and coherent structures,, \emph{Phys. D}, 18 (1986), 267.  doi: 10.1016/0167-2789(86)90187-9.  Google Scholar

[31]

P. Perry, Miura maps and inverse scattering for the Novikov-Veselov equation,, \emph{Analysis & PDE}, ().  doi: 10.2140/apde.2014.7.311.  Google Scholar

[32]

L. Robbiano and C. Zuily, Strichartz estimates for Schrödinger equations with variable coefficients,, \emph{Soci\'et\'e Math\'ematique de France, (2005), 101.   Google Scholar

[33]

G. Staffilani and D. Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients,, \emph{Comm. Partial Differential Equations}, 27 (2002), 1337.  doi: 10.1081/PDE-120005841.  Google Scholar

[34]

T. Tao, Nonlinear Dispersive Equations: Local And Global Analysis,, Regional Conference Series In Mathematics, (2006).   Google Scholar

[35]

N. Tzvetkov, Remark on the local ill-posedness for KdV equation,, \emph{C.R. Acad. Sci. Paris S\'er. I Math.}, 329 (1999), 1043.  doi: 10.1016/S0764-4442(00)88471-2.  Google Scholar

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