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

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

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.
Citation: 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
References:
[1]

L. V. Bogdanov, The Veselov-Novikov equation as a natural generalization of the Korteweg de Vries equation, Teoret. Mat. Fiz., 70 (1987), 309-314. English translation: Theoret. and Math. Phys., 70 (1987), 219-223.

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II: The KdV equation, GAFA, 3 (1993), 107-156. doi: 10.1007/BF01896020.

[3]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, GAFA, 3 (1993), 315-341. doi: 10.1007/BF01896259.

[4]

J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Sel. Math. New. Ser., 3 (1997), 115-159. doi: 10.1007/s000290050008.

[5]

N. Burq, Éstimations de Strichartz pour des parturbations à longue portée de l'opérateur de Schrödinger, Séminaire É.D.P., Exposé No X, 2001-2002.

[6]

A. Carbery, C. E. Kenig and S. Ziesler, Restriction for homogeneous polynomial surfaces in $\mathbb{R}^3$, arXiv:1108.4123 doi: 10.1090/S0002-9947-2012-05685-6.

[7]

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

[8]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.

[9]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012.

[10]

P. G. Grinevich, The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy, Uspekhi Mat. Nauk, 55 (2000), 3-70 (Russian); Russian Math. Surveys, 55 (2000), 1015-1083. doi: 10.1070/rm2000v055n06ABEH000333.

[11]

P. G. Grinevich and S. V. Manakov, Inverse scattering problem for the two-dimensional Schrödinger operator, the $\bar{\partial}$-method and nonlinear equations, Funktsional. Anal. i Prilozhen, 20 (1986), 14-24; English transl., Functional Anal. Appl., 20 (1986), 94-103.

[12]

A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst. - A., 5 (2014), 2061-2068. doi: 10.3934/dcds.2014.34.2061.

[13]

Z. Hani, A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds, Analysis & PDE, 5 (2012), 339-363. doi: 10.2140/apde.2012.5.339.

[14]

A. Kazeykina, Solitons and Large Time Asymptotics of Solutions for the Novikov-Veselov Equation, PhD thesis at \'Ecole Polytechnique, 2012.

[15]

A. Kazeykina and R. G. Novikov, Large time asymptotics for the Grinevich-Zakharov potentials, Bulletin des Sciences Mathématiques, 135 (2011), 374-382. doi: 10.1016/j.bulsci.2011.02.003.

[16]

A. Kazeykina and R. G. Novikov, Absence of exponentially localized solitons for the Novikov-Veselov equation at negative energy, Nonlinearity, 24 (2011), 1821-1830. doi: 10.1088/0951-7715/24/6/007.

[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, Inverse Problems, 28 (2012), 055017. doi: 10.1088/0266-5611/28/5/055017.

[18]

M. Lassas, J. L. Mueller and S. Siltanen, Mapping properties of the nonlinear Fourier transform in dimension two, Comm. Partial Differential Equations, 32 (2007), 591-610. doi: 10.1080/03605300500530412.

[19]

M. Lassas, J. L. Mueller, S. Siltanen and A. Stahel, The Novikov-Veselov equation and the inverse scattering method, Part I, Analysis, Phys. D, 241 (2012), 1322-1335. doi: 10.1016/j.physd.2012.04.010.

[20]

F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339. doi: 10.1137/080739173.

[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, Communications PDE, 35 (2010), 1674-1689. doi: 10.1080/03605302.2010.494195.

[22]

F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 24 (2009), 547-565. doi: 10.3934/dcds.2009.24.547.

[23]

L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371. doi: 10.1016/j.anihpc.2013.12.003.

[24]

L. Molinet, J.-C. Saut and N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Mathematical Journal, 115 (2002), 353-384. doi: 10.1215/S0012-7094-02-11525-7.

[25]

L. Molinet, J.-C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988. doi: 10.1137/S0036141001385307.

[26]

L. Molinet, J.-C. Saut and N. Tzvetkov, Global well-posedness for the KPII equation on the background of non localized solution, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 653-676. doi: 10.1016/j.anihpc.2011.04.004.

[27]

C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis, Vol. 1, Cambridge Studies In Advanced Mathematics 137, 2013.

[28]

R. G. Novikov, Absence of exponentially localized solitons for the Novikov-Veselov equation at positive energy, Phys. Lett. A, 375 (2011), 1233-1235. doi: 10.1016/j.physleta.2011.01.052.

[29]

S. P. Novikov and A. P. Veselov, Finite-gap two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations, Dokl. Akad. Nauk SSSR, 279 (1984), 20-24.. English translation: Soviet Math. Dokl., 30 (1984), 588-591.

[30]

S. P. Novikov and A. P. Veselov, Two-dimensional Schrödinger operator: inverse scattering transform and evolutional equations. Solitons and coherent structures, Phys. D, 18 (1986), 267-273. doi: 10.1016/0167-2789(86)90187-9.

[31]

P. Perry, Miura maps and inverse scattering for the Novikov-Veselov equation, Analysis & PDE, arXiv:1201.2385. doi: 10.2140/apde.2014.7.311.

[32]

L. Robbiano and C. Zuily, Strichartz estimates for Schrödinger equations with variable coefficients, Société Mathématique de France, Paris: Mémoires de la Société Mathématique de France, (2005), 101-102.

[33]

G. Staffilani and D. Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations, 27 (2002), 1337-1372. doi: 10.1081/PDE-120005841.

[34]

T. Tao, Nonlinear Dispersive Equations: Local And Global Analysis, Regional Conference Series In Mathematics, 106, AMS, Providence, RI, 2006.

[35]

N. Tzvetkov, Remark on the local ill-posedness for KdV equation, C.R. Acad. Sci. Paris Sér. I Math., 329 (1999), 1043-1047. doi: 10.1016/S0764-4442(00)88471-2.

show all references

References:
[1]

L. V. Bogdanov, The Veselov-Novikov equation as a natural generalization of the Korteweg de Vries equation, Teoret. Mat. Fiz., 70 (1987), 309-314. English translation: Theoret. and Math. Phys., 70 (1987), 219-223.

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II: The KdV equation, GAFA, 3 (1993), 107-156. doi: 10.1007/BF01896020.

[3]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, GAFA, 3 (1993), 315-341. doi: 10.1007/BF01896259.

[4]

J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Sel. Math. New. Ser., 3 (1997), 115-159. doi: 10.1007/s000290050008.

[5]

N. Burq, Éstimations de Strichartz pour des parturbations à longue portée de l'opérateur de Schrödinger, Séminaire É.D.P., Exposé No X, 2001-2002.

[6]

A. Carbery, C. E. Kenig and S. Ziesler, Restriction for homogeneous polynomial surfaces in $\mathbb{R}^3$, arXiv:1108.4123 doi: 10.1090/S0002-9947-2012-05685-6.

[7]

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

[8]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425. doi: 10.1006/jfan.2000.3687.

[9]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equations, 31 (1995), 1002-1012.

[10]

P. G. Grinevich, The scattering transform for the two-dimensional Schrödinger operator with a potential that decreases at infinity at fixed nonzero energy, Uspekhi Mat. Nauk, 55 (2000), 3-70 (Russian); Russian Math. Surveys, 55 (2000), 1015-1083. doi: 10.1070/rm2000v055n06ABEH000333.

[11]

P. G. Grinevich and S. V. Manakov, Inverse scattering problem for the two-dimensional Schrödinger operator, the $\bar{\partial}$-method and nonlinear equations, Funktsional. Anal. i Prilozhen, 20 (1986), 14-24; English transl., Functional Anal. Appl., 20 (1986), 94-103.

[12]

A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst. - A., 5 (2014), 2061-2068. doi: 10.3934/dcds.2014.34.2061.

[13]

Z. Hani, A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds, Analysis & PDE, 5 (2012), 339-363. doi: 10.2140/apde.2012.5.339.

[14]

A. Kazeykina, Solitons and Large Time Asymptotics of Solutions for the Novikov-Veselov Equation, PhD thesis at \'Ecole Polytechnique, 2012.

[15]

A. Kazeykina and R. G. Novikov, Large time asymptotics for the Grinevich-Zakharov potentials, Bulletin des Sciences Mathématiques, 135 (2011), 374-382. doi: 10.1016/j.bulsci.2011.02.003.

[16]

A. Kazeykina and R. G. Novikov, Absence of exponentially localized solitons for the Novikov-Veselov equation at negative energy, Nonlinearity, 24 (2011), 1821-1830. doi: 10.1088/0951-7715/24/6/007.

[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, Inverse Problems, 28 (2012), 055017. doi: 10.1088/0266-5611/28/5/055017.

[18]

M. Lassas, J. L. Mueller and S. Siltanen, Mapping properties of the nonlinear Fourier transform in dimension two, Comm. Partial Differential Equations, 32 (2007), 591-610. doi: 10.1080/03605300500530412.

[19]

M. Lassas, J. L. Mueller, S. Siltanen and A. Stahel, The Novikov-Veselov equation and the inverse scattering method, Part I, Analysis, Phys. D, 241 (2012), 1322-1335. doi: 10.1016/j.physd.2012.04.010.

[20]

F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339. doi: 10.1137/080739173.

[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, Communications PDE, 35 (2010), 1674-1689. doi: 10.1080/03605302.2010.494195.

[22]

F. Linares and J.-C. Saut, The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 24 (2009), 547-565. doi: 10.3934/dcds.2009.24.547.

[23]

L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371. doi: 10.1016/j.anihpc.2013.12.003.

[24]

L. Molinet, J.-C. Saut and N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Mathematical Journal, 115 (2002), 353-384. doi: 10.1215/S0012-7094-02-11525-7.

[25]

L. Molinet, J.-C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988. doi: 10.1137/S0036141001385307.

[26]

L. Molinet, J.-C. Saut and N. Tzvetkov, Global well-posedness for the KPII equation on the background of non localized solution, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 653-676. doi: 10.1016/j.anihpc.2011.04.004.

[27]

C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis, Vol. 1, Cambridge Studies In Advanced Mathematics 137, 2013.

[28]

R. G. Novikov, Absence of exponentially localized solitons for the Novikov-Veselov equation at positive energy, Phys. Lett. A, 375 (2011), 1233-1235. doi: 10.1016/j.physleta.2011.01.052.

[29]

S. P. Novikov and A. P. Veselov, Finite-gap two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations, Dokl. Akad. Nauk SSSR, 279 (1984), 20-24.. English translation: Soviet Math. Dokl., 30 (1984), 588-591.

[30]

S. P. Novikov and A. P. Veselov, Two-dimensional Schrödinger operator: inverse scattering transform and evolutional equations. Solitons and coherent structures, Phys. D, 18 (1986), 267-273. doi: 10.1016/0167-2789(86)90187-9.

[31]

P. Perry, Miura maps and inverse scattering for the Novikov-Veselov equation, Analysis & PDE, arXiv:1201.2385. doi: 10.2140/apde.2014.7.311.

[32]

L. Robbiano and C. Zuily, Strichartz estimates for Schrödinger equations with variable coefficients, Société Mathématique de France, Paris: Mémoires de la Société Mathématique de France, (2005), 101-102.

[33]

G. Staffilani and D. Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations, 27 (2002), 1337-1372. doi: 10.1081/PDE-120005841.

[34]

T. Tao, Nonlinear Dispersive Equations: Local And Global Analysis, Regional Conference Series In Mathematics, 106, AMS, Providence, RI, 2006.

[35]

N. Tzvetkov, Remark on the local ill-posedness for KdV equation, C.R. Acad. Sci. Paris Sér. I Math., 329 (1999), 1043-1047. doi: 10.1016/S0764-4442(00)88471-2.

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