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

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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

Abstract
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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 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,, \emph{S\'eminaire \'E.D.P.}, (): 2001.
[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.
[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,, \emph{Analysis & PDE}, (). 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,, \emph{S\'eminaire \'E.D.P.}, (): 2001.
[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.
[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,, \emph{Analysis & PDE}, (). 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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