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

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 & 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
[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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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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