American Institute of Mathematical Sciences

Advanced
March  2014, 13(2): 527-542. doi: 10.3934/cpaa.2014.13.527

Well-posedness for the supercritical gKdV equation

Nils Strunk 1,

1. 

Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, 33501 Bielefeld, Germany

Received  September 2012 Revised  July 2013 Published  October 2013

In this paper we consider the supercritical generalized Korteweg-de~Vries equation $\partial_t\psi + \partial_{x x x}\psi + \partial_x(|\psi|^{p-1}\psi) = 0$, where $5 \leq p \in R$. We prove a local well-posedness result in the homogeneous Besov space $\dot B_\infty^{s_p,2}(R)$, where $s_p=\frac12-\frac{2}{p-1}$ is the scaling critical index. In particular local well-posedness in the smaller inhomogeneous Sobolev space $H^{s_p}(R)$ can be proved similarly. As a byproduct a global well-posedness result for small initial data is also obtained.
Keywords: supercritical, Korteweg-de~Vries equation, KdV-like equations, Besov space., Cauchy problem, well-posedness.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B3.
Citation: Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527
References:
[1]

Jöran Bergh and Jörgen Löfström, "Interpolation Spaces. An Introduction", Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223. doi: 10.1007/978-3-642-66451-9.  Google Scholar

[2]

Michael Christ, James E. Colliander and Terence Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293. doi: 10.1353/ajm.2003.0040.  Google Scholar

[3]

Luiz G. Farah, Felipe Linares and Ademir Pastor, The supercritical generalized KdV equation: global well-posedness in the energy space and below, Math. Res. Lett., 18 (2011), 357-377. doi: 10.4310/MRL.2011.v18.n2.a13.  Google Scholar

[4]

Axel Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339.  Google Scholar

[5]

Martin Hadac, Sebastian Herr and Herbert Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941, Erratum: ibid., 3 (2010), 971-972. doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[6]

Carlos E. Kenig, Gustavo Ponce and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.  Google Scholar

[7]

Carlos E. Kenig, Gustavo Ponce and Luis Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[8]

Herbert Koch and Jeremy L. Marzuola, Small data scattering and soliton stability in $\dot H^{-\frac16}$ for the quartic KdV Equation, Anal. PDE, 5 (2012), 145-198. doi: 10.2140/apde.2012.5.145.  Google Scholar

[9]

Herbert Koch and Daniel Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., 58 (2005), 217-284. doi: 10.1002/cpa.20067.  Google Scholar

[10]

Herbert Koch and Daniel Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN, 2007 (2007), Art. ID rnm053, 36. doi: 10.1093/imrn/rnm053.  Google Scholar

[11]

Luc Molinet and Francis Ribaud, On the Cauchy problem for the generalized Korteweg-de Vries equation, Comm. Partial Differential Equations, 28 (2003), 2065-2091. doi: 10.1081/PDE-120025496.  Google Scholar

[12]

Terence Tao, Scattering for the quartic generalised Korteweg-de Vries equation, J. Differential Equations, 232 (2007), 623-651. doi: 10.1016/j.jde.2006.07.019.  Google Scholar

[13]

Norbert Wiener, The quadratic variation of a function and its fourier coefficients., Journ. Math. Phys., 3 (1924), 72-94. Google Scholar

show all references

References:
[1]

Jöran Bergh and Jörgen Löfström, "Interpolation Spaces. An Introduction", Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223. doi: 10.1007/978-3-642-66451-9.  Google Scholar

[2]

Michael Christ, James E. Colliander and Terence Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293. doi: 10.1353/ajm.2003.0040.  Google Scholar

[3]

Luiz G. Farah, Felipe Linares and Ademir Pastor, The supercritical generalized KdV equation: global well-posedness in the energy space and below, Math. Res. Lett., 18 (2011), 357-377. doi: 10.4310/MRL.2011.v18.n2.a13.  Google Scholar

[4]

Axel Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339.  Google Scholar

[5]

Martin Hadac, Sebastian Herr and Herbert Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941, Erratum: ibid., 3 (2010), 971-972. doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[6]

Carlos E. Kenig, Gustavo Ponce and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.  Google Scholar

[7]

Carlos E. Kenig, Gustavo Ponce and Luis Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[8]

Herbert Koch and Jeremy L. Marzuola, Small data scattering and soliton stability in $\dot H^{-\frac16}$ for the quartic KdV Equation, Anal. PDE, 5 (2012), 145-198. doi: 10.2140/apde.2012.5.145.  Google Scholar

[9]

Herbert Koch and Daniel Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math., 58 (2005), 217-284. doi: 10.1002/cpa.20067.  Google Scholar

[10]

Herbert Koch and Daniel Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN, 2007 (2007), Art. ID rnm053, 36. doi: 10.1093/imrn/rnm053.  Google Scholar

[11]

Luc Molinet and Francis Ribaud, On the Cauchy problem for the generalized Korteweg-de Vries equation, Comm. Partial Differential Equations, 28 (2003), 2065-2091. doi: 10.1081/PDE-120025496.  Google Scholar

[12]

Terence Tao, Scattering for the quartic generalised Korteweg-de Vries equation, J. Differential Equations, 232 (2007), 623-651. doi: 10.1016/j.jde.2006.07.019.  Google Scholar

[13]

Norbert Wiener, The quadratic variation of a function and its fourier coefficients., Journ. Math. Phys., 3 (1924), 72-94. Google Scholar

[1]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[2]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097
[3]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032
[4]

Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45
[5]

M. Agrotis, S. Lafortune, P.G. Kevrekidis. On a discrete version of the Korteweg-De Vries equation. Conference Publications, 2005, 2005 (Special) : 22-29. doi: 10.3934/proc.2005.2005.22
[6]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[7]

Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509
[8]

Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655
[9]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
[10]

Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network. Evolution Equations & Control Theory, 2020, 9 (3) : 673-692. doi: 10.3934/eect.2020028
[11]

Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[12]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
[13]

Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control & Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024
[14]

Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure & Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046
[15]

Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857
[16]

John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149
[17]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312
[18]

Mostafa Abounouh, Hassan Al-Moatassime, Sabah Kaouri. Non-standard boundary conditions for the linearized Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021066
[19]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353
[20]

Netra Khanal, Ramjee Sharma, Jiahong Wu, Juan-Ming Yuan. A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations. Conference Publications, 2009, 2009 (Special) : 442-450. doi: 10.3934/proc.2009.2009.442

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences