American Institute of Mathematical Sciences

Advanced
May  2018, 23(3): 1177-1198. doi: 10.3934/dcdsb.2018147

Generalized KdV equation subject to a stochastic perturbation

Annie Millet 1,, and  Svetlana Roudenko 2,

1. 

SAMM, EA 4543, Université Paris 1 Panthéon Sorbonne, 90 Rue de Tolbiac, 75634 Paris Cedex, France and LPSM, Universités Paris 6-Paris 7
2. 

The George Washington University, Department of Mathematics, Phillips Hall, 801 H St NW, Washington, DC 20052, USA

* Corresponding author: Annie Millet

Dedication: In the memory of Igor Chueshov

Received  March 2017 Revised  June 2017 Published  February 2018

Fund Project: The second author is supported by NSF CAREER grant # 1151618.

We prove global well-posedness of the subcritical generalized Korteweg-de Vries equation (the mKdV and the gKdV with quartic power of nonlinearity) subject to an additive random perturbation. More precisely, we prove that if the driving noise is a cylindrical Wiener process on $L^2(\mathbb{R})$ and the covariance operator is Hilbert-Schmidt in an appropriate Sobolev space, then the solutions with $H^1(\mathbb{R})$ initial data are globally well-posed in $H^1(\mathbb{R})$. This extends results obtained by A. de Bouard and A. Debussche for the stochastic KdV equation.

Keywords: Generalized Korteweg de Vries (gKdV) equation, initial value problem, well-posedness, stochastic additive noise.
Mathematics Subject Classification: Primary: 60H15, 35R60, 35Q53; Secondary: 35L75, 37K10.
Citation: Annie Millet, Svetlana Roudenko. Generalized KdV equation subject to a stochastic perturbation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1177-1198. doi: 10.3934/dcdsb.2018147
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces Pure and Applied Mathematics Series, 2nd edition, Academic Press, 2003.  Google Scholar

[2]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.  doi: 10.1007/BF02099299.  Google Scholar

[3]

A. de Bouard and A. Debussche, On the Stochastic Korteweg-de Vries Equation, J. Func. Anal., 154 (1998), 215-251.  doi: 10.1006/jfan.1997.3184.  Google Scholar

[4]

A. de Bouard and A. Debussche, The Korteweg-de Vries equation with multiplicative homogeneous noise, in Stochastic Differential Equations: Theory and Applications (eds. P. H. Baxendale and S. V. Lototsky, Interdisciplinary Math. Sciences, World Scientific, 2 (2007), 113-133.  Google Scholar

[5]

A. de Bouard, A. Debussche and Y. Tsutsumi, White noise driven Korteweg-de Vries Equations, J. Func. Anal., 169 (1999), 532-558.  doi: 10.1006/jfan.1999.3484.  Google Scholar

[6]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $ \mathbb{R} $ and $ \mathbb{T} $, J. Amer. Math. Soc., 16 (2003), 705-749.   Google Scholar

[7]

C. S. Gardner, Korteweg-de Vries equation and generalizations Ⅳ: The Korteweg-de Vries equation as a Hamiltonian system, J. Math. Phys., 12 (1971), 1548-1551.  doi: 10.1063/1.1665772.  Google Scholar

[8]

T. Kato, Quasilinear equations of evolution with applications to partial differential equations, Lecture Notes in Math. , Springer Verlag, Berlin, 448 (1975), 27-50.  Google Scholar

[9]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud. , 8 (1983), Academic Press, New York, 93-128.  Google Scholar

[10]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[11]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Communications on Pure and Applied Mathematics, 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[12]

T. Oh, Periodic stochastic Korteweg-de Vries equation with additive space-time noise, Analysis & PDE, 2 (2009), 281-304.  doi: 10.2140/apde.2009.2.281.  Google Scholar

[13]

G. Richards, Well-posedness of the stochastic KdV-Burgers equation, Stochastic Processes and their Applications, 124 (2014), 1627-1647.  doi: 10.1016/j.spa.2013.12.008.  Google Scholar

[14]

R. Temam, Sur un problème non linéaire, J. Math. Pures Appl., 48 (1969), 159-172.   Google Scholar

[15]

P. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics, 1756. Springer-Verlag, Berlin, 2001.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces Pure and Applied Mathematics Series, 2nd edition, Academic Press, 2003.  Google Scholar

[2]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.  doi: 10.1007/BF02099299.  Google Scholar

[3]

A. de Bouard and A. Debussche, On the Stochastic Korteweg-de Vries Equation, J. Func. Anal., 154 (1998), 215-251.  doi: 10.1006/jfan.1997.3184.  Google Scholar

[4]

A. de Bouard and A. Debussche, The Korteweg-de Vries equation with multiplicative homogeneous noise, in Stochastic Differential Equations: Theory and Applications (eds. P. H. Baxendale and S. V. Lototsky, Interdisciplinary Math. Sciences, World Scientific, 2 (2007), 113-133.  Google Scholar

[5]

A. de Bouard, A. Debussche and Y. Tsutsumi, White noise driven Korteweg-de Vries Equations, J. Func. Anal., 169 (1999), 532-558.  doi: 10.1006/jfan.1999.3484.  Google Scholar

[6]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $ \mathbb{R} $ and $ \mathbb{T} $, J. Amer. Math. Soc., 16 (2003), 705-749.   Google Scholar

[7]

C. S. Gardner, Korteweg-de Vries equation and generalizations Ⅳ: The Korteweg-de Vries equation as a Hamiltonian system, J. Math. Phys., 12 (1971), 1548-1551.  doi: 10.1063/1.1665772.  Google Scholar

[8]

T. Kato, Quasilinear equations of evolution with applications to partial differential equations, Lecture Notes in Math. , Springer Verlag, Berlin, 448 (1975), 27-50.  Google Scholar

[9]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud. , 8 (1983), Academic Press, New York, 93-128.  Google Scholar

[10]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[11]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Communications on Pure and Applied Mathematics, 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[12]

T. Oh, Periodic stochastic Korteweg-de Vries equation with additive space-time noise, Analysis & PDE, 2 (2009), 281-304.  doi: 10.2140/apde.2009.2.281.  Google Scholar

[13]

G. Richards, Well-posedness of the stochastic KdV-Burgers equation, Stochastic Processes and their Applications, 124 (2014), 1627-1647.  doi: 10.1016/j.spa.2013.12.008.  Google Scholar

[14]

R. Temam, Sur un problème non linéaire, J. Math. Pures Appl., 48 (1969), 159-172.   Google Scholar

[15]

P. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics, 1756. Springer-Verlag, Berlin, 2001.  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]

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
[4]

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
[5]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[6]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387
[7]

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
[8]

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
[9]

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 - A, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857
[10]

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
[11]

Boling Guo, Zhaohui Huo. The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in $L^2$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 121-136. doi: 10.3934/dcds.2006.16.121
[12]

Belkacem Said-Houari. Long-time behavior of solutions of the generalized Korteweg--de Vries equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 245-252. doi: 10.3934/dcdsb.2016.21.245
[13]

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
[14]

Giuseppe Maria Coclite, Lorenzo di Ruvo. A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation. Networks & Heterogeneous Media, 2016, 11 (2) : 281-300. doi: 10.3934/nhm.2016.11.281
[15]

Hongjun Gao, Chengfeng Sun. Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3053-3073. doi: 10.3934/dcdsb.2016087
[16]

Zhaosheng Feng, Yu Huang. Approximate solution of the Burgers-Korteweg-de Vries equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 429-440. doi: 10.3934/cpaa.2007.6.429
[17]

Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509
[18]

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
[19]

Terence Tao. Two remarks on the generalised Korteweg de-Vries equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 1-14. doi: 10.3934/dcds.2007.18.1
[20]

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

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (111)
  • HTML views (278)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences