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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  May 2018 Early access  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

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