Advanced Search
Article Contents
Article Contents

Generalized KdV equation subject to a stochastic perturbation

  • * Corresponding author: Annie Millet

    * Corresponding author: Annie Millet 

Dedication: In the memory of Igor Chueshov

The second author is supported by NSF CAREER grant # 1151618

Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 60H15, 35R60, 35Q53; Secondary: 35L75, 37K10.


    \begin{equation} \\ \end{equation}
  • 加载中
  •   R. A. Adams and J. J. F. Fournier, Sobolev Spaces Pure and Applied Mathematics Series, 2nd edition, Academic Press, 2003.
      J. Bourgain , Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994) , 1-26.  doi: 10.1007/BF02099299.
      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.
      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.
      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.
      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. 
      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.
      T. Kato, Quasilinear equations of evolution with applications to partial differential equations, Lecture Notes in Math. , Springer Verlag, Berlin, 448 (1975), 27-50.
      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.
      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.
      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.
      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.
      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.
      R. Temam , Sur un problème non linéaire, J. Math. Pures Appl., 48 (1969) , 159-172. 
      P. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics, 1756. Springer-Verlag, Berlin, 2001.
  • 加载中

Article Metrics

HTML views(1466) PDF downloads(261) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint