American Institute of Mathematical Sciences

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March  2012, 11(2): 709-733. doi: 10.3934/cpaa.2012.11.709

Rough solutions for the periodic Korteweg--de~Vries equation

Massimiliano Gubinelli 1,

1. 

CEREMADE & CNRS (UMR 7534), Université Paris Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris cedex 16, France

Received  March 2010 Revised  April 2011 Published  October 2011

We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg--de Vries (KdV) equation on a periodic domain and with initial condition in $F L^{\alpha,p}$ spaces. We discuss convergence of Galerkin approximations, a modified Euler scheme and the presence of a random force of white-noise type in time.
Keywords: power series solutions., Dispersive equations, rough paths theory.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35D9.
Citation: Massimiliano Gubinelli. Rough solutions for the periodic Korteweg--de~Vries equation. Communications on Pure & Applied Analysis, 2012, 11 (2) : 709-733. doi: 10.3934/cpaa.2012.11.709
References:
[1]

H. Bessaih, M. Gubinelli and F. Russo, The evolution of a random vortex filament, Ann. Probab., 33 (2005), 1825-1855. doi: 10.1214/009117905000000323.  Google Scholar

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.  Google Scholar

[3]

M. Christ, Power series solution of a nonlinear Schrödinger equation, In "Mathematical Aspects of Nonlinear Dispersive Equations," volume 163 of Ann. of Math. Stud., pages 131-155. Princeton Univ. Press, Princeton, NJ, 2007. doi: 10.1353/ajm.2003.0040.  Google Scholar

[4]

M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293. doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$, J. Amer. Math. Soc., 16 (2003), 705-749 (electronic). doi: 10.1007/BF02588080.  Google Scholar

[6]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Symplectic nonsqueezing of the Korteweg-de Vries flow, Acta Math., 195 (2005), 197-252. doi: 10.1007/s004400100158.  Google Scholar

[7]

L. Coutin and A. Lejay, Semi-martingales and rough paths theory, Electron. J. Probab., 10 (2005), 761-785 (electronic).  Google Scholar

[8]

L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields, 122 (2002), 108-140. doi: 10.1007/s004400100158.  Google Scholar

[9]

A. M. Davie, Differential equations driven by rough signals: an approach via discrete approximation,, Appl. Math. Res. Express. AMRX, 2 ().  doi: 10.1093/amrx/abm009.  Google Scholar

[10]

A. De Bouard, A. Debussche and Y. Tsutsumi, Periodic solutions of the Korteweg-de Vries equation driven by white noise,, SIAM J. Math. Anal., 36 (): 815.  doi: 10.1137/S0036141003425301.  Google Scholar

[11]

P. Friz and N. Victoir, A note on the notion of geometric rough paths, Probab. Theory Related Fields, 136 (2006), 395-416. doi: 10.1007/s00440-005-0487-7.  Google Scholar

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P. Friz and N. Victoir, Euler estimates for rough differential equations, J. Differential Equations, 244 (2008), 388-412. doi: 10.1016/j.jde.2007.10.008.  Google Scholar

[13]

P. Friz and N. Victoir, On uniformly subelliptic operators and stochastic area, Probab. Theory Related Fields, 142 (2008), 475-523. doi: 10.1007/s00440-007-0113-y.  Google Scholar

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P. Friz and N. Victoir, Multidimensional stochastic processes as rough paths. Theory and applications, Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010. xiv+656 pp. ISBN: 978-0-521-87607-0  Google Scholar

[15]

J. G. Gaines and T. J. Lyons, Variable step size control in the numerical solution of stochastic differential equations, SIAM J. Appl. Math., 57 (1997), 1455-1484. doi: 10.1137/S0036139995286515.  Google Scholar

[16]

G. Gallavotti, "Foundations of Fluid Dynamics," Texts and Monographs in Physics. Springer-Verlag, Berlin, 2002. Translated from the Italian.  Google Scholar

[17]

J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal., 20 (1989), 1388-1425. doi: 10.1137/0520091.  Google Scholar

[18]

J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Astérisque, (237): Exp. No. 796, 4, 163-187, 1996. Séminaire Bourbaki, Vol. 1994/95.  Google Scholar

[19]

M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140. doi: 10.1016/j.jfa.2004.01.002.  Google Scholar

[20]

M. Gubinelli, Rooted trees for 3D Navier-Stokes equation, Dyn. Partial Differ. Equ., 3 (2006), 161-172.  Google Scholar

[21]

M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs, Potential Anal., 25 (2006), 307-326. doi: 10.1007/s11118-006-9013-5.  Google Scholar

[22]

M. Gubinelli, Ramification of rough paths, J. Differential Equations, 248 (2010), 693-721. doi: 10.1016/j.jde.2009.11.015.  Google Scholar

[23]

M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75. Google Scholar

[24]

Z. Guo, Global well-posedness of korteweg-de vries equation in $H^{-3/4}(R)$, Journal de Mathématiques Pures et Appliqués, 91 (2009), 583-597. doi: 10.1016/j.matpur.2009.01.012.  Google Scholar

[25]

T. Kappeler and P. Topalov, Well-posedness of KdV on $H^{-1}(T)$, In Mathematisches Institut, Georg-August-Universität Göttingen: Seminars 2003/2004, pages 151-155. Universitätsdrucke Göttingen, Göttingen, 2004. doi: 10.1215/S0012-7094-06-13524-X.  Google Scholar

[26]

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

[27]

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

[28]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993) , 1-21. doi: 10.1215/S0012-7094-93-07101-3.  Google Scholar

[29]

C. E. Kenig, G. Ponce and L. 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

[30]

Y. Le Jan and A. S. Sznitman, Stochastic cascades and $3$-dimensional Navier-Stokes equations, Probab. Theory Related Fields, 109 (1997), 343-366. doi: 10.1007/s004400050135.  Google Scholar

[31]

T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310.  Google Scholar

[32]

T. Lyons and Zhongmin Qian, "System Control and Rough Paths," Oxford Mathematical Monographs. Oxford University Press, Oxford, 2002. Oxford Science Publications.  Google Scholar

[33]

T. Nguyen, Power series solution for the modified KdV equation, Electron. J. Differential Equations, 71 (2008), 10 pages.  Google Scholar

[34]

Ya. G. Sinaĭ, A diagrammatic approach to the 3D Navier-Stokes system, Uspekhi Mat. Nauk, 60 (2005), 47-70. doi: 10.1070/RM2005v060n05ABEH003735.  Google Scholar

[35]

Ya. G. Sinaĭ, Power series for solutions of the $3D$-Navier-Stokes system on $R^3$, J. Stat. Phys., 121 (2005), 779-803. doi: 10.1007/s10955-005-8670-x.  Google Scholar

show all references

References:
[1]

H. Bessaih, M. Gubinelli and F. Russo, The evolution of a random vortex filament, Ann. Probab., 33 (2005), 1825-1855. doi: 10.1214/009117905000000323.  Google Scholar

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.  Google Scholar

[3]

M. Christ, Power series solution of a nonlinear Schrödinger equation, In "Mathematical Aspects of Nonlinear Dispersive Equations," volume 163 of Ann. of Math. Stud., pages 131-155. Princeton Univ. Press, Princeton, NJ, 2007. doi: 10.1353/ajm.2003.0040.  Google Scholar

[4]

M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293. doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbbR$ and $\mathbbT$, J. Amer. Math. Soc., 16 (2003), 705-749 (electronic). doi: 10.1007/BF02588080.  Google Scholar

[6]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Symplectic nonsqueezing of the Korteweg-de Vries flow, Acta Math., 195 (2005), 197-252. doi: 10.1007/s004400100158.  Google Scholar

[7]

L. Coutin and A. Lejay, Semi-martingales and rough paths theory, Electron. J. Probab., 10 (2005), 761-785 (electronic).  Google Scholar

[8]

L. Coutin and Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields, 122 (2002), 108-140. doi: 10.1007/s004400100158.  Google Scholar

[9]

A. M. Davie, Differential equations driven by rough signals: an approach via discrete approximation,, Appl. Math. Res. Express. AMRX, 2 ().  doi: 10.1093/amrx/abm009.  Google Scholar

[10]

A. De Bouard, A. Debussche and Y. Tsutsumi, Periodic solutions of the Korteweg-de Vries equation driven by white noise,, SIAM J. Math. Anal., 36 (): 815.  doi: 10.1137/S0036141003425301.  Google Scholar

[11]

P. Friz and N. Victoir, A note on the notion of geometric rough paths, Probab. Theory Related Fields, 136 (2006), 395-416. doi: 10.1007/s00440-005-0487-7.  Google Scholar

[12]

P. Friz and N. Victoir, Euler estimates for rough differential equations, J. Differential Equations, 244 (2008), 388-412. doi: 10.1016/j.jde.2007.10.008.  Google Scholar

[13]

P. Friz and N. Victoir, On uniformly subelliptic operators and stochastic area, Probab. Theory Related Fields, 142 (2008), 475-523. doi: 10.1007/s00440-007-0113-y.  Google Scholar

[14]

P. Friz and N. Victoir, Multidimensional stochastic processes as rough paths. Theory and applications, Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010. xiv+656 pp. ISBN: 978-0-521-87607-0  Google Scholar

[15]

J. G. Gaines and T. J. Lyons, Variable step size control in the numerical solution of stochastic differential equations, SIAM J. Appl. Math., 57 (1997), 1455-1484. doi: 10.1137/S0036139995286515.  Google Scholar

[16]

G. Gallavotti, "Foundations of Fluid Dynamics," Texts and Monographs in Physics. Springer-Verlag, Berlin, 2002. Translated from the Italian.  Google Scholar

[17]

J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal., 20 (1989), 1388-1425. doi: 10.1137/0520091.  Google Scholar

[18]

J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Astérisque, (237): Exp. No. 796, 4, 163-187, 1996. Séminaire Bourbaki, Vol. 1994/95.  Google Scholar

[19]

M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140. doi: 10.1016/j.jfa.2004.01.002.  Google Scholar

[20]

M. Gubinelli, Rooted trees for 3D Navier-Stokes equation, Dyn. Partial Differ. Equ., 3 (2006), 161-172.  Google Scholar

[21]

M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs, Potential Anal., 25 (2006), 307-326. doi: 10.1007/s11118-006-9013-5.  Google Scholar

[22]

M. Gubinelli, Ramification of rough paths, J. Differential Equations, 248 (2010), 693-721. doi: 10.1016/j.jde.2009.11.015.  Google Scholar

[23]

M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75. Google Scholar

[24]

Z. Guo, Global well-posedness of korteweg-de vries equation in $H^{-3/4}(R)$, Journal de Mathématiques Pures et Appliqués, 91 (2009), 583-597. doi: 10.1016/j.matpur.2009.01.012.  Google Scholar

[25]

T. Kappeler and P. Topalov, Well-posedness of KdV on $H^{-1}(T)$, In Mathematisches Institut, Georg-August-Universität Göttingen: Seminars 2003/2004, pages 151-155. Universitätsdrucke Göttingen, Göttingen, 2004. doi: 10.1215/S0012-7094-06-13524-X.  Google Scholar

[26]

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

[27]

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

[28]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993) , 1-21. doi: 10.1215/S0012-7094-93-07101-3.  Google Scholar

[29]

C. E. Kenig, G. Ponce and L. 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

[30]

Y. Le Jan and A. S. Sznitman, Stochastic cascades and $3$-dimensional Navier-Stokes equations, Probab. Theory Related Fields, 109 (1997), 343-366. doi: 10.1007/s004400050135.  Google Scholar

[31]

T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310.  Google Scholar

[32]

T. Lyons and Zhongmin Qian, "System Control and Rough Paths," Oxford Mathematical Monographs. Oxford University Press, Oxford, 2002. Oxford Science Publications.  Google Scholar

[33]

T. Nguyen, Power series solution for the modified KdV equation, Electron. J. Differential Equations, 71 (2008), 10 pages.  Google Scholar

[34]

Ya. G. Sinaĭ, A diagrammatic approach to the 3D Navier-Stokes system, Uspekhi Mat. Nauk, 60 (2005), 47-70. doi: 10.1070/RM2005v060n05ABEH003735.  Google Scholar

[35]

Ya. G. Sinaĭ, Power series for solutions of the $3D$-Navier-Stokes system on $R^3$, J. Stat. Phys., 121 (2005), 779-803. doi: 10.1007/s10955-005-8670-x.  Google Scholar

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