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

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

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.
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. doi: 10.1214/009117905000000323.

[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. doi: 10.1007/BF01895688.

[3]

M. Christ, Power series solution of a nonlinear Schrödinger equation,, In, (2007), 131. doi: 10.1353/ajm.2003.0040.

[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. doi: 10.1090/S0894-0347-03-00421-1.

[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. doi: 10.1007/BF02588080.

[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. doi: 10.1007/s004400100158.

[7]

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

[8]

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

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

P. Friz and N. Victoir, Multidimensional stochastic processes as rough paths. Theory and applications,, Cambridge Studies in Advanced Mathematics, (2010), 978.

[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. doi: 10.1137/S0036139995286515.

[16]

G. Gallavotti, "Foundations of Fluid Dynamics,", Texts and Monographs in Physics. Springer-Verlag, (2002).

[17]

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

[18]

J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain),, Ast\'erisque, (1996), 163.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

Z. Guo, Global well-posedness of korteweg-de vries equation in $H^{-3/4}(R)$,, Journal de Math{\'e}matiques Pures et Appliqu{\'e}s, 91 (2009), 583. doi: 10.1016/j.matpur.2009.01.012.

[25]

T. Kappeler and P. Topalov, Well-posedness of KdV on $H^{-1}(T)$,, In Mathematisches Institut, (2003), 151. doi: 10.1215/S0012-7094-06-13524-X.

[26]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, In Studies in applied mathematics, (1983), 93.

[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. doi: 10.1090/S0894-0347-1991-1086966-0.

[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. doi: 10.1215/S0012-7094-93-07101-3.

[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. doi: 10.1090/S0894-0347-96-00200-7.

[30]

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

[31]

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

[32]

T. Lyons and Zhongmin Qian, "System Control and Rough Paths,", Oxford Mathematical Monographs. Oxford University Press, (2002).

[33]

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

[34]

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

[35]

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

show all references

References:
[1]

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

[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. doi: 10.1007/BF01895688.

[3]

M. Christ, Power series solution of a nonlinear Schrödinger equation,, In, (2007), 131. doi: 10.1353/ajm.2003.0040.

[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. doi: 10.1090/S0894-0347-03-00421-1.

[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. doi: 10.1007/BF02588080.

[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. doi: 10.1007/s004400100158.

[7]

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

[8]

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

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

P. Friz and N. Victoir, Multidimensional stochastic processes as rough paths. Theory and applications,, Cambridge Studies in Advanced Mathematics, (2010), 978.

[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. doi: 10.1137/S0036139995286515.

[16]

G. Gallavotti, "Foundations of Fluid Dynamics,", Texts and Monographs in Physics. Springer-Verlag, (2002).

[17]

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

[18]

J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain),, Ast\'erisque, (1996), 163.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

Z. Guo, Global well-posedness of korteweg-de vries equation in $H^{-3/4}(R)$,, Journal de Math{\'e}matiques Pures et Appliqu{\'e}s, 91 (2009), 583. doi: 10.1016/j.matpur.2009.01.012.

[25]

T. Kappeler and P. Topalov, Well-posedness of KdV on $H^{-1}(T)$,, In Mathematisches Institut, (2003), 151. doi: 10.1215/S0012-7094-06-13524-X.

[26]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, In Studies in applied mathematics, (1983), 93.

[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. doi: 10.1090/S0894-0347-1991-1086966-0.

[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. doi: 10.1215/S0012-7094-93-07101-3.

[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. doi: 10.1090/S0894-0347-96-00200-7.

[30]

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

[31]

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

[32]

T. Lyons and Zhongmin Qian, "System Control and Rough Paths,", Oxford Mathematical Monographs. Oxford University Press, (2002).

[33]

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

[34]

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

[35]

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

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