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

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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

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

Abstract
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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. 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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