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On the blow-up boundary solutions of the Monge -Ampére equation with singular weights
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 |
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. |
[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. |
[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. |
[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. |
[5] |
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 (electronic).
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), 197-252.
doi: 10.1007/s004400100158. |
[7] |
L. Coutin and A. Lejay, Semi-martingales and rough paths theory, Electron. J. Probab., 10 (2005), 761-785 (electronic). |
[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. |
[9] |
A. M. Davie, Differential equations driven by rough signals: an approach via discrete approximation, Appl. Math. Res. Express. AMRX, 2, Art. ID abm009, 40 pp.
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 (2004/05), 815-855 (electronic).
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), 395-416.
doi: 10.1007/s00440-005-0487-7. |
[12] |
P. Friz and N. Victoir, Euler estimates for rough differential equations, J. Differential Equations, 244 (2008), 388-412.
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), 475-523.
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, 120. Cambridge University Press, Cambridge, 2010. xiv+656 pp. ISBN: 978-0-521-87607-0 |
[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. |
[16] |
G. Gallavotti, "Foundations of Fluid Dynamics," Texts and Monographs in Physics. Springer-Verlag, Berlin, 2002. Translated from the Italian. |
[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. |
[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. |
[19] |
M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140.
doi: 10.1016/j.jfa.2004.01.002. |
[20] |
M. Gubinelli, Rooted trees for 3D Navier-Stokes equation, Dyn. Partial Differ. Equ., 3 (2006), 161-172. |
[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. |
[22] |
M. Gubinelli, Ramification of rough paths, J. Differential Equations, 248 (2010), 693-721.
doi: 10.1016/j.jde.2009.11.015. |
[23] |
M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[31] |
T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. |
[32] |
T. Lyons and Zhongmin Qian, "System Control and Rough Paths," Oxford Mathematical Monographs. Oxford University Press, Oxford, 2002. Oxford Science Publications. |
[33] |
T. Nguyen, Power series solution for the modified KdV equation, Electron. J. Differential Equations, 71 (2008), 10 pages. |
[34] |
Ya. G. Sinaĭ, A diagrammatic approach to the 3D Navier-Stokes system, Uspekhi Mat. Nauk, 60 (2005), 47-70.
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), 779-803.
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), 1825-1855.
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), 209-262.
doi: 10.1007/BF01895688. |
[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. |
[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. |
[5] |
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 (electronic).
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), 197-252.
doi: 10.1007/s004400100158. |
[7] |
L. Coutin and A. Lejay, Semi-martingales and rough paths theory, Electron. J. Probab., 10 (2005), 761-785 (electronic). |
[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. |
[9] |
A. M. Davie, Differential equations driven by rough signals: an approach via discrete approximation, Appl. Math. Res. Express. AMRX, 2, Art. ID abm009, 40 pp.
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 (2004/05), 815-855 (electronic).
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), 395-416.
doi: 10.1007/s00440-005-0487-7. |
[12] |
P. Friz and N. Victoir, Euler estimates for rough differential equations, J. Differential Equations, 244 (2008), 388-412.
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), 475-523.
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, 120. Cambridge University Press, Cambridge, 2010. xiv+656 pp. ISBN: 978-0-521-87607-0 |
[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. |
[16] |
G. Gallavotti, "Foundations of Fluid Dynamics," Texts and Monographs in Physics. Springer-Verlag, Berlin, 2002. Translated from the Italian. |
[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. |
[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. |
[19] |
M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140.
doi: 10.1016/j.jfa.2004.01.002. |
[20] |
M. Gubinelli, Rooted trees for 3D Navier-Stokes equation, Dyn. Partial Differ. Equ., 3 (2006), 161-172. |
[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. |
[22] |
M. Gubinelli, Ramification of rough paths, J. Differential Equations, 248 (2010), 693-721.
doi: 10.1016/j.jde.2009.11.015. |
[23] |
M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[31] |
T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. |
[32] |
T. Lyons and Zhongmin Qian, "System Control and Rough Paths," Oxford Mathematical Monographs. Oxford University Press, Oxford, 2002. Oxford Science Publications. |
[33] |
T. Nguyen, Power series solution for the modified KdV equation, Electron. J. Differential Equations, 71 (2008), 10 pages. |
[34] |
Ya. G. Sinaĭ, A diagrammatic approach to the 3D Navier-Stokes system, Uspekhi Mat. Nauk, 60 (2005), 47-70.
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), 779-803.
doi: 10.1007/s10955-005-8670-x. |
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