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Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system
Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes
1. | Department of Mathematics, University of Bergen, 5008 Bergen, Norway |
References:
[1] |
A. Biswas, Local existence and Gevrey regularity of 3-D Navier-Stokes equations with $l_p$ initial data, J. Differential Equations, 215 (2005), 429-447.
doi: 10.1016/j.jde.2004.12.012. |
[2] |
A. Biswas and D. Swanson, Existence and generalized Gevrey regularity of solutions to the Kuramoto-Sivashinsky equation in $R^n$, J. Differential Equations, 240 (2007), 145-163.
doi: 10.1016/j.jde.2007.05.022. |
[3] |
M. Bjørkavåg and H. Kalisch, Exponential Convergence of a Spectral Projection of the KdV Equation, Physics Letters A, 365 (2007), 278-283.
doi: 10.1016/j.physleta.2006.12.085. |
[4] |
J. L. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV-equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 22 (2005), 783-797.
doi: 10.1016/j.anihpc.2004.12.004. |
[5] |
A. de Bouard, N. Hayashi and K. Kato, Gevrey regularizing effect for the (generalized) Korteweg-de Vries equation and nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 6 (1995), 673-725. |
[6] |
J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Parts II, Geometric Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[7] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}^{N}$ and $T$, J. Amer. Math. Soc., 16 (2003), 705-749.
doi: 10.1090/S0894-0347-03-00421-1. |
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations and applications, J. Functional Anal., 211 (2004), 173-218.
doi: 10.1016/S0022-1236(03)00218-0. |
[9] |
A. B. Ferrari and E. S. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, 23 (1998), 1-16.
doi: 10.1080/03605309808821336. |
[10] |
C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Functional Anal., 87 (1989), 359-369.
doi: 10.1016/0022-1236(89)90015-3. |
[11] |
J. Gorsky and A. A. Himonas, Construction of non-analytic solutions for the generalized KdV equation, J. Math. Anal. Appl., 303 (2005), 522-529.
doi: 10.1016/j.jmaa.2004.08.055. |
[12] |
Z. Grujić, Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation, J. Dynam. Differential Equations, 12 (2000), 217-227.
doi: 10.1023/A:1009002920348. |
[13] |
Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential Integral Equations, 15 (2002), 1325-1334. |
[14] |
N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation, SIAM J. Math. Anal., 22 (1991), 1738-1743.
doi: 10.1137/0522107. |
[15] |
H. Hayashi, Solutions of the (generalized) Korteweg-de Vries equation in the Bergman and Szegö spaces on a sector, Duke Math. J., 62 (1991), 575-591.
doi: 10.1215/S0012-7094-91-06224-1. |
[16] |
H. Kalisch, Rapid convergence of a Galerkin projection of the KdV equation, C. R. Math. Acad. Sci. Paris, 341 (2005), 457-460.
doi: 10.1016/j.crma.2005.09.006. |
[17] |
H. Kalisch and X. Raynaud, On the rate of convergence of a collocation projection of the KdV equation, M2AN Math. Model. Numer. Anal., 41 (2007), 95-110.
doi: 10.1051/m2an:2007010. |
[18] |
T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 3 (1986), 455-467. |
[19] |
C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equations, J. Amer. Math. Soc., 9 (1996), 573-603.
doi: 10.1090/S0894-0347-96-00200-7. |
[20] |
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. |
[21] |
I. Kukavica and V. Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc., 137 (2009), 669-677.
doi: 10.1090/S0002-9939-08-09693-7. |
[22] |
G. Staffilani, On solutions for periodic generalized KdV equations, Internat. Math. Res. Notices, 18 (1997), 899-917.
doi: 10.1155/S1073792897000585. |
[23] |
T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 890-908.
doi: 10.1353/ajm.2001.0035. |
show all references
References:
[1] |
A. Biswas, Local existence and Gevrey regularity of 3-D Navier-Stokes equations with $l_p$ initial data, J. Differential Equations, 215 (2005), 429-447.
doi: 10.1016/j.jde.2004.12.012. |
[2] |
A. Biswas and D. Swanson, Existence and generalized Gevrey regularity of solutions to the Kuramoto-Sivashinsky equation in $R^n$, J. Differential Equations, 240 (2007), 145-163.
doi: 10.1016/j.jde.2007.05.022. |
[3] |
M. Bjørkavåg and H. Kalisch, Exponential Convergence of a Spectral Projection of the KdV Equation, Physics Letters A, 365 (2007), 278-283.
doi: 10.1016/j.physleta.2006.12.085. |
[4] |
J. L. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV-equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 22 (2005), 783-797.
doi: 10.1016/j.anihpc.2004.12.004. |
[5] |
A. de Bouard, N. Hayashi and K. Kato, Gevrey regularizing effect for the (generalized) Korteweg-de Vries equation and nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 6 (1995), 673-725. |
[6] |
J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Parts II, Geometric Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[7] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}^{N}$ and $T$, J. Amer. Math. Soc., 16 (2003), 705-749.
doi: 10.1090/S0894-0347-03-00421-1. |
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations and applications, J. Functional Anal., 211 (2004), 173-218.
doi: 10.1016/S0022-1236(03)00218-0. |
[9] |
A. B. Ferrari and E. S. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, 23 (1998), 1-16.
doi: 10.1080/03605309808821336. |
[10] |
C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Functional Anal., 87 (1989), 359-369.
doi: 10.1016/0022-1236(89)90015-3. |
[11] |
J. Gorsky and A. A. Himonas, Construction of non-analytic solutions for the generalized KdV equation, J. Math. Anal. Appl., 303 (2005), 522-529.
doi: 10.1016/j.jmaa.2004.08.055. |
[12] |
Z. Grujić, Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation, J. Dynam. Differential Equations, 12 (2000), 217-227.
doi: 10.1023/A:1009002920348. |
[13] |
Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential Integral Equations, 15 (2002), 1325-1334. |
[14] |
N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation, SIAM J. Math. Anal., 22 (1991), 1738-1743.
doi: 10.1137/0522107. |
[15] |
H. Hayashi, Solutions of the (generalized) Korteweg-de Vries equation in the Bergman and Szegö spaces on a sector, Duke Math. J., 62 (1991), 575-591.
doi: 10.1215/S0012-7094-91-06224-1. |
[16] |
H. Kalisch, Rapid convergence of a Galerkin projection of the KdV equation, C. R. Math. Acad. Sci. Paris, 341 (2005), 457-460.
doi: 10.1016/j.crma.2005.09.006. |
[17] |
H. Kalisch and X. Raynaud, On the rate of convergence of a collocation projection of the KdV equation, M2AN Math. Model. Numer. Anal., 41 (2007), 95-110.
doi: 10.1051/m2an:2007010. |
[18] |
T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 3 (1986), 455-467. |
[19] |
C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equations, J. Amer. Math. Soc., 9 (1996), 573-603.
doi: 10.1090/S0894-0347-96-00200-7. |
[20] |
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. |
[21] |
I. Kukavica and V. Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc., 137 (2009), 669-677.
doi: 10.1090/S0002-9939-08-09693-7. |
[22] |
G. Staffilani, On solutions for periodic generalized KdV equations, Internat. Math. Res. Notices, 18 (1997), 899-917.
doi: 10.1155/S1073792897000585. |
[23] |
T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 890-908.
doi: 10.1353/ajm.2001.0035. |
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