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Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes

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  • Motivated by the work of Grujić and Kalisch, [Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential and Integral Equations 15 (2002) 1325--1334], we prove the local well-posedness for the periodic KdV equation in spaces of periodic functions analytic on a strip around the real axis without shrinking the width of the strip in time.
    Mathematics Subject Classification: Primary: 35Q53; Secondary: 35A07.

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  • [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 $R$ 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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