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Gain of regularity for equations of KdV type. (English) Zbl 0764.35021

Les auteurs montrent des résultats de régularité pour l’équation de Korteweg-de Vries nonlinéaire: \(\partial_ +u-f(\partial_ x^ 3 u,\partial_ x^ 2 u,\partial_ x u,u,x,t)=0\), \(f\) étant \(C^ \infty\). Si \(f\) vérifie \({\partial f\over\partial y_ 3}\geq c>0\) et \({\partial u \over \partial y_ 2}\leq 0\) (\(f(y_ 3,y_ 2,y,y_ 0,x,t)\)) ainsi que des propriètés sur elle et ses dérivées d’être bornée dans de bonnes-conditions, si, \(u\) vérifie une certaine régularité celle-ci peut-être améliorée (Théorèmes 2.1 et 2.2). Ils établissent également un théorème d’existence et d’unicité du problème de Cauchy associé (Théorème 3.2).

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations

Keywords:

regularity
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References:

[1] Cohen, A., Solutions of the Korteweg-de Vries Equation from Irregular Data, Duke Math. J., Vol. 45, 149-181 (1978) · Zbl 0372.35022
[2] Constantin, P.; Saut, J. C., Local Smoothing Properties of Dispersive Equations, J. A.M.S., Vol. 1, 413-439 (1988) · Zbl 0667.35061
[3] Craig, W.; Goodman, J., Linear Dispersive Equations of Airy Type, J. Diff. Equ., Vol. 87, 38-61 (1990) · Zbl 0709.35090
[4] Craig, W.; Kappeler, T.; Strauss, W., Infinite Gain of Regularity for Dispersive Evolution Equations, Microlocal Analysis and Nonlinear Waves, Vol. 30, 47-50 (1991), Springer, I.M.A. · Zbl 0767.35076
[6] Hayashi, N.; Ozawa, T., Smoothing Effect for Some Schrödinger Equations, J. of Funct. Anal., Vol. 85, 307-348 (1989) · Zbl 0681.35079
[7] Hayashi, N.; Nakamitsu, K.; Tsutsumi, M., On Solutions of the Initial Value Problem for the Nonlinear Schrödinger Equation in One Space Dimension, Math. Z., Vol. 192, 637-650 (1986) · Zbl 0617.35025
[8] Hayashi, N.; Nakamitsu, K.; Tsutsumi, M., On Solutions of the Initial Value Problem for the Nonlinear Schrödinger Equation, J. Funct. Anal., Vol. 71, 218-245 (1987) · Zbl 0657.35033
[9] Kato, T., On the Cauchy Problem for the (Generalized) Korteweg-de Vries Equation, Adv. in Math. Suppl. Studies; Studies in Appl. Math., Vol. 8, 93-128 (1983)
[10] Kruzhkov, S. N.; Faminskii, A. V., Generalized Solutions to the Cauchy Problem for the Korteweg-de Vries Equation, Math. U.S.S.R. Sbornik, vol. 48, 93-138 (1984)
[11] Ponce, G., Regularity of Solutions to Nonlinear Dispersive Equations, Diff Equ., Vol. 78, 122-135 (1989) · Zbl 0699.35036
[12] Sjölin, P., Regularity of Solutions to the Schrödinger Equation, Duke Math. J., Vol. 55, 699-715 (1987) · Zbl 0631.42010
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