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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:

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