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Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. (English) Zbl 1048.35101
The authors study the continuous dependence of the solution of defocusing equations with respect to the initial data. Existence of global and smooth solutions of such equations is indeed known if the initial data belongs to the Schwarz space of smooth and rapidly decreasing functions. The authors start with the 1D nonlinear Schrödinger equation : \(-iu_{t}+u_{xx}= | u| ^{2}u\) posed in \([ -T,T] \times \mathbb{R}\) with the initial data : \(u( 0,x) =u_{0}( x) \). They prove that this Cauchy problem is not well-posed in \(H_{x}^{s}\) for every \( s<0\). Instead of the known approach dealing with soliton or breather solutions, the authors here introduce the method introduced by T. Ozawa [Commun. Math. Phys. 139, No.3, 479- 493 (1991; Zbl 0742.35043)]. Notice that they have to consider the different cases \(s<-1/2\), \(s=-1/2\) and \(s>-1/2\). Moving to the modified KdV equation : \(u_{t}+u_{xxx}=6u^{2}u_{x}\) they prove that the Cauchy problem is not locally well-posed in \(H_{x}^{s}\) for every \(-1/4<s<1/4\). The proof is essentially based on the use of the spatial Fourier transform, on appropriate changes of scale and on estimates (either Strichartz ones or energy ones). Concerning the KdV equation : \( u_{t}+u_{xxx}=6uu_{x}\), they prove that it is not locally well-posed in \( H_{x}^{s}\) for every \(-1\leq s<-3/4\). The proof here essentially requires Miura’s tranform which links the modified and the non modified KdV equations R. M. Miura [J. Math. Phys. 9, 1202-1204 (1968; Zbl 0283.35018)]. The authors also discuss the case obtained when the problem is posed on \( \mathbb{R}\times \mathbb{T}\), where \(\mathbb{T}\) is the torus.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
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