A priori estimates and weak solutions for the derivative nonlinear Schr枚dinger equation on torus below

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• 作者：Hideo Takaoka¹ ; ^{takaoka@math.sci.hokudai.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35Q55 ; secondary ; 42B37
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 January 2016
• 年：2016
• 卷：260
• 期：1
• 页码：818-859
• 全文大小：508 K

文摘

We propose a priori estimates for a weak solution to the derivative nonlinear Schrödinger equation (DNLS) on torus with small L²$L^2$-norm datum in low regularity Sobolev spaces. These estimates allow us to show the existence of solutions in H^s(T)$H^s(\mathbb{T})$ with some s<1/2$s<1/2$ in a relatively weak sense. Furthermore we make some remarks on the error estimates arising from the finite dimensional approximation solutions of DNLS using the Fourier–Lesbesgue type as auxiliary spaces, despite the fact that Nahmod, Oh, Rey-Bellet and Staffilani [12] have already seen them.