Global wellposedness for KdV below \(L^2\)
Consider the initial value problem for the Korteweg-de Vries equation (KdV) \[ \partial_tu+\partial_x^3u+ {\textstyle\frac 12}\partial_xu^2=0,\quad u(0)=\phi\in H^s({\mathbb R}),\qquad {\textstyle-\frac 13}<s<0, \tag{1} \] where \(\partial_t=\partial/\partial t\), \(\partial_x=\partial/\partial x\), \(\partial_x^3=\partial^3/\partial x^3\) and \(H^s({\mathbb R})\) is the usual Sobolev space. The wellposedness theory shows that, given \(s>-\frac 34\), there exists a unique solution \(u\in C([0,T];H^s({\mathbb R}))\) continuously depending on the initial data \(\phi\) [cf. \textit{J. Bourgain}, Geom. Funct. Anal. 3, 107-156, 209-262 (1993; Zbl 0787.35097 and Zbl 0787.35098), and \textit{C. E. Kenig, G. Ponce} and \textit{L. Vega}, J. Am. Math. 9, No. 2, 573-603 (1996; Zbl 0848.35114)]. The problem (1) is said to be globally wellposed if one can take \(T=\infty\). For \(s\), \(a\in{\mathbb R}\) define \(H^{s,a}({\mathbb R})=\{\varphi:\|\varphi\|_{H^{s,a}({\mathbb R})}<\infty\}\) where \[ \|\varphi\|_{H^{s,a}({\mathbb R})}=\Biggl(\int_{\mathbb R} \Bigl[(1+|k|)^s\chi_{\{|k|\geq 1\}}(k)+|k|^a\chi_{\{|k|<1\}}(k)\Bigr]^2 |\widehat\varphi(k)|^2 dk\Biggr)^{1/2}, \] \(\chi_A(k)\) is the characteristic function of the set \(A\) and \(\widehat\varphi\) is the Fourier transform of \(\varphi\). The main result of the paper is the following: Theorem. The initial value problem (1) with \(\phi\in H^{s,a}({\mathbb R})\) is globally wellposed in \(H^{s,a}\) for \(s\in(s_0(a),0]\) with \(s_0(a)=a/12<0\) for appropriate \(a<0\). Moreover, for \(\phi\in H^{s,a}({\mathbb R})\) with \(s\in(s_0(a),0]\), \(u(t)-e^{-t\partial_x^3}\phi\) belongs to \(L^2({\mathbb R})\) for all \(t\).
scientific article; zbMATH DE number 1489517