%ALaurens, Thierry%BJournal Name: Communications in Mathematical Physics; Journal Volume: 397; Journal Issue: 3; Related Information: CHORUS Timestamp: 2023-01-30 01:12:26
%D2022%ISpringer Science + Business Media
%JJournal Name: Communications in Mathematical Physics; Journal Volume: 397; Journal Issue: 3; Related Information: CHORUS Timestamp: 2023-01-30 01:12:26
%K
%MOSTI ID: 10381012
%PMedium: X
%TGlobal Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics
%XAbstract
Given a suitable solutionV(t, x) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data$$u(0,x) \in V(0,x) + H^{-1}(\mathbb {R})$$$u(0,x)\in V(0,x)+{H}^{-1}\left(R\right)$. Our conditions onVdo include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles$$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$$V(0,x)\in {H}^{5}(R/Z)$satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022.https://doi.org/10.1088/1361-6544/ac37f5) we show that smooth step-like initial data also satisfy our hypotheses. We employ the method of commuting flows introduced in Killip and Vişan (Ann. Math. (2) 190(1):249–305, 2019.https://doi.org/10.4007/annals.2019.190.1.4) where$$V\equiv 0$$$V\equiv 0$. In that setting, it is known that$$H^{-1}(\mathbb {R})$$${H}^{-1}\left(R\right)$is sharp in the class of$$H^s(\mathbb {R})$$${H}^{s}\left(R\right)$spaces.

%0Journal Article