Global well-posedness for the KdV equations on the real line with low regularity forcing terms. (English) Zbl 1115.35115
The author considers the initial value problem for the forced KdV equation
\[
\begin{cases} \partial _t u + \partial _x^3 u + u\partial _x u = f, \quad (x,t) \in \mathbb R\times[0,\infty ), \\ u(x,0) = u_0 (x) \in H^s(\mathbb R), \end{cases}
\]
where \(u(x,t)\) and \(f(x)\) are real valued functions. The time local well-posedness with \(f(x) \in H^\sigma\), \(\sigma \geq - 3\) and the time global well-posedness with \(f = p\delta^{'}(x)\) or\(f(x) \in H^\sigma\), \(\sigma \geq - 3/2\) are studied. The main tools are the Fourier restriction norm method and \(I\)-method.
Reviewer: Temuri A. Jangveladze (Tbilisi)
MSC:
35Q53 | KdV equations (Korteweg-de Vries equations) |
35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
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