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The Korteweg-de Vries equation on the half-line with Robin and Neumann data in low regularity spaces. (English) Zbl 1492.35265

Summary: The well-posedness of the initial-boundary value problem (ibvp) for the Korteweg-de Vries equation on the half-line is studied for initial data \(u_0 (x)\) in spatial Sobolev spaces \(H^s (0, \infty)\), \(s > - 3 / 4\), and Robin and Neumann boundary data \(\varphi (t)\) in the temporal Sobolev spaces suggested by the time regularity of the Cauchy problem for the corresponding linear equation. First, linear estimates in Bourgain spaces are derived by utilizing the Fokas solution formula of the ibvp for the forced linear equation. Then, using these and the needed bilinear estimates, it is shown that the iteration map defined by the Fokas solution formula is a contraction in an appropriate solution space.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35G31 Initial-boundary value problems for nonlinear higher-order PDEs
35G16 Initial-boundary value problems for linear higher-order PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35B65 Smoothness and regularity of solutions to PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
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