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Periodic Korteweg-de Vries equation with measures as initial data. (English) Zbl 0891.35138
Summary: The main result of the paper is that the periodic KdV equation \(y_t+ \partial^3_xy+ yy_x=0\) has a unique global solution for initial data \(y(0)\) given by a measure \(\mu\in M(\mathbb{T})\) of sufficiently small norm \(|\mu|\). There are two main ingredients in the proof. The first is the study of the local well-posedness problem in terms of the space-time Fourier-norms. At the end the estimates eventually depend on a uniform estimate in terms of the Fourier coefficients \[ \sup_{n\in\mathbb{Z},t\in\mathbb{R}} |\widehat y(n)(t)|< C. \] Such a priori bound (in the space of pseudo-measures) on the solution may be derived from spectral theory and more precisely from the preservation of the periodic spectrum of a potential evolving according to KdV, which is the second ingredient. Thus the result at this stage depends heavily on integrability features of this particular equation. We also sketch an argument establishing almost periodicity properties of these solutions.
This work is in spirit closely related to [J. Bourgain, Geom. Funct. Anal. 3, No. 2, 107-156 (1993; Zbl 0787.35097)]. Natural questions suggested by these investigations are an extension of the result (at least for the IVP local in time) to a more general nonintegrable setting as well as to what extent the estimates on Fourier coefficients by spectral invariants and vice versa remain valid in distributional spaces.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35P05 General topics in linear spectral theory for PDEs
35B45 A priori estimates in context of PDEs
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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