Periodic Korteweg-de Vries equation with measures as initial data.

*(English)*Zbl 0891.35138Summary: 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.

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.) |

##### Keywords:

periodic KdV equation; unique global solution; local well-posedness; a priori bound; periodic spectrum; integrability; almost periodicity
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\textit{J. Bourgain}, Sel. Math., New Ser. 3, No. 2, 115--159 (1997; Zbl 0891.35138)

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##### References:

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