Periodic Korteweg de Vries equation with measures as initial data

Abstract

The main result of the paper is that the periodic KdV equation $y_t + \partial^3_x y + yy_x = 0$ has a unique global solution for initial data y(0) given by a measure $\mu\in M({\Bbb T})$ of sufficiently small norm $\parallel\mu\parallel$. 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 as exploited in [Bo] and also subsequent work such as [K-P-V₂]. At the end the estimates eventually depend on a uniform estimate in terms of the Fourier coefficients¶¶${{\rm sup}\atop{n\in{\Bbb Z},\,t\in{\Bbb R}}}|\hat{y}(n)(t)| < C .$.¶¶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 [Bo]. Natural questions suggested by these investigations is 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 remains valid in distributional spaces.