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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. GAFA 3 (1993), no. 2, 107–156.
C. Kenig, G. Ponce and L. Vega. Wellposedness and scattering results for the generalized Korteweg-de Vries equation via the extraction principle. Comm. Pure Appl. Math. 46 (1993), 527–560.
C. Kenig, G. Ponce and L. Vega. A bilinear estimate with application to the KdV equation. J. of the AMS 9 (1996), 573–603.
H. McKean and E. Trubowitz. Hill’s operator and hyperelliptic function theory in the presence of infinitely many branchpoints. Comm. Pure Appl. Math. 29 (1976), 143–226.
E. Trubowitz. The inverse problem for periodic potentials. Comm. Pure Appl. Math. 30 (1977), 325–341.
S. Kuksin. Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDE’s. Comm. Math. Phys. 167 (1995), 531–552.
J. Pöschel, E. Trubowitz. Inverse Spectral Theory. Academic Press, 1987.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bourgain, J. Periodic Korteweg de Vries equation with measures as initial data. Sel. math., New ser. 3, 115–159 (1997). https://doi.org/10.1007/s000290050008
Published:
Issue Date:
DOI: https://doi.org/10.1007/s000290050008