2003/11/13 by Michael Christ, James Colliander, J. Colliander +4 · 2 citations
Engineering · Mathematics · #35L70 #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations #advanced mathematical theories #math.AP #msc:35L70 #msc:35Q55
paper · pdf · doi:10.48550/arxiv.math/0311227
11 pages
arxiv created 2003/11/13 · openalex publication_date 2003/11/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the periodic non-linear Schrodinger equations with odd integer power nonlinearities, for initial data which are assumed to be small in some negative order Sobolev space, but which may have large L2 mass. These equations are known to be illposed in Hs for all negative s, in the sense that the solution map fails to be uniformly continuous from Hs to itself, even for short times and small norms. Here we show that these equations are even more unstable. For the cubic equation, the solution map is discontinuous from Hs to even the space of distributions. For the quintic and higher order nonlinearities, there exist pairs of solutions which are uniformly bounded in Hs, are arbitrarily close in any CN norm at time zero, and fail to be close in the distribution topology at an arbitrarily small positive time.