Nonuniqueness of weak solutions of the nonlinear Schroedinger equation
Michael Christ
TL;DR
The paper shows nonuniqueness of generalized/weak solutions to the 1D periodic nonlinear Schrödinger equation for certain nonlinearities, proving that for any $s<0$ there exist nonzero generalized solutions varying continuously in the Sobolev space $H^s$ with identically vanishing initial data. It develops a modified Cauchy problem with a nonlinear term $N(u) = (|u|^2 - 2 \mu(|u|^2))u$ (where $\mu(f) = (2\pi)^{-1}\int_T f\,dx$) and uses Fourier-space truncations and driven high-frequency content to generate a reverse energy cascade, yielding nonuniqueness of weak solutions even from zero initial data and defining extended-sense weak solutions via truncation limits. The analysis rewrites the problem in Fourier space as an infinite system of ODEs for the Fourier coefficients, introduces the phase-rotated variable $y_n(t)=e^{in^2 t}\widehat{u}(t,n)$, and constructs a corrected evolution $y=x+h$ with a carefully designed high-frequency perturbation $h$ vanishing to infinite order at $t=0$, ensuring a small remainder and controlled support. The results extend to spaces $\mathcal{H}^p$ and to quadratic nonlinearities, illustrating a robust nonuniqueness phenomenon, and point to possible extensions to other PDEs such as KdV and Navier–Stokes, while highlighting interpretations of nonlinear terms for rough data.
Abstract
Generalized solutions of the Cauchy problem for the one-dimensional periodic nonlinear Schr\"odinger equation, with certain nonlinearities, are not unique. For any $s<0$ there exist nonzero generalized solutions varying continuously in the Sobolev space $H^s$, with identically vanishing initial data.