Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schr\"odinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities. In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces,in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the $L^2$ topology, despite the existence of global weak solutions at this regularity. Finally, we investigate the KdV equation at the endpoint regularity $H^{-3/4}$, and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval $[-T,T]$ and a locally Lipschitz continuous map taking the initial data in $H^{-3/4}$ to a distributional solution $u \in C^0 ([-T,T];$H^{-3/4})$ which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV.Comment: minor edit

A remark on normal forms and the "upside-down" I-method for periodic NLS: growth of higher Sobolev norms

We study growth of higher Sobolev norms of solutions to the one-dimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish \|u(t)\|_{H^s} \lesssim (1+|t|)^{\alpha (s-1)+} with \alpha = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with \alpha = 1/2 via the space-time estimate due to Bourgain [4], [5]. In the cubic case, we concretely compute the terms arising in the first few steps of the normal form reduction and prove the above estimate with \alpha = 4/9. These results improve the previously known results (except for the quintic case.) In Appendix, we also show how Bourgain's idea in [4] on the normal form reduction for the quintic nonlinearity can be applied to other powers.Comment: 24 pages. Small modification in Section 1, to appear in J. Anal. Mat