Global well-posedness for the generalized derivative nonlinear Schrödinger equation
Ben Pineau, Mitchell A. Taylor
TL;DR
The paper proves global well-posedness for the generalized derivative NLS $iu_t+u_{xx}=i|u|^{2\sigma}u_x$ in low to high Sobolev spaces, focusing on the challenging regime $\sigma<1$. The authors introduce a dyadic paradifferential gauge transformation and a dyadic function space framework $(X^s_T, Y^s_T)$ with frequency envelopes to overcome derivative loss and rough nonlinearities, establishing $H^s$ well-posedness for $\sigma\in(\sqrt{3}/2,1)$ and $s\in[1,4\sigma)$, and high-regularity results up to $s<4\sigma$ for all $\sigma\in(\frac12,1)$. A key innovation is combining time-differentiated energy estimates with modulation analysis to balance the nonlinearity’s Hölder regularity, enabling global control via energy and mass conservation. The results improve previously known local results to global well-posedness in broad $H^s$ ranges and lay groundwork for applying the developed techniques to other dispersive quasilinear equations with rough nonlinearities. Overall, the work demonstrates robust low and high regularity theories for a quasilinear dispersive model lacking strong decay and requiring delicate para-differential tools.
Abstract
We study the well-posedness of the generalized derivative nonlinear Schrödinger equation (gDNLS) $$iu_t+u_{xx}=i|u|^{2σ}u_x,$$ for small powers $σ$. We analyze this equation at both low and high regularity, and are able to establish global well-posedness in $H^s$ when $s\in [1,4σ)$ and $σ\in (\frac{\sqrt{3}}{2},1)$. Our result when $s=1$ is particularly relevant because it corresponds to the regularity of the energy for this problem. To our knowledge, this is the first low regularity well-posedness result for a quasilinear dispersive model where the nonlinearity is both rough and lacks the decay necessary for global smoothing type estimates. These two features pose considerable difficulty when trying to apply standard tools for closing low-regularity estimates. While the tools developed in this article are used to study gDNLS, we believe that they should be applicable in the study of local well-posedness for other dispersive equations of a similar character. It should also be noted that the high regularity well-posedness presents a novel issue, as the roughness of the nonlinearity limits the potential regularity of solutions. Our high regularity well-posedness threshold $s<4σ$ is twice as high as one might naïvely expect, given that the function $z\mapsto |z|^{2σ}$ is only $C^{1,2σ-1}$ Hölder continuous. Moreover, although we cannot prove $H^1$ well-posedness when $σ\leq \frac{\sqrt{3}}{2}$, we are able to establish $H^s$ well-posedness in the high regularity regime $s\in (2-σ,4σ)$ for the full range of $σ\in (\frac{1}{2},1)$. This considerably improves the known local results, which had only been established in either $H^2$ or in weighted Sobolev spaces.