Global well-posedness for the generalized derivative nonlinear Schrödinger equation

The HADES seminar on Tuesday, October 26th, will be given by Benjamin Pineau at 5 pm in 740 Evans.

Speaker: Benjamin Pineau

Abstract: In this talk, we study the well-posedness of the generalized derivative nonlinear Schrödinger equation (gDNLS)
for small powers $\sigma$. We analyze this equation at both low and high regularity, and are able to establish global well-posedness in $H^s$ when $s\in [1,4\sigma)$ and $\sigma \in (\frac{\sqrt{3}}{2},1)$. Our result when $s=1$ is particularly relevant because it corresponds to the regularity of the energy for this problem. Moreover, a theorem of Liu, Simpson and Sulem (~2013) establishes the orbital stability of the gDNLS solitons, provided that there is a suitable $H^1$ well-posedness theory.

To our knowledge, this is the first low regularity well-posedness result for a quasilinear dispersive model where the nonlinearity is both rough and is of lower than cubic order. These two features pose considerable difficulty when trying to apply standard tools for closing low-regularity estimates. While the tools we developed 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 $s<4\sigma$ is twice as high as one might naively expect, given that the function $z\mapsto |z|^{2\sigma}$ is only $C^{1,2\sigma-1}$ Hölder continuous. Moreover, although we cannot prove $H^1$ well-posedness when $\sigma\leq \frac{\sqrt{3}}{2}$, we are able to establish $H^s$ well-posedness in the high regularity regime $s\in (2-\sigma,4\sigma)$ for the full range of $\sigma\in (\frac{1}{2},1)$. This considerably improves the known local results, which had only been established in either $H^2$ or in weighted Sobolev spaces. This is joint work with Mitchell Taylor.

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