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Optimal Local Well-Posedness for the Periodic Derivative Nonlinear Schrödinger Equation

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Abstract

We prove local well-posedness for the periodic derivative nonlinear Schrödinger equation, which is \(L^2\) critical, in Fourier-Lebesgue spaces which scale like \({H}^s({\mathbb {T}})\) for \(s>0\). Our result is optimal in the sense that it covers the full subcritical regime. In particular we close the existing gap in the subcritical theory by improving the result of Grünrock and Herr (SIAM J Math Anal 39(6):1890–1920, 2008), which established local well-posedness in Fourier-Lebesgue spaces which scale like \({H}^s({\mathbb {T}})\) for \(s >\frac{1}{4}\). We achieve this result by a delicate analysis of the structure of the solution and the construction of an adapted nonlinear submanifold of a suitable function space. Together these allow us to construct the unique solution to the given subcritical data. This constructive procedure is inspired by the theory of para-controlled distributions developed by Gubinelli et al. (Forum Math Pi, 3:75, 2015) and Catellier and Chouk (Ann Probab 46(5):2621–2679, 2018) in the context of stochastic PDE. Our proof and results however, are purely deterministic.

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Notes

  1. Here and henceforth we mean that the homogenous part of the Fourier-Lebesgue norm scales like the corresponding homogeneous Sobolev norm.

  2. More precisely the trilinear estimates containing as one of their inputs the derivative term.

  3. The operator \((i\partial _t + \partial ^2_x)^{-1}\) is just the Duhamel operator I as in (2.14). For technical reasons, here we in fact use the truncated version of the Duhamel operator \({\mathcal {I}}\) as in (2.16).

  4. Note that for data with finite mass, Eq. (1.1) is in fact gauged equivalent to (1.5).

  5. Local well-posedness for the gauged equation (1.5) implies local existence, uniqueness and continuity of the flow map for (1.1) [27, 40].

  6. When \(p=q=2\) these spaces coincide with Bourgain’s Fourier restriction norm spaces associated to the Schrödinger equation, and are simply denoted by \(X^{s, b}\).

  7. In principle, it might be possible that a trilinear estimate holds in some exotic Banach space not of form \(X_{p,q}^{s,b}\) but, if not unlikely, this would at least require a rather sophisticated construction.

  8. See also, later work by Burq and Tzvetkov in the context of nonlinear wave equations [9].

  9. More precisely, here the solution v is supposed to contain three parts: w, a term paracontrolled by \(\partial _x\overline{w}\) and a term paracontrolled by \(\partial _x\overline{w} \, w\) as in (1.13).

  10. Note that the decomposition of v is nonlinear both in w and in the para-controlled terms.

  11. More precisely for \(\delta >0\) and \(q = \frac{1}{4\delta }\), the right space is \(X^{\frac{1}{2}, 1-2 \delta }_{p, q}\). See Sect. 3 for precise definitions.

  12. Here “high” and “low” are with respect to the frequencies \(k_2\) and \(k_3\).

  13. Note that in [27] this same space is denoted by \(\widehat{H}^s_{p^{\prime }}({\mathbb {T}})\) where \(\frac{1}{p}+ \frac{1}{p^{\prime }} = 1\).

  14. We note that a weaker version of Lemma 3.4 here already appeared in [27], Lemma 3.1.

  15. The difference between \(E_L\) and \(E_L^Y\)—same for \(E_N\)—which we call \(E^X_L\) as defined in (4.4) will appear in the Eq. (4.16) for w.

  16. The divisor bound applies when \(\Delta \ne 0\); however when \(\Delta =0\) we must have \(k=k_1=k_2=k_3\) by the definition of \({\mathbb {V}}_3\), so the bound is still true.

  17. Here we mean pairings as defined in Lemma 3.4.

  18. This kernel depends on \(k_j\) and \(\lambda _j\), but we will write it as \(R(\lambda ,\sigma )\) for simplicity.

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Yu Deng & Haitian Yue

  2. Department of Mathematics, University of Massachusetts, 710 N. Pleasant Street, Amherst, MA, 01003, USA

    Andrea R. Nahmod

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  1. Yu Deng
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Andrea R. Nahmod is partially supported by NSF-DMS-1463714 and NSF-DMS-1800852.

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Deng, Y., Nahmod, A.R. & Yue, H. Optimal Local Well-Posedness for the Periodic Derivative Nonlinear Schrödinger Equation. Commun. Math. Phys. 384, 1061–1107 (2021). https://doi.org/10.1007/s00220-020-03898-8

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