The unconditional uniqueness for the energy-supercritical NLS

We consider the nonlinear Schrödinger equation (NLS)

where $\Lambda^{d}=\mathbb{R}^{d}$ or $\mathbb{T}^{d}$ and $\pm$ denotes defocusing/focusing. In Euclidean spaces, the NLS $(\ref{equ:NLS})$ enjoys the scaling invariance

which preserves the homogeneous Sobolev norm $\|u_{0}\|_{\dot{H}^{s_{c}}}$ where the critical scaling exponent is given by

Accordingly, the initial value problem $(\ref{equ:NLS})$ for $u_{0}\in\dot{H}^{s_{c}}$ can be classified as energy subcritical, critical or supercritical depending on whether the critical Sobolev exponent $s_{c}$ lies below, equal to or above the energy exponent $s=1$.

In this paper, we focus on the cubic and quintic cases under the energy-supercritical setting $(s_{c}>1)$ where

In the energy-supercritical setting, the global well-posedness of $(\ref{equ:NLS})$ is fully open, away from the classical local well-posedness and $L^{2}$-supercritical blowup results [cazenave1990the, glassey1977on]. But it has been, for a long time, believed that, even under the energy-supercritical setting, the defocusing version of $(\ref{equ:NLS})$ is globally well-posed and the solution scatters when $\Lambda=\mathbb{R}$, just like the energy-critical and subcritical cases, especially after the breakthrough [bourgain1999global, colliander2008global, grillakis2000on, kenig2006global, ryckman2007global]¹¹1See [dodson2019defocusing] for a more detailed survey. on the $\mathbb{R}^{d}$ energy-critical cubic and quintic cases. (See, for example [killip2010energy].) Surprisingly, the recent work [merle2019blow] unexpectedly constructed the first instance of finite time blowup solution for the defocusing energy-supercritical NLS. Thus it is of interest to know if there could exist a scattering global solution in $\dot{H}^{s_{c}}$ but may not be in $C([0,T);\dot{H}^{s_{c}})\bigcap L_{t,x}^{p(d+2)/2}$ when blowups of this type exist.

There are certainly multiple routes for such a problem. But one way is the classical unconditional uniqueness theorem in $\dot{H}^{s_{c}}$ which itself has remained open at least for $\mathbb{T}^{d}$. With an unconditional uniqueness result, we know that there could be at most one solution in $C([0,T);\dot{H}^{s_{c}})$ regardless of auxiliary spaces. One application is to prove that blowup solutions of the type in [merle2019blow] is the only possible $C([0,T);\dot{H}^{s_{c}})$ solution if exist in these domains. In this paper, we prove the $\dot{H}^{s_{c}}$ unconditional uniqueness for $(\ref{equ:NLS})$ as follows and address this issue.

The fundamental concept of unconditional uniqueness was first raised by Kato in [kato1995on, kato1996correction] when proving well-posedness in Strichartz type spaces had made vast progress. In $\mathbb{R}^{d}$, these unconditional uniqueness problems at critical regularity are usually proved by showing any solution must agree with the Strichartz solution, if exists, using the inhomogeneous (retarded) Strichartz estimate. Such a method has been proven to be successful even in the $\mathbb{R}^{3}$ quintic energy-critical case, see for example [colliander2008global]. (This is a very active field, see for example [babin2011on, furioli2003unconditional, guo2013normal, kishimoto2019unconditional, kwon2020normal, molinet2018unconditional, mosincat2020unconditional, zhou1997uniqueness] and the reference within for work on other dispersive equations along this line.)

However, such arguments for the Euclidean setting are no longer effective if $(\ref{equ:NLS})$ is posed on $\mathbb{T}^{d}$, as the Strichartz estimate is rather weak in the periodic case. The $L_{x}^{2}$ Strichartz estimate does not hold in the periodic case and hence the dual Strichartz estimate also fails. On the other hand, the well-posedness on $\mathbb{T}^{d}$ is more intricate, such as using the $X_{s,b}$ space [bourgain1993fourier] and the atomic $U^{p}$ and $V^{p}$ spaces [herr2011global, ionescu2012the]. Thus the unconditional uniqueness problems on $\mathbb{T}^{d}$ under the critical setting are much more difficult to handle. Nevertheless, a unified method has recently unexpectedly arisen from the study of the derivation of (1.1) on the $\mathbb{T}^{d}$ case in [herr2019unconditional] and under the energy-critical setting in [chen2019the, chen2020unconditional].⁴⁴4We mention [herr2019unconditional] 1st here and in the related places in the rest of the paper. Even though [chen2019the] was posted on arXiv one month before [herr2019unconditional], X. Chen and Holmer were not aware of the unconditional uniqueness implication of [chen2019the] until [herr2019unconditional].

We find that one could use the scheme of [chen2020unconditional] to perfectly solve the unconditional uniqueness problem under the energy-supercritical setting for both $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$. The proof comes from the Gross-Pitaevskii(GP) hierarchy, which seems to be weaker than the NLS analysis, as it originates from the derivation of NLS. However, we will see that such an argument is also powerful and worthy for further study. Here, we focus on the quintic GP hierarchy, also see [chen2020unconditional] for the cubic case. The quintic GP hierarchy is a sequence $\left\{\gamma^{(k)}(t)\right\}_{k=1}^{\infty}$ which satisfies the infinitely coupled hierarchy of equations:

where $b_{0}$ is some coupling constant, $\pm$ denotes defocusing/focusing. Given any solution $u$ of $(\ref{equ:NLS})$, it generates a solution to $(\ref{equ:gp hierarchy,quintic})$ by letting

in operator form or

in kernel form where $\mathbf{x}_{k}=(x_{1},...,x_{k})$.

The hierarchy approach was first suggested by Spohn [spohn1980kinetic] for the derivation of NLS from quantum many-body dynamic. Around 2005, it was Erdös, Schlein, and Yau who first rigorously derived the 3D cubic defocusing NLS from a 3D quantum many-body dynamic in their fundamental papers [erdos2006derivation, erdos2007derivation, erdos2007rigorous, erdos2009rigorous, erdos2010derivation]. The proof for the uniqueness of the GP hierarchy was the principal part and also surprisingly dedicate due to the fact that it is a system of infinitely many coupled equations over an unbounded number of variables. With a sophisticated Feynman graph analysis in [erdos2007derivation], they proved the $H^{1}$-type unconditional uniqueness of the $\mathbb{R}^{3}$ cubic GP hierarchy. The first series of ground breaking papers have motivated a large amount of work.

Subsequently in 2007, with imposing an additional a-prior condition on space-time norm, Klainerman and Machedon [klainerman2008on], inspired by [erdos2007derivation, klainerman1993space], gave an another proof of the uniqueness of the GP hierarchy in a different space of density matrices defined by Strichartz type norms. They provided a different combinatorial argument, the now so-called Klainerman-Machedon (KM) board game argument, to combine the inhomogeneous terms effectively reducing their numbers and then derived a space-time estimate to control these terms. At that time, it was open to prove that the limits coming from the $N$-body dynamics satisfy the space-time bound. Nonetheless, [klainerman2008on] has made the delicate analysis of the GP hierarchy approachable from the perspective of PDE. Later, Kirkpatrick, Schlein, and Staffilani [kirkpatrick2011derivation] obtained the KM space-time bound via a simple trace theorem in both $\mathbb{R}^{2}$ and $\mathbb{T}^{2}$ and derived the 2D cubic defocusing NLS from the 2D quantum many-body dynamic. Such a scheme also motivated many works [chen2011the, chen2016collapsing, chen2016focusing, chen2017focusing, gressman2014on, shen2021the, sohinger2016local, xie2015derivation] for the uniqueness of GP hierarchies.

Later in 2008, T. Chen and Pavlović [chen2011the] initiated the study of the quintic GP hierarchy and provided a proof for the quintic KM board game argument, which laid the foundation for the further study of the quintic GP hierarchy. They also showed that the 2D quintic case, which is usually considered the same as the 3D cubic case since they share the same scaling criticality, satisfied the KM space-time bound while it was still open for the 3D cubic case at that time. To attack the problem, they also considered the well-posedness theory with more general data in [chen2010on, chen2013a, chen2014higher]. (See also [chen2010energy, mendelson2019poisson, mendelson2020a, mendelson2019an, sohinger2016local, sohinger2015randomization]). Then in 2011, they proved that the 3D cubic KM space-time bound holds for the defocusing $\beta<1/4$ case in [chen2014derivation]. The result was quickly improved to $\beta<2/7$ by X. Chen in [chen2013on] and then extended to the almost optimal case, $\beta<1$, by X. Chen and Holmer in [chen2016correlation, chen2016on]. Around the same period of time, Gressman, Sohinger, and Staffilani [gressman2014on] studied the uniqueness of the GP hierarchy on $\mathbb{T}^{3}$ and proved that the sharp space-time estimate on $\mathbb{T}^{3}$ needed an additional $\varepsilon$ derivatives than the $\mathbb{R}^{3}$ setting in which one derivative is needed. Later, Herr and Sohinger generalized this fact to more general cases in [herr2016the].

In 2013, by introducing quantum de Finetti theorem from [lewin2014derivation], T. Chen, Hainzl, Pavlović and Seiringer [chen2015unconditional] provided a simplified proof of the $L_{T}^{\infty}H_{x}^{1}$-type 3D cubic uniqueness theorem in [erdos2007derivation]. With the quantum de Finetti theorem, one can replace the space-time estimates by Sobolev multilinear estimates. The scheme in [chen2015unconditional], which consists of the KM board game argument, the quantum de Finetti theorem and the Sobolev multilinear estimates, is robust to deal with such uniqueness problems. Following the scheme in [chen2015unconditional], Sohinger [sohinger2015a] solved the aforementioned $\varepsilon$-loss problem for the defocusing $\mathbb{T}^{3}$ cubic case. In [hong2015unconditional], Hong, Taliaferro, and Xie used the scheme to obtain unconditional uniqueness theorems in $\mathbb{R}^{d}$, $d=1,2,3$, with regularities matching the NLS analysis. Then in [hong2016uniqueness], they proved $H^{1}$ small solution uniqueness for the $\mathbb{R}^{3}$ quintic case. For other refined uniqueness theorems, see also [chen2014on].

The uniqueness analysis of GP hierarchy started to unexpectedly yield new NLS results with regularity lower than the NLS analysis all of a sudden since [herr2019unconditional] and [chen2019the, chen2020unconditional]. In [herr2019unconditional], with the scheme in [chen2015unconditional], Herr and Sohinger discovered new unconditional uniqueness results for the cubic NLS on $\mathbb{T}^{d}$, which covered the full scaling-subcritical regime for $d\geq 4$. (See also the later work [kishimoto2021unconditional] using NLS analysis.)

On the other hand, the $\mathbb{T}^{3}$ quintic energy-critical case at $H^{1}$ regularity was not known until recently [chen2019the]. By discovering the new hierarchical uniform frequency localization (HUFL) property for the GP hierarchy, X. Chen and Holmer established a new $H^{1}$-type uniqueness theorem for the $\mathbb{T}^{3}$ quintic energy-critical GP hierarchy. The new uniqueness theorem, though neither conditional nor unconditional for the GP hierarchy implies the $H^{1}$ unconditional uniqueness result for the $\mathbb{T}^{3}$ quintic energy-critical NLS. Then in [chen2020unconditional], they proved the unconditional uniqueness for the $\mathbb{T}^{4}$ cubic energy-critical case by working out new combinatorics and extending the KM board game argument. As the previously used Sobolev multilinear estimates fail on $\mathbb{T}^{4}$, they develop the new combinatorics which enable the application of $U$-$V$ multilinear estimates, which is indeed weaker than Sobolev multilinear estimates. The scheme in [chen2020unconditional], which effectively combines the quantum de Finetti theorem, the $U$-$V$ space techniques, the multilinear estimates proved by using the scale invariant Strichartz estimates / $l^{2}$-decoupling theorem and the HUFL properties, provides a unified proof of the large solution uniqueness.