Consider the mass-critical defocusing NLS in dimension $d\geq 1$:
$$iu_{t}+\Delta u = |u|^{4/d}u, \quad (t,x) \in I\times\mathbb{R}^{2}$$
Define the mass $M(u)$ and scattering size $S(u)$ of the solution $u$ respectively by
$$M(u):=\|u\|_{L_{x}^{2}(\mathbb{R}^{d})}^{2}, \quad S(u):=\|u\|_{L_{t,x}^{\frac{2(d+2)}{d}}(I\times\mathbb{R}^{d})}^{\frac{2(d+2)}{d}}$$ For a time $t_{0}$, we also define the quantities $$S_{\leq t_{0}}(u):=\int_{I\cap (-\infty,t_{0}]}\int_{\mathbb{R}^{d}}|u(t,x)|^{\frac{2(d+2)}{d}}dxdt, \quad S_{\geq t_{0}}(u):=\int_{I\cap[t_{0},+\infty)}|u(t,x)|^{\frac{2(d+2)}{d}}dxdt$$
For mass $m\geq 0$, define the quantity $A(m)$ by $$A(m) := \sup\{S(u) : M(u)\leq m\},$$ where the supremum is taken over all maximal lifespan solutions $u$ with mass $M(u)\leq m$. One can show that $A: [0,+\infty) \rightarrow [0,+\infty]$ is monotone nondecreasing, left-continuous, is finite for small $m$. Therefore there exists a unique critical mass $0<m_{0}=m_{0}(d)\leq +\infty$, such that $A(m)$ is finite for all $m<m_{0}$ and $A(m)$ is infinite for all $m\geq m_{0}$.
Conjecture 1. $m_{0} = +\infty$
I should remark that this conjecture actually has been resolved by Benjamin Dodson in all dimensions in both the focusing (where $m_{0}=M(Q)$, $Q$ being the ground state) and defocusing cases, but I pose it anyway for the sake of the question.
In the paper "Minimal-Mass Blowup Solutions" by T. Tao, M. Visan, and X. Zhang, the authors prove using concentration compactness, in particular a Palais-Smale condition modulo the symmetry group of the equation, that if $m_{0}<+\infty$, then there exists a maximal lifespan solution of critical mass which blows up forward and backward in time and has "some compactness" to it. More precisely,
Theorem 2. If $m_{0}<+\infty$, then there exists a maximal lifespan solution $u \in C_{t,loc}^{0}L_{x}^{2}(I\times\mathbb{R}^{d})$ of mass exactly $m_{0}$ and $S_{\geq 0}(u)=S_{\leq 0}(u)=+\infty$. Moreover, there exist functions $x:I\rightarrow\mathbb{R}^{d}$, $\xi:I\rightarrow\mathbb{R}^{d}$ and $N:I\rightarrow (0,\infty)$ such that $$\forall \eta>0, \exists C(\eta) > 0, \int_{|x-x(t)|\geq C(\eta)/N(t)} |u(t,x)|^{2}dx \leq \eta \tag{1}$$ $$\ \int_{|\xi-\xi(t)|\geq C(\eta)N(t)}|\widehat{u}(t,\xi)|^{2}d\xi \leq\eta \tag{2}$$
In a remark (see pg. 6, the paragraph below Corollary 1.20), the authors claim that one can modify the earlier "induction on energy" arguments of Bourgain, CKSTT, and Ryckman and Visan to give an alternate proof of Theorem 1. In this quantitative approach, one fixes parameters $1\gg \eta_{0}\gg \eta_{1}\gg\eta_{2}>0$, where each $\eta_{j}$ is allowed to depend on $m_{0}$ and all of the previous $\eta's$.
Definition 3. A minimal-mass almost blowup solution is a solution $u$ on a compact time interval $I_{*}$ such that $M(u)=m_{0}$ and $S(u)>1/\eta_{2}$.
One can show, as is sketched in Section 8 of the cited Tao, Visan, and Zhang paper, that the frequency localization and spatial localization conditions (1) and (2) hold not for all $\eta>0$ but instead all $\eta\geq \eta_{1}$. Using these weaker localization properties, I believe that I can show the following:
Theorem 3. A maximal lifespan solution $u:I\times\mathbb{R}^{d}\rightarrow\mathbb{C}$ with $M(u)=m_{0}$ and $S(u)=+\infty$ satisfies conditions (1) and (2) for all $\eta>0$.
The problem is that a priori I do not know that--or at least it's not clear to me--such a minimal mass blowup solution exists, assuming of course Conjecture 1 is false. I merely know that there exist minimal mass solutions of arbitrarily large scattering size. The argument I have in mind to prove Theorem 3 proceeds by contradiction. Therefore there exists a minimal-mass blowup solution $u$ and an $\eta>0$ for which conditions (1) or (2) fail. The assumption that $S(u)=+\infty$ allows one to take the compact interval $I_{*}\subset I$ larger and larger to achieve arbitrarily large scattering size $S_{I^{*}}(u)$, without having to change functions $u$ and therefore $\eta$.
Does anyone have any ideas/suggestions how to get around this issue or prove Theorem 2 using the induction on mass approach, as the authors suggest one can do?
