Mass concentration and blow-up dynamics above the ground-state threshold for the focusing mass-critical nonlinear Schrödinger equation

The source paper (Killip–Tao–Vișan, J. Eur. Math. Soc. 2009) proved the mass-critical nonlinear Schrödinger equation (NLS) scattering conjecture for radial data in two dimensions and, alongside its main theorem, recorded adjacent open questions: removing the radial symmetry from the conjecture, the analogous energy-critical problem, and a more precise description of mass concentration and blow-up behaviour for solutions with mass at least the ground-state mass $M(Q)$. The symmetry-removal question is fully resolved by Dodson's series of papers (2012–2016: defocusing in dimensions one, two, and three-and-higher; focusing below the ground state in all dimensions; all without symmetry), as restated authoritatively by Fan–Killip–Vișan–Zhao (2025); the source's own remark then extends its lim-sup mass-concentration corollary to all dimensions without symmetry. The energy-critical analogue is resolved in dimensions five and higher (Killip–Vișan, 2010), claimed in dimension four in an unpublished preprint, and without a published resolution in dimension three. The surviving open core, active as of 2026, is the blow-up side at masses at or above $M(Q)$: (i) upgrading the known lim-sup-along-a-sequence concentration of at least $M(Q)$ to a lim-inf statement for arbitrary $L^2$ data — the source identifies quadratic oscillation as the obstruction — and (ii) determining the sharp universal blow-up rate and concentration profile for general data outside the near-ground-state regime: Kwak–Kwon (2026) proved that finite-time blow-up never occurs at exactly the self-similar rate (any dimension, no symmetry) and established the sharp log-log window bound under radial symmetry (dimensions two and higher), leaving the non-radial sharp bound and all lim-inf statements open.

Background

The nonlinear Schrödinger equation (NLS) with mass-critical nonlinearity is

$$i\,\partial_t u+\Delta u=\mu\,|u|^{4/d}u,\qquad u:\ \mathbb{R}\times\mathbb{R}^d\to\mathbb{C},$$

where $d\ge 1$ is the space dimension, $\mu=-1$ is the focusing (attractive) sign and $\mu=+1$ the defocusing (repulsive) sign; the source paper treats the cubic case $d=2$. The equation conserves the mass $M(u(t))=\int_{\mathbb{R}^d}|u(t,x)|^2\,dx$ and is invariant under the scaling $u(t,x)\mapsto\lambda^{d/2} u(\lambda^2 t,\lambda x)$, which also preserves the mass; this is the meaning of mass-critical: the conserved quantity controls the solution size exactly at the scaling threshold, so the interplay between dispersion and self-focusing is decisive. The natural data class is the rough space $L^2(\mathbb{R}^d)$, where the conserved energy (a second functional involving $|\nabla u|^2$ and $|u|^{2+4/d}$, available only for more regular data) need not be finite. A solution scatters if it is global in time and asymptotically behaves like a solution of the free linear Schrödinger equation; it blows up if the scale-invariant spacetime norm $\|u\|_{L^{2(d+2)/d}_{t,x}([0,T^*)\times\mathbb{R}^d)}$, the norm controlled by the local theory, is infinite — either because the maximal forward time of existence $T^*\le\infty$ is finite (finite-time blow-up), or because $T^*=\infty$ and the solution fails to scatter (infinite-time blow-up).

Let $Q$ denote the ground state, the unique positive radial decaying solution of the elliptic equation $\Delta Q+|Q|^{4/d}Q=Q$; its mass $M(Q)$ is the sharp threshold for the focusing equation. The pseudoconformal transform — an additional symmetry of the mass-critical equation acting on sufficiently regular solutions — applied to the standing wave $e^{it}Q(x)$ produces a finite-time blow-up solution of mass exactly $M(Q)$, so blow-up exists at the threshold. The mass-critical scattering conjecture (Conjecture 1.8 in the source) predicted that all $L^2$ data yield global scattering solutions, except that in the focusing case scattering should hold exactly for data of mass strictly below $M(Q)$.

Before the source paper, Bourgain proved the defocusing conjecture for radial two-dimensional data and, in the focusing direction, the first mass-concentration result for merely $L^2$ data: some small amount of mass must concentrate in windows of the self-similar (parabolic) size $(T^*-t)^{1/2}$, at least along a sequence of times. Killip–Vișan–Zhang settled the radial defocusing problem in dimensions three and higher, and Keraani proved concentration results for masses close to the critical one. The source paper proved the full conjecture for radial data in two dimensions (Theorem 1.10) and drew two blow-up corollaries: Corollary 1.11 — a radial blow-up solution of mass below $2M(Q)$ concentrates at least $M(Q)$ at essentially all late times (the lim-inf sense), in any window slightly larger than the self-similar scale; and Corollary 1.12 — an arbitrary radial blow-up solution concentrates at least $M(Q)$ along some sequence of times tending to blow-up (the lim-sup sense). The authors identify the obstruction to the lim-inf statement at general masses as the spectre of quadratic oscillation: in the concentration-compactness arguments — the standard framework in which sequences of rescaled solutions are decomposed into their weak limits, called profiles — the profiles may drift and oscillate at the level of the scaling, so compactness can be recovered only along a subsequence of times. For finite-energy data, a lim-inf statement with a moving center is classical (the references are summarized in the source paper's introduction); the open issue is data merely in $L^2$.

The symmetry-removal question recorded in the source was resolved within a few years by Dodson, who introduced frequency-localized interaction Morawetz estimates (a monotonicity identity for the interaction of two frequency-localized pieces of the solution) and proved the conjecture in full generality: defocusing in dimensions $d\ge 3$ (2012), $d=2$ (2016), and $d=1$ (2016), and focusing with mass below the ground state in every dimension $d\ge 1$ without any symmetry assumption (2015). A simpler proof of the non-radial focusing case below the ground state was given by Dodson–Murphy (2018). Fan–Killip–Vișan–Zhao (2025), co-authored by two authors of the source paper, state Dodson's theorem as the standard result: for all $d\ge 1$ and all data in $L^2(\mathbb{R}^d)$ — in the focusing case with mass below $M(Q)$ — the solution is global and scatters, with no symmetry hypothesis. The source paper already noted that its Corollary 1.12 argument works in any dimension and without spherical symmetry once the scattering conjecture holds; since the conjecture is now settled, the lim-sup mass-concentration statement for blow-up solutions holds unconditionally in all dimensions and without symmetry.

The description of blow-up at and above the ground-state mass is far less complete. At exactly the threshold $M(Q)$, blow-up solutions are completely classified: up to the symmetries of the equation they are precisely the pseudoconformal images of the ground state — this is due to Merle (1993) for radial data, extended to general data by Tao–Vișan–Zhang (2008) and Killip–Li–Vișan–Zhang (2009), and revisited by Dodson (2023; 2024). Slightly above the threshold, the Merle–Raphaël program (2004–2006) gives an essentially complete description for finite-energy data with mass in a small interval $(M(Q),M(Q)+\alpha^*]$ above the threshold: the blow-up profile is $Q$ and the rate is either the log-log rate — the frequency scale $\lambda(t)^{-1}$ satisfies $\|\nabla u(t)\|_{L^2}\sim\sqrt{\log|\log(T^*-t)|/(T^*-t)}$, slightly faster than the self-similar rate $(T^*-t)^{-1/2}$ — or a strictly faster one; and the concentrated mass is quantized near the threshold (2005). The log-log rate was first constructed rigorously by Perelman (2001), for one-dimensional data close to a soliton; this example shows that the self-similar window cannot be shrunk by more than the square root of a double-logarithmic factor, so the log-log scale is the best possible window for any general concentration statement. For general masses and data merely in $L^2$, much less is known. Bourgain–Wang (1997) constructed finite-energy blow-up solutions with prescribed regular behaviour away from the singular point, in which the mass in excess of the concentrating bubble does not concentrate, so one cannot hope that all the mass collapses. Martel–Raphaël (2018) constructed, in two dimensions, strongly interacting multi-bubble solutions concentrating $K\cdot M(Q)$ at a rate faster than the pseudoconformal one. Fan–Mendelson (2023) constructed log-log blow-up solutions at pure $L^2$ regularity in two dimensions, by probabilistic perturbations of more regular data. Fan (2021) and Dodson (2021; 2022) proved that certain classes of non-scattering solutions above the threshold converge, along sequences of times and after rescaling, to the ground state — weak, sequential analogues of the desired concentration statement. Most recently, Kwak–Kwon (2026) proved that blow-up never occurs at exactly the self-similar rate: for finite blow-up times, general $L^2$ data, any dimension $d\ge 1$, and no symmetry, the mass in windows that are arbitrarily small fixed multiples of the self-similar window reaches at least $M(Q)$ along a sequence of times (their Theorem 1.1). Under radial symmetry in dimensions $d\ge 2$, again for finite blow-up times, they established the sharp log-log scale (their Theorem 1.2 and Corollary 1.4): an asymptotic blow-up profile exists, the effective radius of the blow-up core obeys the log-log bound on a time average, and, in the mass-concentration form, for every sufficiently large fixed $r$ the mass in windows of radius $r\sqrt{(T^*-t)/\log|\log(T^*-t)|}$ reaches at least $\int_{|x|\le r/\sqrt{2\pi}}|Q(x)|^2\,dx$ — that is, $M(Q)$ up to a defect exponentially small in $r$ — along a sequence of times; the constants are optimal (the sharp factor $\sqrt{2\pi}$ matches the near-soliton log-log solutions of Merle–Raphaël), and the log-log correction to the window scale is sharp by Perelman-type constructions. Their analysis, as they emphasize, relies on no ansatz or variational structure; they also note that whether the self-similar bounds could be improved for general data had remained an open question emphasized in survey articles (see the survey of Martel–Merle–Raphaël–Szeftel, 2014), and that apart from the near-soliton regime the nature of general blow-up dynamics remains largely unknown.

Problem Statement

Setting. Consider the focusing mass-critical nonlinear Schrödinger equation

$$i\,\partial_t u+\Delta u=-|u|^{4/d}u\quad\text{on }\mathbb{R}\times\mathbb{R}^d$$

(the source case is $d=2$, the cubic equation; the question is posed for general $d\ge 1$). Let the initial data $u_0$ lie in $L^2(\mathbb{R}^d)$ with mass $M(u_0)\ge M(Q)$, where $Q$ is the ground state, and let $u$ be the corresponding maximal-lifespan solution, assumed to blow up forward in time at $0<T^*\le\infty$: the spacetime norm $\|u\|_{L^{2(d+2)/d}_{t,x}([0,T^*)\times\mathbb{R}^d)}$ is infinite, either because $T^*<\infty$ is a finite blow-up time, or because $T^*=\infty$ and the solution fails to scatter.

Part 1 (strong mass concentration). Prove or disprove: there is a concentration center $x(t)\in\mathbb{R}^d$ such that

$$\liminf_{t\nearrow T^*}\ \int_{|x-x(t)|\le R(t)}|u(t,x)|^2\,dx\ \ge\ M(Q)$$

for every scale function $R(t)$ slightly larger than the self-similar window, i.e. with $(T^*-t)^{-1/2}R(t)\to\infty$ as $t\nearrow T^*$ (when $T^*=\infty$, $t^{-1/2}R(t)\to\infty$). This is the lim-inf upgrade of the known lim-sup statement — concentration of at least $M(Q)$ only along a sequence of times. The lim-sup statement in windows slightly larger than the self-similar window is now known unconditionally in all dimensions and without symmetry, for finite and infinite blow-up times alike: the source's Corollary 1.12 covers both cases, and its argument, as the source remarks, works in any dimension and without spherical symmetry once the scattering conjecture holds — a hypothesis settled by Dodson. For finite blow-up times, Kwak–Kwon (2026) strengthened the lim-sup statement to windows that are arbitrarily small fixed multiples of the self-similar window. Known partial cases of the lim-inf statement: it holds for finite-energy data (classical, with fixed radius and moving center) and for radial $L^2$ data of mass below $2M(Q)$ (Keraani; Corollary 1.11 of the source). The open regime is general masses with non-radial data, or any $L^2$ data at sufficiently large mass; the source identifies quadratic oscillation as the obstruction. A disproof would amount to constructing a blow-up solution of mass at least $M(Q)$ such that for every choice of center $x(t)$ there is an admissible scale $R(t)$ for which the lim-inf of the windowed mass is strictly below $M(Q)$ — a genuinely new, oscillatory blow-up mechanism.

Part 2 (universal rate and concentration profile above the threshold). Outside the near-threshold regime where the Merle–Raphaël theory applies (finite-energy data with mass in $(M(Q),M(Q)+\alpha^*]$), determine the blow-up dynamics for general data of mass at least $M(Q)$: (a) prove, without radial symmetry, the analogue of the sharp log-log bounds that Kwak–Kwon (2026) established for radial solutions in dimensions $d\ge 2$ with finite blow-up time — in the mass-concentration form of their Corollary 1.4: for every sufficiently large fixed $r$, mass at least $\int_{|x|\le r/\sqrt{2\pi}}|Q(x)|^2\,dx$, i.e. $M(Q)$ up to a defect exponentially small in $r$, concentrates along a sequence of times (a lim-sup statement) in windows of radius $r\sqrt{(T^*-t)/\log|\log(T^*-t)|}$ centered at a concentration center $x(t)\in\mathbb{R}^d$ (for radial solutions the center is the origin; for non-radial solutions the statement may equivalently be written with a supremum over centers, as in the non-radial Theorem 1.1 of Kwak–Kwon); equivalently, in the time-averaged rate form of their Theorem 1.2, the effective radius of the blow-up core is bounded by the log-log window, corresponding to a blow-up rate at least of order $(T^*-t)^{-1/2}\sqrt{\log|\log(T^*-t)|}$ — or construct a non-radial blow-up solution whose concentration scale is larger than the log-log window by an unbounded factor: a blow-up rate slower than the log-log rate $(T^*-t)^{-1/2}\sqrt{\log|\log(T^*-t)|}$ yet faster than the self-similar rate $(T^*-t)^{-1/2}$, consistent with the now-known nonexistence of exactly-self-similar-rate blow-up (the log-log scale itself is sharp even under symmetry, by Perelman-type constructions, so no smaller window can be expected in general); and (b) determine which amounts of mass can concentrate at blow-up for given initial mass at least $M(Q)$ — near threshold the concentrated mass is quantized in multiples of $M(Q)$, multi-bubble constructions realize $K\cdot M(Q)$, and Bourgain–Wang-type solutions show the excess mass need not concentrate at all — and classify the corresponding blow-up profiles (bubble towers, their scales and interaction regimes).

Parts 1 and 2 are two facets of one objective: the quantitative description of mass concentration at blow-up for general data above the ground-state threshold. Together they instantiate the open direction recorded in the source paper — a more precise description of mass concentration for general blow-up solutions at masses at or above $M(Q)$, which the source poses in general terms — in the precise form left open by the 2012–2026 literature. The question therefore comprises the three conjunctive sub-goals Part 1, Part 2(a) and Part 2(b). Part 2(b) is asked beyond the near-threshold facts recalled in it — for general $L^2$ blow-up data of mass at least $M(Q)$ in any dimension $d\ge 1$ — and asks for the set of achievable concentrated masses and, for each achievable value, the corresponding blow-up profile together with its concentration scale and interaction regime, up to the symmetries of the equation. The symmetry-removal part recorded in the same source is resolved (see Background).

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