Mass-Supercritical & Energy-Subcritical Regime

• The mass-supercritical and energy-subcritical regime is defined by parameters that exceed the mass-critical threshold yet remain below the energy-critical level, influencing solution stability and blow-up behavior.
• It is pivotal in fields such as nonlinear Schrödinger equations, fluid dynamics, and quantum electrodynamics, impacting existence theory and stability analyses.
• Recent advances utilize variational methods, minimax principles, and phase-space decomposition to address challenges in compactness and threshold phenomena.

The mass-supercritical and energy-subcritical regime characterizes a wide array of nonlinear dispersive and variational systems across mathematical physics, fluid dynamics, and mathematical analysis. This regime refers broadly to the situation in which system parameters (such as mass, pressure, or nonlinearity exponent) exceed the mass-critical threshold but remain below the energy-critical threshold—a distinction governing the qualitative dynamics, existence theory, and stability of solutions. The precise definitions, mathematical structure, and physical manifestations of this regime are strongly context-dependent, as illustrated by its roles in nonlinear Schrödinger equations (NLS), transcritical two-phase flows, pseudo-relativistic models, and heavy-ion pair creation phenomena (Chen et al., 2022, Beceanu et al., 2018, Noris et al., 2013, Poblador-Ibanez et al., 2021, Dulaev et al., 2023, Jin et al., 2021, Fang et al., 2011).

1. Definition of the Mass-Supercritical, Energy-Subcritical Regime

In nonlinear dispersive PDEs, notably the NLS on $\mathbb{R}^N$ or bounded domains, the criticality is governed by the scaling invariance of the equation. The $L^2$-critical, or mass-critical, exponent is $p_c = 2 + \frac{4}{N}$; the energy-critical exponent for $H^1$ is $p_e = 2^* = \frac{2N}{N-2}$ (when $N \geq 3$). The regime is described as:

• Mass-supercritical: $p > p_c$
• Energy-subcritical: $p < p_e$

For instance, for $N \geq 3$: $2 + \frac{4}{N} < p < \frac{2N}{N-2}$ This regime is critical for the existence, stability, and long-time dynamics of solitary waves, and for the precise control of blow-up and scattering phenomena, as well as posing additional difficulties in constructing solutions due to the lack of convexity or coercivity of associated functionals (Chen et al., 2022, Noris et al., 2013, Fang et al., 2011).

In two-phase fluid dynamics, the terminology is adapted to thermodynamic variables: "mass-supercritical" may denote $p > p_{\text{crit}}$ (pressure exceeding pure-component critical pressure), while "energy-subcritical" is enforced by $T < T_{\text{crit}}$ (temperature below mixture critical temperature), ensuring the persistence of a sharp interface and phase-coexistence under strong nonidealities (Poblador-Ibanez et al., 2021).

2. Variational and Dynamical Implications in Nonlinear Dispersive PDEs

For the mass-supercritical, energy-subcritical NLS,

$i\partial_{t} u + \Delta u = |u|^{p-1} u$

with mass and energy conserved, classic results establish:

• Global well-posedness and scattering for data below an explicit threshold—typically characterized by the ground state $Q$ via the potential well $K$:

$$M[u]^{1-s_c} E[u]^{s_c} < M[Q]^{1-s_c} E[Q]^{s_c}, \quad \|u\|_{L^2}^{1-s_c} \|\nabla u\|_{L^2}^{s_c} < \|Q\|_{L^2}^{1-s_c} \|\nabla Q\|_{L^2}^{s_c}$$

where $s_c = N/2 - 2/(p-1)$ (Fang et al., 2011).

• Orbital stability of ground states on bounded domains within a maximal mass $\rho^*$, via a constrained minimization of the energy at fixed mass—stability holds for all $\rho < \rho^*$ (Noris et al., 2013).
• Singular minimization and threshold phenomena: The energy functional is not bounded below on the mass-constraint in this regime, requiring additional methods for existence, such as variational problems with two constraints, concentration-compactness techniques, or refined GN inequalities (Noris et al., 2013, Chen et al., 2022, Jin et al., 2021).

3. Construction, Compactness, and New Minimax Approaches

A central difficulty in the supercritical regime is to construct solutions and establish compactness of Palais-Smale (PS) sequences under the lack of coercivity. Techniques include:

• Deformation and minimax principles on constrained manifolds: New minimax theorems on $S(c)$ (the sphere of prescribed $L^2$ norm) and $S(c) \times \mathbb{R}$ leverage scaling symmetries and abstract critical point theory, circumventing the need for classical Ambrosetti-Rabinowitz or Nehari-type monotonicity conditions (Chen et al., 2022).
• Pohozaev constraint and concentration-compactness: Augmenting PS sequences by enforcing the Pohozaev identity

$$P(u) = \|\nabla u\|_2^2 - \int_{\mathbb{R}^N} [f(u)u - 2F(u)]dx = 0$$

secures boundedness in $H^1$ (Chen et al., 2022).

• Incoming/outgoing phase-space analysis: For large, rough, radial data away from the origin, a phase-space decomposition into outgoing (or incoming) parts enables the construction of global solutions in spaces below critical regularity (Beceanu et al., 2018).
• Variational problems with kinetic- or energy-constraints: In models such as the pseudo-relativistic NLS, a cap on kinetic energy restores compactness in otherwise unbounded-from-below minimization (Jin et al., 2021).

4. Applications in Sharply-Interfaced Supercritical Multiphase Flow

In real-fluid simulations for transcritical injection,

• Mass-supercritical, energy-subcritical ("transcritical") regime: $p > p_\text{crit}$ (chamber pressure above pure-component critical pressure), $T < T_\text{crit}$ (interface temperature subcritical), ensures coexistence of two dense, strongly non-ideal phases, with a sharp interface sustained by local thermodynamic equilibrium (Poblador-Ibanez et al., 2021).
• Volume-of-Fluid (VOF) models: The interface dynamics require conservative advection with explicit phase-change terms, real-fluid equations of state (such as Soave-Redlich-Kwong), and jump conditions enforcing phase equilibrium, momentum, and energy conservation.
• Enhanced dissolution and instability: High pressure increases gas solubility in the liquid, modifies local fluid properties near the interface, lowers surface tension, and accelerates atomization via Kelvin-Helmholtz and Weber instabilities.
• Computational techniques: Efficient FFT-based Poisson solvers for the low-Mach-number pressure field leverage the uniform-coefficient splitting induced by the regime's physical constraints (Poblador-Ibanez et al., 2021).

5. Spectral, Stability, and Threshold Phenomena

The regime also governs spectral stability and bifurcation phenomena:

• Ground-state branch uniqueness and stability: On bounded domains, for each $0<\rho<\rho^*$ there is a unique least-energy solution, which is orbitally stable if the Lagrange multiplier curve $\mu'(\alpha)>0$ (ensured for the ground state before the maximal mass) (Noris et al., 2013).
• Spectral coercivity: For pseudo-relativistic models, imposing a kinetic energy cap enables spectral estimates ensuring that linearized operators have the necessary coercivity for orbital stability (Jin et al., 2021).
• Scattering and blow-up dichotomy: The existence of a sharp threshold (e.g., mass-energy relation to $Q$) separates global/scattering solutions from possible blow-up, especially prominent in focusing NLS (Fang et al., 2011).
• Energy spectrum and angular distributions in quantum electrodynamics: In slow heavy-ion collisions, the transition from subcritical ($2Z<Z_{\text{cr}}$) to supercritical ($2Z>Z_{\text{cr}}$) is detectable via reversal of positron creation yield as a function of collision energy and via the differential energy spectrum. The angular distribution remains nearly isotropic in both regimes due to the symmetry at small internuclear distances (Dulaev et al., 2023).

6. Summary Table: Criticality Conditions and Main Applications

Context	Mass-Supercritical Condition	Energy-Subcritical Condition	Key Phenomena/Applications
NLS on $\mathbb{R}^N$/$B_1$	$p > 2 + 4/N$	$p < 2^* = 2N/(N-2)$	Existence, stability, scattering, blow-up
Pseudo-relativistic NLS (1D)	$p > 3$ ($L^2$ scaling-crit.)	$p < 5$ ($H^{1/2}$ theory)	Stability of solitary waves via constrained variational
Real-fluid two-phase (VOF)	$p > p_\text{crit}$ (pressure/thermo)	$T < T_\text{crit}$	Transcritical injection, phase-change, atomization
Heavy-ion QED (positron pair creation)	$2Z > Z_{\text{cr}}$ (critical charge)	—	Onset of spontaneous pair creation

7. Outlook and Further Directions

The mass-supercritical and energy-subcritical regime remains a primary area for research across applied analysis, computational methods, and experimental physical sciences:

• Further relaxation of analytical constraints: Novel abstract minimax and deformation results, such as those circumventing AR-conditions or topological families, broaden the landscape of treatable nonlinearities (Chen et al., 2022).
• Sharp experimental detection: In slow heavy-ion collisions, observables such as reversals in energy and angular spectra provide direct experimental indicators of the supercritical transition (Dulaev et al., 2023).
• Transcritical atomization: Enhanced atomization and interface dynamics underlie engineering advances in supercritical injection and combustion.
• Nonlinear stability theory: Grillakis–Shatah–Strauss frameworks and advanced concentration-compactness methods continue to elucidate the fine structure of stability and instability, especially regarding the orbital stability of “ground states” or solitary waves at and above critical mass/energy thresholds (Noris et al., 2013, Jin et al., 2021).
• Numerical methodology: Advances in fast and robust FFT-based solvers, phase-field/VOF interface tracking, and high-fidelity real-fluid models address the uniquely coupled instabilities characteristic of this regime (Poblador-Ibanez et al., 2021).

The regime encapsulates intricate balance conditions, separating fundamentally distinct dynamical behaviors, and its theoretical structure underpins a significant portion of contemporary nonlinear analysis and applied modeling.