Stable Non-Vacuum Steady States

• The paper establishes rigorous criteria for asymptotic stability in non-vacuum steady states across various nonlinear PDE models using energy functionals and spectral analysis.
• It employs advanced techniques like weighted Sobolev spaces, Hardy inequalities, and semigroup methods to demonstrate global attractor properties and explicit decay rates.
• The analysis impacts diverse fields such as fluid dynamics, plasma physics, and quantum many-body theory by elucidating the persistence and robustness of nontrivial equilibrium states.

An asymptotically stable non-vacuum steady state refers to a time-independent solution of a dynamical system—modeled via partial differential equations (PDEs) from fluid dynamics, kinetic theory, quantum many-body theory, or mathematical biology—that is not identically zero (“non-vacuum”) and such that any sufficiently small initial perturbation around this steady state decays to zero as time tends to infinity. This concept plays a central role in the qualitative theory of nonlinear PDEs governing complex systems, as it underpins the robustness and physical relevance of nontrivial equilibrium and stationary states. The following sections systematically review the rigorous theory and diverse contexts for asymptotically stable non-vacuum steady states, drawing on advanced results from PDE analysis, applied mathematics, and mathematical physics.

1. Mathematical Formulation and Context

The precise mathematical characterization depends on the underlying system, but generically a steady state $U_*$ for a PDE $U_t = \mathcal{F}(U)$ is said to be asymptotically stable if, for initial data $U_0$ sufficiently close to $U_*$ (in an appropriate norm), the solution $U(t)$ exists globally and

$\lim_{t \to \infty} \|U(t) - U_*\| = 0.$

In the non-vacuum case, $U_*$ encodes a non-vanishing physical quantity—such as positive fluid density, non-zero velocity field, or strictly positive chemical concentration.

Example settings include:

• Thin film equations with soluble surfactant: The triple $(h_*, \Gamma_*, C_{0*})$ is spatially constant and strictly positive, corresponding to a positive film height and surfactant concentrations (Escher et al., 2010).
• Stationary solutions to compressible Euler–Maxwell or Navier–Stokes equations: $n_*$, $u_*$, $E_*$, $B_*$, where $n_*$ (density) is positive and generally spatially inhomogeneous (Liu et al., 2014, Wu et al., 2021).
• Classical Landau flows and homogeneous axisymmetric solutions in Navier–Stokes theory: Velocity fields with explicit, non-zero profiles invariant under scaling (Karch et al., 2011, Li et al., 2019).
• Non-equilibrium steady states (NESS) in quantum and kinetic field theory: $h(x,v)$ or density operators $\gamma_f$ representing nontrivial fermion densities or phase-space filling (Hack et al., 2018, Hadama, 2023).

Asymptotic stability ensures that such physical steady states persist and attract perturbations, providing a mechanism for global selection of physically relevant macroscopic patterns.

2. Model Classes and Non-Vacuum Steady States

A broad range of model PDEs and systems admit asymptotically stable non-vacuum steady states:

• Parabolic and Degenerate Parabolic Systems: Thin film flows with soluble surfactants (Escher et al., 2010) and multidimensional reaction–diffusion equations in combustion (Li et al., 2022). Analyses show that uniform non-vacuum steady states are global attractors, with stability often proven via energy functionals exploiting degeneracy or weighted Sobolev spaces.
• Navier–Stokes and Euler–Maxwell Systems: Explicit axisymmetric Landau solutions (Karch et al., 2011), their (–1)-homogeneous generalizations (Li et al., 2019), and critically decaying flux carriers in 2D exterior domains (Guillod, 2016) all provide concrete, singular or spatially decaying, nontrivial steady-state profiles. Asymptotic stability for general initial data in $L^2$ or subcritical Sobolev spaces is deduced using refined Hardy inequalities, semigroup theory, and energy methods.
• Free Boundary and Physical Vacuum Problems: Lane–Emden profiles (polytropic equilibria) and gravitational hydrostatic equilibria in compressible Navier–Stokes–Poisson models show stability even up to physical vacuum interfaces (Luo et al., 2015, Luo et al., 2022), provided specific conditions on the adiabatic exponent are met, and initial data satisfy smallness and compatibility constraints.
• Plasma and Kinetic Models: Non-spatially-uniform, inflow-generated steady states for Vlasov–Poisson (Kim, 2022) and relativistic Vlasov–Poisson with magnetic field and gravity (Jin et al., 2023) are constructed via characteristic flows and shown to be exponentially stable. Stability crucially depends on the structure of the steady states (e.g., velocity Lipschitz regularity) and the presence of strong confining fields.
• Quantum Many-Body and Field Theories: Stationary density operators in the Hartree equation for infinitely many fermions—including the zero-temperature Fermi gas—are proven asymptotically stable in Schatten–Sobolev spaces. Here, perturbative density function reformulation and advanced Strichartz-type estimates are central (Hadama, 2023).
• Non-equilibrium Steady States (NESS) in QFT: Mode-wise protocols gluing KMS states for different baths yield steady states robust to local perturbations, with the mode-KMS structure crucial for stability (Hack et al., 2018).

3. Stability Methodologies

The core analytical tools for establishing asymptotic stability are:

• Energy Functionals: Construction of Lyapunov or entropy functionals that are coercive near the steady state, with the time derivative yielding a negative-semidefinite dissipation term. For instance, in thin film flow (Escher et al., 2010), an energy identity exploiting the surface tension law is central.
• Linearized Spectral Analysis and Semigroup Theory: Linearizing about the steady state and demonstrating that the linearized operator generates an analytic semigroup with negative spectral bound. Application of the principle of linearized stability allows extension to the full nonlinear (often quasilinear or semilinear) system (Karch et al., 2011, Liu et al., 2014).
• Weighted Sobolev and Exponentially Weighted Spaces: Shifting the spectrum of spatially constant-coefficient linearizations into the stable half-plane through exponential weights, usually necessary when the essential spectrum meets the imaginary axis (as in reaction–diffusion front end‐states) (Li et al., 2022).
• Hardy-type and Bootstrap Inequalities: Precise control of nonlinear terms via Hardy inequalities (for handling singular steady flows in Navier–Stokes (Li et al., 2019, Guillod, 2016)), as well as weighted Hardy inequalities tailored to domains with boundaries or phase transitions (Hong et al., 2020).
• A Priori Assumptions and Continuation Methods: Hypotheses ensuring smallness of solution norms are closed via bootstrapping energy or derivative estimates, thus deducing global existence and decay.
• Dynamical System and Fixed Point Approaches: In quantum and kinetic settings, reformulating the dynamics as fixed point problems for perturbations or densities allows the application of contraction mapping with sophisticated multilinear estimates (Hadama, 2023).
• Characteristic Flows and Lagrangian Formulation: Exploiting explicit characteristic methods for kinetic equations (Kim, 2022) and for free boundary compressible flows (Luo et al., 2015), mapping the original PDE into fixed domains and enabling uniform control even near degeneracies at the vacuum boundary.

4. Physical Contexts and Implications

Asymptotically stable non-vacuum steady states reflect fundamental physical phenomena in multiple contexts:

• Fluid and Plasma Stability: Persistence of explicit, parameterized flows (e.g., Landau and least singular axisymmetric solutions) underlies the reliability of physical models for jets, wakes, and boundary-driven shear flows, even in the presence of singularities or critical decay (Karch et al., 2011, Li et al., 2019, Guillod, 2016).
• Astrophysical Fluid Dynamics: The stability of Lane–Emden and hydrostatic equilibria informs the global behavior and long-time evolution of stellar objects—both polytropes ($\gamma > 4/3$) and white dwarfs ($y = \lim_{s \to 0+} s p'(s)/p(s) > 4/3$) (Luo et al., 2015, Luo et al., 2022).
• Non-equilibrium Thermodynamics: NESS arising from genuinely nonequilibrium situations, e.g. thermal gradients in quantum field scenarios or CFTs under black brane evolution (Hack et al., 2018, Amado et al., 2015), exhibit universal scaling laws for fluxes and pressures, validated by both rigorous analysis and numerical simulation.
• Biological and Chemotactic Systems: Systems modeling vascular network formation or chemotactic aggregation admit nonconstant phase-transition profiles, whose stability elucidates the robust emergence of spatial patterns (Hong et al., 2020).
• Kinetic and Quantum Systems: Lysos-profiled non-uniform steady states for plasmas with self-consistent fields and mechanism for stabilization via gravitational confinement and boundaries (Kim, 2022, Jin et al., 2023). In quantum many-body systems, the proof of stability for singular states such as the Fermi sea marks a major step in understanding the dynamics of infinitely many-particle systems (Hadama, 2023).

5. Decay Rates, Regularity, and Limitations

Quantitative asymptotic stability results often include explicit rates:

• Exponential decay: Established for several compressible and incompressible fluid systems under boundedness and/or positivity constraints on density (e.g., $L^2$ and $L^\infty$ exponential convergence (Wu et al., 2021, Liu et al., 2014)). The spectrum-shifting effect of exponential weights in reaction--diffusion systems similarly ensures exponential relaxation in weighted norms (Li et al., 2022).
• Polynomial/Algebraic decay: In physical vacuum and chemotaxis systems, the degeneracy at the vacuum boundary or phase interface yields decay rates depending polynomially on time, determined by the interplay of pressure law exponents and weighted a priori estimates (Luo et al., 2015, Luo et al., 2022, Hong et al., 2020).

Regularity of solutions is controlled by:

• Weighted Sobolev and Lipschitz estimates: Ensuring uniform bounds up to boundaries or inhomogeneous regions (Kim, 2022, Jin et al., 2023).
• Physical vacuum/regularity conditions: Hölder regularity of the sound speed across degenerate interfaces (Luo et al., 2015).

Limitations of current results, as clearly stated in the literature, include:

• Assumptions of smallness on initial data (energy norms, weighted spaces).
• Applicability restricted to thin film or asymptotic regimes (lubrication limit (Escher et al., 2010)).
• Limitations on allowable parameter ranges (e.g., adiabatic exponent, pressure law, flux size in 2D Navier-Stokes (Guillod, 2016)).
• In kinetic or quantum models, techniques may not fully capture thermalization or global (large perturbation) dynamics (Hack et al., 2018, Hadama, 2023).

6. Analytical and Numerical Techniques

The stability analyses synthesize:

• Energy and Lyapunov techniques tailored for degenerate/parabolic or hyperbolic-parabolic systems.
• Semigroup, bootstrapping, and functional analysis to control spectral properties, handle singular steady fields, and close nonlinear estimates.
• Weighted inequalities, Sobolev algebra properties, and elliptic regularity for control in unbounded or exterior domains.
• Lagrangian (or characteristic) approaches to exploit physical structure and boundary conditions, essential for handling vacuum and interface problems.
• Perturbative expansions and multilinear estimates (especially in quantum and kinetic settings), often relying on deep results in harmonic analysis and operator theory (e.g., Strichartz estimates, Christ–Kiselev–Gohberg–Kreĭn theorems (Hadama, 2023)).

Numerical simulations have supported the universality and accuracy of NESS scaling laws in black brane evolution, matching analytical predictions within 1% for a wide range of parameters (Amado et al., 2015).

7. Broader Impact and Future Directions

The theory of asymptotically stable non-vacuum steady states:

• Anchors mathematical models in physically plausible, robust nontrivial solutions.
• Validates and guides theoretical and numerical approaches to pattern formation, confinement, and long-time selection in fluids, plasmas, and condensed matter.
• Highlights the subtle interplay of dissipation, nonlinearity, and symmetry (e.g., central symmetry in 2D flows, conservation laws in CFT steady states) in determining global dynamics.
• Illuminates open problems regarding large perturbations, full (non-perturbative) thermalization, and the role of singularities or degeneracies in complex, high-dimensional systems.

Continued research aims to relax structural or smallness assumptions, extend stability to broader functional classes, and unify methods across physical and mathematical disciplines.