Larson-Penston Collapsing Solution Explained

• The Larson–Penston solution is a self-similar profile describing the gravitational collapse of an isothermal, spherically symmetric fluid with a central density singularity.
• It exhibits monotonic density and velocity profiles that enable robust construction of Lyapunov functionals and precise nonlinear stability analysis under radial perturbations.
• High-order weighted energy methods and computer-assisted spectral analysis confirm its role as the dynamically selected attractor in protostellar collapse scenarios.

The Larson–Penston collapsing solution is a fundamental self-similar profile describing the gravitational implosion of an isothermal, spherically symmetric fluid under Newtonian gravity. Its mathematical structure, physical emergence as an attractor, and stability properties have made it central to the theory of star formation, protostellar collapse, and the paper of singularity formation in compressible flows. The nonlinear stability of the Larson–Penston solution—specifically, its persistence under radially symmetric perturbations—is a core result for the dynamical theory of gravitational collapse (Guo et al., 15 Sep 2025).

1. Mathematical Structure of the Larson–Penston Solution

The original Larson–Penston solution solves the isothermal Euler–Poisson system for a self-gravitating fluid with the equation of state $p = k\rho$ (with $k > 0$), in spherical symmetry. In similarity variables,

$$\rho(t, r) = (-t)^{-2} \tilde{\rho}(y), \quad u(t, r) = \sqrt{k}\,\tilde{u}(y), \quad y = \frac{r}{\sqrt{-k t}}$$

the problem is reduced to a coupled nonlinear ODE for $\tilde{\rho}(y)$ and an associated velocity variable. The Larson–Penston solution is the unique regular, analytic self-similar solution passing through the sonic point $y^*$ (where $1 - y^* \tilde{\omega}(y^*) = 0$), and which exhibits a central density singularity as $t\to 0^-$ (Guo et al., 2020).

2. Ground State Character and Monotonicity

A key property of the Larson–Penston solution is its "ground state" character: it is the unique physically relevant self-similar attractor among a wider family of solutions, exhibiting global spatial monotonicity. The density profile $\tilde{\rho}(y)$ is positive and decays monotonically; the infall velocity profile is everywhere negative and strictly monotonic. These monotonicity properties are crucial analytic features because they enable the construction of Lyapunov functionals and provide coercive control for stability analysis (Guo et al., 15 Sep 2025).

3. Nonlinear Stability Framework for Radial Perturbations

The nonlinear stability analysis is formulated for radially symmetric perturbations of the Larson–Penston background, denoting the perturbed solution by

$$(\zeta, \mu) = (\zeta_{LP}, \mu_{LP}) + (\theta, \phi)$$

where $(\theta, \phi)$ are small (in a suitable norm), time-dependent perturbations. The coupled system for $(\theta, \phi)$ is quasilinear and, after normalization, takes the form: $\begin{split} \theta_s &= -\Lambda \theta + \theta + \phi \ \phi_s &= -\Lambda \phi + \text{(op. term in %%%%9%%%%)} + \text{nonlinearities} \end{split}$ with $\Lambda$ a radial scaling operator (essentially, a weighted derivative mimicking the similarity generator), and where $(\theta, \phi)$ are governed by a quasilinear evolutionary system which is degenerate near the singularity.

4. High-Order Weighted Energy Methods

To achieve nonlinear stability, the analysis employs a two-tiered high-order energy method:

• Lower-order estimates: Leveraging the dissipative semigroup generated by the linearized problem, an estimate via Duhamel’s formula provides decay for the lower-order perturbation energy, but typically suffers a loss of derivatives due to non-self-adjoint terms.
• Top-order (high-order) energy estimate: For the highest derivatives, the proof commutes the perturbation equations with weighted differential operators $\mathcal{D}^{2j}$ specifically adapted to both the interior degeneracy near $z=0$ and the asymptotic flattening of the far-field ($z \gg 1$). The high-order energies are defined as: $$E_{2j}[\theta, \phi] = \int \chi_{2j}(z) \left( \frac{|\mathcal{D}^{2j}\theta|^2}{\zeta^2} + |\mathcal{D}^{2j}\phi|^2 \right) dz$$ where $\chi_{2j}(z)$ are position-dependent weights whose growth or decay is tied to the regularity and localization properties required for the problem.

Core technical achievements include:

• Proving dissipativity of the commuted linearized operator—i.e., in the energy identity,

$$\frac{d}{ds} E_{2j} + k_0 j\, E_{2j} \leq \text{lower order terms}$$

for sufficiently large $j$ and positive $k_0$, with the critical point that this dissipativity is uniformly valid globally in space (even on arbitrarily large backward light cones emanating from the singular region), due to the global monotonicity of the Larson–Penston profile.

• Achieving boundedness of the nonlinear commutator and error terms via careful interior/exterior splitting and the use of Hardy-type inequalities and localized interior weights.

5. Mode-Stability and Spectral Analysis

A significant obstacle is the non-self-adjoint linearized spectral problem. The system is subject to a trivial instability due to time translation symmetry, but otherwise all complex unstable eigenvalues must be ruled out. The proof proceeds by:

• Directly analyzing low- and high-frequency regimes by exploiting monotonicity and the structure of the non-self-adjoint operator.
• Applying rigorous computer-assisted spectral analysis in the intermediate frequency regime to exclude nontrivial instabilities.
• Employing a supersymmetric formulation to facilitate exclusion of eigenvalues with large imaginary part (see Lemma 6.7 in (Guo et al., 15 Sep 2025)).

The spectral stability result is "mode-stability" in that all perturbations, except those corresponding to infinitesimal time shifts along the similarity family, decay.

6. Nonlinear Closure and Global Existence

Having established semigroup decay at low order and dissipativity without derivative loss at high order, the final step is to close the nonlinear estimates via a fixed-point argument. The high-order energy inequality ensures that nonlinear remainders can be controlled (absorbed) by the dissipative part. Lipschitz continuity for differences of solutions is obtained, which is essential for the contraction mapping and for suppressing the neutral direction caused by the time-translation symmetry. The precise final collapse time is selected via the Brouwer fixed-point theorem.

7. Implications for Gravitational Collapse and Compressible Flow Theory

This nonlinear stability result provides the first global-in-similarity-time nonlinear control of radially imploding, self-similar compressible flows in the isothermal Euler–Poisson system. The ground state character of the Larson–Penston solution and its global monotonicity are shown to be not only central to its attractor property but also essential for analytic control in both linear and nonlinear regimes. The methodology of high-order weighted energy estimates, global coercivity, and precise semigroup control sets a technical benchmark for future stability results in gravitational collapse and for understanding singularity formation in compressible flows with self-gravity.

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Summary Table: Key Features in the Nonlinear Stability Analysis (Guo et al., 15 Sep 2025)

Ingredient	Mathematical/Physical Role	Implementation
Ground state monotonicity	Enables global coercivity, Lyapunov bounds	Non-oscillatory LP profile
Top-order weighted energy method	Avoids derivative loss; global decay	Weighted derivatives, interior-exterior splitting
Mode-stability proof	Rules out nontrivial instabilities	Energy method + computer-assisted analysis
Semigroup decay for linearized evolution	Drives convergence to similarity profile	Duhamel formula, lower-order estimate
Nonlinear closure (fixed point, modulation)	Identifies collapse time, suppresses neutral mode	Brouwer theorem, difference estimate

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The full nonlinear stability of the Larson–Penston family of self-similarly collapsing solutions under radially symmetric perturbations establishes it as the dynamically selected outcome for isothermal gravitational collapse, and provides a rigorous analytic framework underpinning singularity formation and attractor dynamics in the context of Newtonian self-gravitating fluids.