Development of Implosions of Solutions to the Three-Dimensional Degenerate Compressible Navier-Stokes Equations

Abstract: A fundamental open problem in the theory of the multidimensional compressible Navier-Stokes equations is whether smooth solutions can develop singularities in finite time. For constant viscosity coefficients, recent remarkable results show that there exist smooth initial data for which the corresponding smooth solutions of the barotropic flow undergo finite-time implosion at the origin, with the density blowing up to infinity. In contrast, when the viscosity coefficients depend linearly on the density (as in the shallow water case), it has been established that, for general large spherically symmetric initial data, the solutions remain globally regular. These results indicate that the qualitative behavior of multidimensional solutions is sensitive to the structure of the viscosity coefficients. In this paper, we investigate the case of nonlinear viscosity coefficients with power-law density dependence. We identify a threshold value, depending on the adiabatic exponent, such that, for any power below this threshold, there exists a class of smooth initial data with strictly positive density for which the corresponding smooth solutions implode in finite time at the origin. The key issue is to show that, in this regime, the degenerate viscous terms are not sufficiently strong to suppress the convective mechanism driving the implosion. Establishing this rigorously is highly nontrivial due to the degenerate structure of the Navier-Stokes equations. To overcome this difficulty, we first derive a pointwise estimate for the density and then obtain spatial decay estimates for the velocity gradient via carefully constructed weighted high-order energy estimates and interpolation inequalities. The resulting decay rate is sufficiently rapid to compensate for the singular growth of the density, leading to uniform-in-time control of the viscous terms and ultimately to the formation of implosion.

• The paper constructs smooth, strictly positive-density solutions that implode in finite time for 0<δ<δ*(γ)<1/2 and 1<γ<1+2/√3, with density and velocity diverging near the origin.
• The authors use self-similar variables, spatial decay estimates, weighted high-order energies, and finite-dimensional unstable-mode selection to stabilize an imploding Euler profile despite degenerate viscosity.
• The result applies to several physically motivated gas models and periodic domains, while leaving the intermediate regime δ*(γ)≤δ<1 and larger adiabatic exponents unresolved.

Context and motivation

This paper by Chen, Liu, and Zhu addresses singularity formation for the three-dimensional barotropic compressible Navier–Stokes (CNS) equations with density-dependent viscosity coefficients of power-law type, $\mu(\rho)=a_1\rho^\delta$, $\lambda(\rho)=a_2\rho^\delta$, with $\delta>0$. The governing system is the standard mass and momentum balance in $\mathbb{R}^3$ with polytropic pressure $P=A\rho^\gamma$, $\gamma>1$, subject to the physical constraint $a_1>0$, $2a_1+3a_2\geq 0$.

The motivating landscape is sharply divided. For constant viscosity ($\delta=0$), Merle–Raphaël–Rodnianski–Szeftel, Buckmaster–Cao-Labora–Gómez-Serrano–Shkoller–Vicol, and Cao-Labora–Gómez-Serrano–Shi–Staffilani constructed smooth spherically symmetric (and later non-symmetric) initial data whose solutions implode in finite time, with $\rho\to\infty$ at the origin. By contrast, for linearly density-dependent viscosity ($\lambda(\rho)=a_2\rho^\delta$0, as in the viscous Saint-Venant/shallow-water system), Chen–Zhang–Zhu proved global regularity of large spherically symmetric smooth solutions in two and three dimensions, even allowing vacuum states and physical vacuum free boundaries. These two regimes suggest that solution behavior depends sensitively on the viscosity exponent $\lambda(\rho)=a_2\rho^\delta$1, and they leave open precisely which intermediate exponents permit implosion.

A further conceptual point motivates the setup: prior blow-up results for degenerate viscosity with $\lambda(\rho)=a_2\rho^\delta$2 relied on initial data vanishing on an open set, where the continuum hypothesis underlying hydrodynamics fails. The authors deliberately impose $\lambda(\rho)=a_2\rho^\delta$3 everywhere so that any implosion produced is not an artifact of vacuum.

Main results

The central theorem identifies a threshold exponent $\lambda(\rho)=a_2\rho^\delta$4 such that implosion persists for all $\lambda(\rho)=a_2\rho^\delta$5. For $\lambda(\rho)=a_2\rho^\delta$6,

$\lambda(\rho)=a_2\rho^\delta$7

For every such $\lambda(\rho)=a_2\rho^\delta$8, there exist $\lambda(\rho)=a_2\rho^\delta$9 initial data with $\delta>0$0 (and a finite-codimension family thereof) such that the corresponding smooth solution blows up at some time $\delta>0$1:

$\delta>0$2

with convergence to the self-similar Euler profile along rescaled variables. The same conclusion holds for the periodic problem on a torus of period 10. The threshold satisfies $\delta>0$3 as $\delta>0$4 and $\delta>0$5 as $\delta>0$6, so the admissible range shrinks to nothing as $\delta>0$7 approaches the upper endpoint; the result is restricted to $\delta>0$8 throughout.

The result has direct physical applicability. For cutoff inverse-power-force models derived from the Chapman–Enskog expansion, $\delta>0$9 with $\mathbb{R}^3$0. The theorem then covers rigid elastic spherical molecules ($\mathbb{R}^3$1, requiring $\mathbb{R}^3$2), Maxwellian molecules ($\mathbb{R}^3$3, requiring $\mathbb{R}^3$4), and ionized gases ($\mathbb{R}^3$5, requiring $\mathbb{R}^3$6). Thus implosion is established within genuinely physical parameter regimes rather than only for mathematically convenient viscosity laws.

Reformulation and reduction to profile stability

The proof follows the now-standard strategy introduced by Merle–Raphaël–Rodnianski–Szeftel. Writing $\mathbb{R}^3$7 with $\mathbb{R}^3$8, the authors apply the self-similar change of variables

$\mathbb{R}^3$9

under which CNS becomes a perturbed dynamical system for $P=A\rho^\gamma$0 containing a dissipative term

$P=A\rho^\gamma$1

where $P=A\rho^\gamma$2. The condition $P=A\rho^\gamma$3—which is exactly what forces the upper bound $P=A\rho^\gamma$4—ensures that viscosity decays exponentially in self-similar time, so it cannot prevent convergence to the inviscid imploding profile. This is the structural reason the convective mechanism wins over dissipation in the regime considered.

The analysis is reduced to global-in-time stability of the truncated self-similar Euler profile $P=A\rho^\gamma$5, where $P=A\rho^\gamma$6 is the smooth spherically symmetric steady profile constructed for $P=A\rho^\gamma$7, and $P=A\rho^\gamma$8 is a spatial cutoff. The perturbation equations split into a linear part $P=A\rho^\gamma$9, quadratic nonlinearities $\gamma>1$0, profile-dependent forcing $\gamma>1$1, and the degenerate dissipative term $\gamma>1$2. A truncated operator $\gamma>1$3, equal to $\gamma>1$4 inside $\gamma>1$5 and augmented by a strong damping term $\gamma>1$6 outside, is maximally dissipative modulo a finite-dimensional unstable subspace $\gamma>1$7.

Proof architecture

The stability argument combines four ingredients, closed via a bootstrap argument on a finite-codimensional manifold of initial data.

Spatial decay estimates. The key obstacle relative to the constant-viscosity case is that dissipation strengthens where density is large, so one must show the velocity gradient decays fast enough to absorb the singular weight $\gamma>1$8. The authors first prove pointwise bounds on $\gamma>1$9 along characteristics $a_1>0$0 via a region-segmentation method, obtaining

$a_1>0$1

They then derive gradient decay $a_1>0$2, faster than what naive interpolation between $a_1>0$3 bounds and the weighted energy would give. This accelerated decay is essential: it controls terms of the form $a_1>0$4 appearing when differentiating the degenerate viscous term, which a direct interpolation cannot handle.

Lower-order temporal decay. Inside $a_1>0$5, the semigroup generated by $a_1>0$6 contracts on the stable subspace, and Duhamel estimates against the forcing terms yield exponential decay of the perturbation and its derivatives up to fourth order. In the exterior region, transport along characteristics combined with Gagliardo–Nirenberg interpolation gives matching decay, including $a_1>0$7 control of fourth derivatives.

Higher-order weighted energy. For the top-order energy $a_1>0$8 with a carefully chosen weight $a_1>0$9 growing like $2a_1+3a_2\geq 0$0 at infinity, the symmetric hyperbolic structure of the differentiated system plus the repulsivity of the radial and angular components of the profile yields

$2a_1+3a_2\geq 0$1

The commutator estimates for the convection terms exploit spherical symmetry to isolate radial and angular derivative contributions, and the repulsivity conditions absorb them into the damping. The degenerate dissipation term itself is handled by integration by parts and the spatial decay estimates; its coefficient $2a_1+3a_2\geq 0$2 decays, but the decay rate is slow enough (via $2a_1+3a_2\geq 0$3) to remain harmless.

Unstable mode selection. Since $2a_1+3a_2\geq 0$4 is finite-dimensional and invariant, the unstable coefficients satisfy a finite ODE driven by the projected forcing. Using an outgoing property of trajectories near the boundary of a small ball in a suitable metric $2a_1+3a_2\geq 0$5 on $2a_1+3a_2\geq 0$6, a Brouwer fixed-point argument selects initial coefficients $2a_1+3a_2\geq 0$7 keeping the unstable component bounded by $2a_1+3a_2\geq 0$8. This produces the finite-codimension set of imploding initial data.

Local well-posedness of the reformulated problem is established separately, including a higher-order weighted regularity estimate obtained by approximating the weight with compactly supported functions $2a_1+3a_2\geq 0$9 and passing to the limit via Fatou's lemma.

Limitations and open questions

Several restrictions are intrinsic to the argument as presented. First, the adiabatic exponent is confined to $\delta=0$0; whether implosion occurs for larger $\delta=0$1 with degenerate viscosity remains open, as does the sharpness of $\delta=0$2 itself—the paper does not establish global regularity for $\delta=0$3, so the threshold is a sufficient condition, not a proven dichotomy. Second, the construction requires a finite-codimension set of initial data selected through the unstable-mode fixed-point argument; genericity of these data is not addressed. Third, the results are proved for spherically symmetric profiles; the extension to non-symmetric imploding solutions, analogous to the torus construction of Cao-Labora–Gómez-Serrano–Shi–Staffilani in the constant-viscosity case, is carried out only for the periodic setting via remarks rather than a full treatment. Finally, the gap between the implosion regime $\delta=0$4 and the global regularity known at $\delta=0$5 is not bridged: the behavior of solutions for $\delta=0$6 with strictly positive initial density is left undetermined.

Conclusion

The paper establishes finite-time implosion of smooth, strictly positive-density solutions of the three-dimensional compressible Navier–Stokes equations with power-law degenerate viscosity, for all exponents $\delta=0$7 below an explicit $\delta=0$8-dependent threshold $\delta=0$9. The technical core is a demonstration that, in this regime, the degenerate viscous terms—despite strengthening in high-density regions—decay sufficiently fast in self-similar time to be absorbed by carefully constructed weighted energy and spatial decay estimates, leaving the convective self-similar blow-up mechanism intact. Together with the global regularity known at $\rho\to\infty$0, the result delineates part of the phase diagram of the multidimensional degenerate CNS problem while leaving the intermediate range $\rho\to\infty$1 unresolved.