Axisymmetric Compressible Navier–Stokes Equations

• Three-dimensional axisymmetric compressible Navier–Stokes equations are a set of PDEs modeling cylindrical flows invariant under rotation with challenges from compressibility, vacuum, and complex viscosity effects.
• The system incorporates a power-law bulk viscosity and slip boundary conditions that enable the establishment of global strong solutions even for large initial data and the presence of vacuum.
• Rigorous analysis reveals exponential decay rates and convergence to the incompressible limit under high viscosity, providing practical insights for simulations and engineering applications.

The three-dimensional axisymmetric compressible Navier–Stokes equations describe the motion of compressible Newtonian fluids that are invariant under rotations about a preferred axis, restricting the dependence of all field variables to radius, axial position, and time, and enforcing invariance with respect to the azimuthal coordinate (possibly with nontrivial swirl). These equations are fundamental for modeling and analysis in fluid dynamics, especially for flows exhibiting geometric symmetry, and their analysis involves unique mathematical and physical challenges due to compressibility, possible vacuum, boundary effects, and viscosity structure. Recent advances provide a rigorous global well-posedness theory—remarkably even for large initial data and in the presence of vacuum—for physically realistic viscosity laws and natural boundary conditions.

1. Mathematical Formulation of the Axisymmetric Compressible Navier–Stokes System

In cylindrical coordinates $(r, \theta, z)$, the compressible Navier–Stokes system for a barotropic fluid, assuming axisymmetry (i.e., all quantities independent of $\theta$) and possibly nonzero swirl, can be expressed as: $$\begin{aligned} & \rho_t + \operatorname{div}(\rho \mathbf{u}) = 0, \ & (\rho \mathbf{u})_t + \operatorname{div}(\rho \mathbf{u} \otimes \mathbf{u}) + \nabla P(\rho) = \mu \Delta \mathbf{u} + \nabla \left((\mu + \lambda(\rho)) \operatorname{div} \mathbf{u} \right). \end{aligned}$$ The viscosity coefficients include a constant shear viscosity $\mu>0$ and a bulk viscosity $\lambda(\rho)=b \rho^\beta$, $\beta>4/3$, $b>0$. The domain is a cylinder (excluding the axis), with the fields periodic in $z$ and subject to slip boundary conditions (Navier type) on the lateral boundary. The pressure admits a polytropic or more general state equation, such as $P(\rho)=a \rho^\gamma$ for isentropic flows. The system is equipped with initial conditions that may admit vacuum states and allow large oscillations.

Boundary conditions are typically specified as: $$\mathbf{u} \cdot \mathbf{n} = 0, \qquad \mathbf{u} \times \mathbf{n} = -K \mathbf{u} \quad \text{on } \partial \Omega,$$ where $\mathbf{n}$ is the outward normal and $K$ a symmetric matrix on the boundary, enforcing impermeability and tangential slip damping.

2. Existence Theory in the Large Data and Vacuum Regimes

A major development, exemplified in (Lei, 14 Sep 2025) and (Lei, 16 Sep 2025), is the global existence of strong (and, in the presence of additional regularity, classical) solutions for the axisymmetric compressible Navier–Stokes system with large initial data and possible vacuum, under realistic (non-degenerate) viscosity structure. The main findings are:

• Strong/weak solution existence: For arbitrary (possibly large) initial data—$0 \leq \rho_0 \in W^{1,q}$ ($q>3$ for strong solutions), $u_0 \in \widetilde{H}^1$ (the usual Sobolev space incorporating the slip boundary condition)—and vacuum-allowing initial density, there exists a unique global strong solution $(\rho,u)$ in the cylindrical domain, periodic in the axis, for all $t>0$, provided either $\lambda(\rho)=b\rho^\beta$, $\beta>4/3$ (with $b>0$, constant $\mu>0$) (Lei, 14 Sep 2025), or the total bulk viscosity combination $\nu=2\mu+\lambda$ is sufficiently large (Lei, 16 Sep 2025). If only $0\leq \rho_0\in L^\infty$ and $u_0\in \widetilde{H}^1$, global weak solutions are obtained.
• Uniform-in-time bounds: The density $\rho$ remains uniformly bounded in $L^\infty$, preventing blow-up of solutions; in the case of vacuum, certain higher gradients (e.g., $\nabla \rho$ if vacuum persists) may exhibit exponential-in-time growth, but $\rho$ stays globally bounded.
• Exponential decay: For any $p\in[1,\infty)$, exponential decay rates are established: $$\|\rho(\cdot,t) - \rho_\infty\|_{L^p} + \|u(\cdot,t)\|_{L^p} \leq C e^{-\alpha_0 t},$$ where $\rho_\infty$ is the asymptotic constant density, $C$, $\alpha_0$ depend on problem parameters and data norms.
• Independence from data size (for suitably strong viscosity): In both frameworks, no smallness restriction on the initial data is needed—large data are fully allowed.

3. A Priori Estimates and the Role of Bulk Viscosity

Central to these results are energy and higher-order a priori estimates exploiting the viscosity structure:

• Effective viscous flux: Define the "effective viscous flux" $$G=(2\mu+\lambda(\rho))\operatorname{div}u-(P(\rho)-P(\rho_0))$$. Uniform $L^p$ or $L^\infty$ estimates for $\rho$ are derived by controlling $G$ using elliptic estimates and suitable commutator bounds, leveraging the strong ellipticity from the bulk viscosity.
• Uniform density bounds: The density bound is specifically achieved using the power-law bulk viscosity (with $\beta>4/3$), which ensures enough dissipation in the div$\,u$ component to counteract compressibility-induced growth. If the bulk viscosity is sufficiently large ($\nu=2\mu+\lambda \geq \nu_1$ for some threshold $\nu_1$), density oscillations are uniformly damped.
• Decay rates: Both the energy inequality and higher-order estimates (involving $\nabla u$, $\nabla^2 u$, and $\dot{u}$, where $\dot{u}=u_t+u\cdot\nabla u$) incorporate exponential time-decay factors inversely proportional to the viscosity, with explicit rates $\alpha_0=D_0/\nu$ for $D_0>0$.
• Vacuum handling: If initial density vanishes somewhere, solutions exist and are regular, but the spatial density gradient may grow exponentially. However, the $L^\infty$ control ensures global well-posedness regardless of initial vacuum regions.
• Boundary regularity: The slip boundary condition is compatible with all analysis stages, avoiding technical difficulties posed by non-slip in axisymmetric domains and ensuring the energy method's viability even near the boundary.

4. Incompressible Limit under Large Bulk Viscosity

The incompressible regime is rigorously reached as the bulk viscosity dominates:

• Convergence to inhomogeneous incompressible system: As $\nu=2\mu+\lambda(\rho)\rightarrow\infty$, solutions $(\rho^\nu,u^\nu)$ converge (possibly up to a subsequence) to $(\rho,u)$ solving the inhomogeneous incompressible Navier–Stokes system,

$$\begin{aligned} & \rho_t + \div(\rho u) = 0, \ & u_t + u\cdot\nabla u - \mu\Delta u + \nabla\pi = 0, \ & \div\,u = 0, \end{aligned}$$

with the same slip boundary condition and initial density, modulo strong topology as regularity allows (Lei, 16 Sep 2025).

• Quantitative convergence: In the special case $u_0=0$, the convergence rate is $u^\nu = \mathcal{O}(\nu^{-1/2})$ in $$L^2(\Omega\times(0,\infty))\cap L^\infty(0,\infty;L^2(\Omega))$$.
• Global strong solution convergence: If $\rho_0\in W^{1,q}$, $q>2$, all solutions converge to the unique global strong solution of the incompressible limit.

5. Relation to Prior and Parallel Theories

These results stand in sharp contrast to classical compressible Navier–Stokes theory, where global strong solutions in 3D with large initial data and vacuum are not available except under more restrictive or degenerate viscosity assumptions. The following features are emphasized:

• Comparison to Lions–Feireisl theory: The uniform-in-time $L^\infty$ control of density relies essentially on the power-law or large bulk viscosity; without it, ill-posedness or finite-time blowup may occur.
• Axisymmetry as a technical advantage: By working in a domain excluding the axis and assuming axisymmetry, the analysis reduces to two spatial variables, facilitating elliptic estimates and estimates for the effective viscous flux.
• Robustness to vacuum: The methods do not require a lower density bound at initial time; propagation of regularity and exponential energy decay hold even starting from vacuum.
• Applicability to numerics and physics: The existence, uniqueness, and explicit decay properties shown provide theoretical justification for the stable simulation and analysis of high-speed, axisymmetric compressible flows in engineering and physical applications, especially when vacuum or strong compressibility effects are present.

6. Summary Table: Main Theoretical Outcomes

Parameter	Condition	Analytical Consequence
Bulk viscosity $\lambda$	$\lambda(\rho)=b\rho^\beta$, $\beta>4/3$ ($b>0$); or $\nu:=2\mu+\lambda \gg 1$	Global strong/weak solution existence—even with vacuum and large data; exponential decay to equilibrium
Initial density $\rho_0$	$0\leq \rho_0\in W^{1,q}/L^\infty$, possibly vanishing	No lower bound needed; regularity propagation ensured
Initial velocity $u_0$	$u_0\in \widetilde{H}^1$, axisymmetric, periodic in $x_3$	Compatibility with slip boundary; full a priori estimate machinery available
Long-time regime	Any $p\in[1,\infty)$, $t\to\infty$	$\|\rho-\rho_\infty\|_{L^p}$, $\|u\|_{L^p}$ decay exponentially: $Ce^{-\alpha_0 t}$

7. Perspectives and Outlook

The global theory for the three-dimensional axisymmetric compressible Navier–Stokes equations with large initial data and vacuum is now established when a strong enough density-dependent bulk viscosity (or sufficiently large total bulk viscosity) is present (Lei, 14 Sep 2025, Lei, 16 Sep 2025). The sliding boundary condition (slip) is mathematically tractable and physically reasonable for cylindrical domains, preventing boundary layers from obstructing energy dissipation arguments. The explicit exponential decay rates toward equilibrium and the rigorous incompressible limit analysis under large bulk viscosity further elucidate the dissipative mechanisms in compressible fluids with axial symmetry. A plausible implication is that similar strategies—leveraging weighted estimates, elliptic regularity, and effective viscous flux methods—can be adapted to other symmetry-reduced or high-viscosity flows where classical compressible theory remains incomplete.