Radial Viscous Shallow Water Equations

• Radially symmetric viscous shallow water equations are compressible fluid models with radial symmetry, incorporating density-dependent viscosity and BD entropy structures.
• The system is derived from the compressible isentropic Navier–Stokes equations with variable viscosity, leading to effective simplifications in both analysis and numerical implementation.
• Recent results confirm global existence and instantaneous shock regularization, demonstrating the robustness of the model even with large, discontinuous initial data.

The radially symmetric viscous shallow water equations describe the evolution of compressible, viscous fluid flows with surface layers, subject to the closure constraint of radial (spherical or cylindrical) symmetry. These systems are derived from the compressible isentropic Navier–Stokes equations with density-dependent viscosities, representing the endpoint case of the Bresch–Desjardins (BD) entropy structure for the specific scaling $\mu(\rho)=\rho$, $\lambda(\rho)=0$. Recent research has established the global existence and remarkable regularization properties of solutions to these equations in both two- and three-dimensional radially symmetric settings, including the persistence and smoothing of initial discontinuities and the global regularity for arbitrarily large initial data (Haspot, 2019, Huanga et al., 17 Dec 2025).

1. Mathematical Formulation and Symmetry Reduction

The global dynamics are governed by the isentropic compressible Navier–Stokes system with variable viscosity: $\begin{aligned} &\partial_t\rho + \div(\rho \bu) = 0\ &\partial_t(\rho\bu) + \div(\rho\bu\otimes\bu) + \nabla p(\rho) = \div(\mu(\rho)\nabla\bu) + \nabla(\lambda(\rho)\div\bu) \end{aligned}$ with pressure law $p(\rho) = \rho^\gamma$ ($\gamma > 1$), viscosity coefficients $\mu(\rho)=\rho$, $\lambda(\rho)=0$.

Imposing radial symmetry,

$\rho(x,t) = H(r,t), \quad \bu(x,t) = u(r,t)\frac{x}{r},\quad r=|x|,$

transforms the system to one spatial dimension (in $r$). For $N=2$ (planar/radial) and $N=3$ (spherical) symmetries, the equations respectively become: $$\partial_t H + \frac{1}{r^{N-1}}\partial_r\left(r^{N-1} H u\right) = 0$$

$$\partial_t(H u) + \frac{1}{r^{N-1}}\partial_r\left(r^{N-1} H u^2\right) + \partial_r\left(\frac{1}{\gamma} H^\gamma\right) = \frac{1}{r^{N-1}}\partial_r\left(r^{N-1} H \partial_r u\right) - (N-1)\frac{H u}{r^2}$$

This reduction is fundamental: the vorticity $\operatorname{curl} \bu$ vanishes, and the system is closed under scalar or radial vector fields, simplifying both the analysis and numerical implementation.

2. Entropy Structures and Effective Velocity

Key analytical control mechanisms arise from the total energy and BD (Bresch–Desjardins) entropy. The effective velocity is introduced as

$v = u + 2\mu \nabla \ln \rho.$

For radially symmetric flows with $\mu(\rho) = \rho$, the BD entropy becomes

$$\mathcal{E}(t) = \int \left[ \frac{1}{2}H (u+\partial_r \ln H)^2 + \frac{1}{\gamma -1}H^\gamma \right] dr,$$

and dissipates with time,

$$\frac{d\mathcal{E}}{dt} + \int H^{\gamma-2}|\partial_r H|^2 dr = 0.$$

Both classical energy and BD entropy facilitate global estimates and regularization effects unique to the variable viscosity case (Haspot, 2019, Huanga et al., 17 Dec 2025). The case $\alpha=1$ in the viscous law $\mu(\rho)=\rho^\alpha$ is a critical endpoint: it allows for strong entropy dissipation without further structure on the bulk viscosity.

3. Global Existence and Regularity Results

Extensive recent advances have established the global existence of weak and classical solutions of the radially symmetric viscous shallow water system, even for arbitrarily large, possibly discontinuous initial data:

• For initial data $H_0 \in H^3(\Omega)$, $u_0 \in H^3(\Omega)$, $H_0 \ge H_*>0$, there exists a unique global classical solution $(H,u) \in C([0, \infty); H^3(\Omega))$ with $H(r,t) \ge H_*/2>0$ $\forall\ r,t$ (Huanga et al., 17 Dec 2025).
• Global weak solutions exist when the initial momentum $m_0 \in \mathrm{bmo}^{-1}(\mathbb{R}^N)$ and the initial density is merely bounded, potentially allowing for initial shocks (Haspot, 2019).
• For well-prepared, sufficiently regular radially symmetric initial data, strong solutions persist globally in time without smallness conditions (Haspot, 2019, Huanga et al., 17 Dec 2025).

A significant feature is the instantaneous Lipschitz regularization of the density: for any $t>0$, $\rho(t, \cdot) \in W^{1, \infty}$. This implies that any initial shock in the density is smoothed out immediately, an effect not present in the constant-viscosity case.

4. Analytical Techniques and Functional Framework

The proofs invoke several advanced analytical tools:

• Parabolic $\mathbf{W}^{1, \infty}$ smoothing: The continuity equation with an effective drift and the parabolic regularization structure yield instantaneous gain of spatial regularity in density.
• Entropy inequalities: Energy and BD entropy yield uniform-in-time bounds for relevant Sobolev and Orlicz norms.
• Koch–Tataru framework: For weak solutions, the initial momentum is placed in $\mathrm{bmo}^{-1}$ and controlled in the corresponding $E_T$ space using heat-kernel energy estimates and fixed-point iteration.
• Weighted Sobolev and interpolation inequalities: In 2D, crucial radial-embedding theorems of the form $H^{1/2}(\Omega)\hookrightarrow L^p(\Omega)$ for all $p<\infty$ enable $L^\infty$ control.
• Lagrangian coordinates: Transformation to mass coordinates provides sharp lower bounds on the fluid depth $H$ and enables control near vacuum.

The radial symmetry plays a critical role, simplifying drift terms and making the maximum principle and certain compactness arguments effective for large data.

5. Central Estimates and Their Implications

The analysis is underpinned by a set of robust a priori inequalities:

• Uniform $L^\infty$ bounds for density via maximum principles and control of effective velocity divergence.
• Upper bounds for energy and entropy with time-integrated dissipation.
• Interpolation estimates (e.g., for $L^\infty$ norms with $r$-weighted $L^2$ derivatives in radial coordinates).
• Lower bounds for positivity of density via Lagrangian arguments.

These allow continuation of solutions to arbitrary times, precluding finite-time breakdown from vacuum, blow-up of $\|\rho-1\|_{L^\infty}$, or loss of integrability of the momentum.

Estimate	Main Ingredient	Role in Analysis
$L^\infty$ density	Parabolic max principle	Prevents blow-up, ensures positivity
$W^{1,\infty}$ regularity	De Giorgi/Schauder, entropy	Smoothing of initial shocks
$E_T$ momentum bounds	Koch–Tataru iteration	Controls infinite-energy initial data
BD entropy	Variable viscosity + structure	Ensures regularity, compactness

6. Comparison with Prior Work

Early theories of compressible Navier–Stokes equations, e.g., the Lions–Feireisl framework, require stronger integrability and regularity (such as $\rho_0 \in H^1$) and constant viscosity, thereby excluding both large discontinuities in density and the BD entropy mechanism (Haspot, 2019). Hoff seminally treated $L^\infty$ initial data but without Lipschitz regularization or BD entropy. Danchin, in the context of critical-Besov spaces, addresses small data but not global well-posedness for large or discontinuous data.

The introduction of variable viscosity and effective velocity, coupled with radially symmetric formulations, enables the treatment of both large and rough data, with strong regularization of shocks—a substantial generalized extension.

7. Open Problems and Future Prospects

Current directions and open problems include:

• Extension to general (non-radial) flows, which requires advanced vorticity control and handling the coupling between vorticity and the effective velocity.
• Analysis of shock front dynamics and discontinuity propagation in higher dimensions.
• Degenerate viscosity laws beyond the $\mu(\rho)=\rho$ endpoint and the inclusion and effect of vacuum.
• Asymptotic stability, convergence rates to equilibrium, and the large-time structure of global solutions.

The radially symmetric viscous shallow water system provides a canonical and technically tractable platform for investigating these deep questions. The intersection of BD entropy, variable viscosities, and radial symmetry currently represents the most advanced understanding of regularization, global existence, and shock dynamics in compressible viscid flows (Haspot, 2019, Huanga et al., 17 Dec 2025).