Critical Blow-Up Curve in Radial Navier-Stokes

• Critical Blow-Up Curve is defined as the precise threshold of bulk viscosity exponents that separate uniform density bounds from potential blow-up in radial compressible flows.
• The analysis employs ODE comparisons, Hardy-type inequalities, and energy dissipation arguments to derive these sharp criteria under Dirichlet boundary conditions.
• Results improve previous criteria by showing that uniform density bounds hold if β exceeds max{1, (γ+2)/4}, thereby guiding strong solution theory and fluid simulation models.

The critical blow-up curve, within the context of radially symmetric compressible Navier-Stokes equations on two-dimensional solid balls, denotes the precise boundary in the space of viscosity law exponents beyond which the solution’s key regularity property transitions—specifically, the uniform boundedness in time of the mass density $\rho$. Its determination is fundamental to global strong solvability and lies at the heart of the dichotomy between finite-time breakdown versus global regularity in nonlinear compressible flows subject to radial symmetry and density-dependent viscosity (Huang et al., 2023).

1. Radially Symmetric Setup and Reduction

The analysis is carried out for the compressible isentropic Navier-Stokes system posed on the two-dimensional ball $\Omega = \{x \in \mathbb{R}^2 : |x| < R\}$, with velocity Dirichlet boundary condition $u=0$ on $|x|=R$. The unknowns are the density $\rho(x,t)$ and velocity $u(x,t)$ which are constrained to be radially symmetric: $$\rho(x,t) = \rho(r,t), \quad u(x,t) = u(r,t)\,\frac{x}{r}, \quad r = |x|.$$ Initial data are prescribed as scalar functions on $[0,R]$, with regularity $\,\rho_0 \in W^{1,q}(\Omega),\,u_0 \in H^1(\Omega)$ for $q>2$, and strict compatibility at boundary and center, notably $u_0(0) = u_0(R) = 0$. The pressure law is $P(\rho) = A\rho^\gamma$, shear viscosity is constant ($\mu$), and bulk viscosity polynomial: $\lambda(\rho) = b\rho^\beta$.

Under the radial ansatz, the system reduces to

$$\begin{aligned} &\rho_t + (\rho u)_r + \frac{1}{r}\rho u = 0, \ &(\rho u)_t + (\rho u^2)_r + \frac{1}{r}\rho u^2 + P_r = \left[(2\mu + \lambda(\rho))(u_r + \frac{u}{r})\right]_r, \end{aligned}$$

with initial–boundary setup as above.

2. Critical Exponent and Blow-Up Curve: Definition and Calculation

The critical blow-up curve is defined by the set of viscosity exponents $\beta$ in the bulk viscosity law $\lambda(\rho) = b\rho^\beta$ that separate global-in-time uniform boundedness of the density from potential blow-up (loss of control).

The main theorem in (Huang et al., 2023) establishes that:

• Global strong solutions exist, for arbitrary large initial smooth data, if $\beta>1$ and $\gamma>1$.
• Uniform $L^\infty$ density bounds are ensured only if

$\beta > \max\left\{1,\,\frac{\gamma+2}{4}\right\},$

which is an improved criterion compared to previous results for general domains where $\beta>\frac{4}{3}$ was needed.

This curve (i.e., the function $\beta_c(\gamma) = \max\{1,\,(\gamma+2)/4\}$) precisely tracks the transition between global regularity and possible density blow-up.

Table: Critical Blow-Up Curve for Uniform Density Bounds

Adiabatic Exponent $\gamma$	Critical Bulk Viscosity Exponent $\beta_c(\gamma)$
$1 < \gamma \leq 2$	$\frac{\gamma+2}{4}$
$\gamma \geq 2$	$1$

This boundary emerges from detailed ODE-comparison and energy dissipation arguments in the proof structure. When $\beta > \beta_c(\gamma)$, uniform-in-time $L^\infty$ control on the density follows, precluding finite-time blow-up scenarios.

3. Analytical Mechanism for Blow-Up Versus Uniform Boundedness

The global strong solvability proof leverages the structure of the radial reduction. Key steps include:

• The derivation of a “density–moment” ODE for

$$D_t [ 2\rho \ln \rho + (\rho^\beta-1) ] = -P(\rho-1) + \text{boundary terms}$$

• Control of weighted commutator terms specific to radial symmetry, via Hardy-type inequalities (e.g., bounding $r^{-1}u$ in terms of $u'$, given $u(0)=0$).
• Boundary integral estimates in $r$, relying on the vanishing velocity at $r=0,R$ and singularity smoothing due to the radial weight.
• Application of Zlotnik-type differential inequality arguments, exploiting the above ODE and explicit interpolation constants derived from the parameters ($\gamma$, $\beta$).
• The threshold $\beta_c(\gamma)$ falls out as the minimal bulk viscosity exponent for which the accumulation of boundary terms is dominated by dissipation, precluding blow-up.

4. Comparison with General-Domain and Related Models

The critical blow-up curve obtained for radially symmetric 2D balls improves existing results for more general two-dimensional domains. For example, prior global existence required $\beta>\frac{4}{3}$ for arbitrary domains, but the radial geometry allows sharp, reduced constraints.

Moreover, the exact value of $\beta_c(\gamma)$ depends on dimension, symmetry, and boundary conditions. In higher dimensions, or in the presence of swirling flows or periodic/non-radial data, the blow-up curve may shift, and additional technical difficulties arise related to concentration phenomena and singularity formation.

Analogous radial-critical curves appear in related models such as the compressible MHD system (Huang et al., 2023), ultra-relativistic hydrodynamics (Kunik et al., 2024), and pressureless gas dynamics (Nedeljkov et al., 2016), though their criteria involve different structural balances.

5. Significance and Applications

Understanding and precisely determining the critical blow-up curve has several direct implications:

• It explicitly quantifies the threshold for global regularity under density-dependent viscosity, providing sharp existence and blow-up criteria for strong solutions.
• It underlies the mechanism preventing shock or singularity formation due to density concentration in radially symmetric compressible flows.
• It guides design and analysis of physically meaningful bulk viscosity laws for simulation and modeling of real fluids, where radial symmetry is often an imposed feature (e.g., spherical implosions, astrophysical phenomena).

6. Generalizations and Open Directions

While (Huang et al., 2023) establishes the critical curve for 2D radial flows under Dirichlet conditions, several open directions emerge:

• Extension to non-radial geometries or flows with swirl/tangential components, where the critical curve may differ or become non-explicit.
• Influence of boundary conditions: Neumann or mixed boundary prescriptions alter the analytic structure.
• Coupling with additional physics (magnetic fields, radiation, multi-phase effects), which may shift the critical exponents or admit new blow-up phenomena.
• Rigorous characterization of borderline ($\beta=\beta_c(\gamma)$) behavior, where uniform boundedness may fail or only weaker forms of existence persist.

7. Literature Context

The critical blow-up curve for radially symmetric compressible Navier-Stokes equations is rooted in the analytic tradition launched by Kazhikhov, further strengthened by the sharp $L^\infty$ bounds and global strong existence proofs of Huang–Su–Yan–Yu (Huang et al., 2023). Their work marks a substantial advancement over earlier results and provides a concrete functional delineation for practitioners and theorists exploring strong solution regularity in compressible fluid systems.