Global existence and optimal decay for a three-dimensional penalized Navier--Stokes system with biharmonic damping

Technical Summary: Global Existence and Optimal Decay for a Three-Dimensional Penalized Navier–Stokes System with Biharmonic Damping

Problem Statement
This paper investigates a three-dimensional parabolic system serving as a hyperviscous and penalized approximation of the incompressible Navier–Stokes equations. The model addresses the mathematical challenges of the classical Navier–Stokes equations (1.1), particularly the open problem of global regularity and uniqueness for arbitrary initial data in three dimensions. The proposed system (1.3) combines three mechanisms:

1. Classical viscous diffusion ($-\nu\Delta u$).
2. Biharmonic (hyperviscous) damping ($\beta\Delta^2 u$), providing higher-order dissipation.
3. Divergence penalization ($-\frac{1}{\varepsilon}\nabla \text{div} u$), which relaxes the strict incompressibility constraint ($\text{div} u = 0$) by enforcing near-incompressibility as the parameter $\varepsilon \to 0$.

Crucially, the nonlinear convection term is modified using a Temam-type skew-symmetric correction ($N(u) = (u \cdot \nabla)u + \frac{1}{2}u \text{div} u$). This correction compensates for the weak compressibility introduced by the penalization, restoring the $L^2$-cancellation property of the transport term, which is typically lost in compressible formulations.

Methodology
The analysis relies on a combination of energy methods, semigroup theory, and Fourier splitting techniques, with a central focus on deriving estimates that are uniform with respect to the penalization parameter $\varepsilon > 0$.

• Weak Solutions (Theorem 1.3): The existence of global weak solutions for arbitrary $L^2$ initial data is established via a Friedrichs-type Galerkin approximation. The authors project the equation onto Fourier balls using cut-off operators $J_n$ rather than the Leray projector. This distinction is critical; using the Leray projector would annihilate the penalization term, altering the limit problem. Uniform a priori estimates are derived from the energy identity, and compactness is obtained via the Aubin–Lions–Simon lemma.
• Strong Solutions (Theorem 1.4): For small initial data in $H^2(\mathbb{R}^3)$, global strong solutions are proven using a fixed-point argument on the mild formulation associated with the linear semigroup $S_\varepsilon(t) = e^{tL_\varepsilon}$. The linear operator $L_\varepsilon = \nu\Delta - \beta\Delta^2 + \frac{1}{\varepsilon}\nabla\text{div}$ generates a semigroup with smoothing properties. The proof hinges on a uniform $H^2$ energy estimate where the nonlinear terms are absorbed by the viscous and biharmonic dissipation.
• Decay Estimates (Theorem 1.6): Optimal large-time decay rates are derived using the Fourier splitting method (Schonbek). The analysis separates low and high frequencies. The low-frequency behavior is controlled via the Duhamel formula and the specific decomposition of the nonlinearity, while high frequencies are managed through differentiated energy inequalities and Gagliardo–Nirenberg interpolation.

Key Contributions and Results

1. Global Weak Existence: The paper proves the global existence of weak solutions for any initial data $u_0 \in L^2(\mathbb{R}^3)$. The solutions satisfy specific regularity classes, including $L^\infty(0, T; L^2)$, $L^2(0, T; H^2)$, and $L^4(0, T; L^4)$.
2. Global Strong Solutions with Uniform Smallness: For sufficiently small initial data in $H^2(\mathbb{R}^3)$, the system admits a unique global strong solution. A significant contribution is that the smallness threshold $\delta_0$ and the associated energy bounds are independent of $\varepsilon$. This uniformity ensures the results apply to the entire family of penalized systems, not just a fixed instance.
3. Optimal Decay Rates: For initial data in $L^1(\mathbb{R}^3) \cap H^2(\mathbb{R}^3)$, the authors derive optimal decay estimates:
   $$\|u(t)\|_{L^2} \leq C(1+t)^{-3/4}$$
   $$\|\nabla^k u(t)\|_{L^2} \leq C(1+t)^{-3/4 - k/2}, \quad k=1, 2$$
   These rates match the asymptotic decay of the classical heat equation in three dimensions. The analysis confirms that while the biharmonic term dominates high-frequency dissipation, the large-time decay is governed by the heat-flow behavior of low frequencies.
4. Uniformity in $\varepsilon$: A defining feature of the analysis is that all constants and estimates remain uniform as $\varepsilon \to 0$. The penalization term $\frac{1}{\varepsilon}\|\text{div} u\|^2$ is treated as a non-negative dissipation term that is either retained or discarded, never bounded from above by a quantity involving $1/\varepsilon$.

Significance and Claims
The paper claims to provide a "stable and rigorous analytical foundation" for the penalized approximation of incompressible flows. By incorporating the biharmonic damping and the Temam correction, the model achieves improved well-posedness results compared to the classical Navier–Stokes equations, specifically regarding the availability of unconditional $L^2$ energy estimates and higher-order smoothing.

The authors note that their uniform estimates suggest that as $\varepsilon \to 0$, the strong solutions converge to a limit solving an incompressible hyperviscous Navier–Stokes system. However, they explicitly state that identifying the limit system and proving convergence rates in $\varepsilon$ are directions left for future work. Furthermore, while global regularity for arbitrary data is known for the incompressible hyperviscous system (due to Lions), the authors do not claim such a result for the penalized system with large data, noting that the skew-symmetric compressible structure modifies higher-order energy estimates in a way that prevents a direct extension of the large-data result.

The work distinguishes itself from previous studies on damped Navier–Stokes equations (e.g., with absorption terms) by focusing on a higher-order damping mechanism and a penalty method that eliminates the pressure term, thereby simplifying the weak formulation by removing the divergence-free constraint on test functions.