2D HV-CNS: Hypo-Viscous Compressible Navier–Stokes

• The paper demonstrates that hypo-viscous dissipation with a fractional Laplacian (α ∈ (0,1)) leads to non-uniqueness and intricate asymptotic behaviors in compressible flows.
• It employs advanced harmonic analysis and convex integration techniques to establish global well-posedness and sharp decay rates for strong solutions under subcritical regimes.
• The research details the vanishing viscosity limit, showing convergence to rarefaction wave profiles and highlighting the balance between hyperbolic dynamics and weak dissipative effects.

The two-dimensional isentropic hypo-viscous compressible Navier–Stokes equations (2D HV-CNS) describe the dynamics of a compressible barotropic fluid on the torus or in the whole plane, incorporating a fractional Laplacian viscous dissipation operator with exponent $\alpha \in (0,1)$. Unlike the classical viscous case ($\alpha=1$), this hypo-viscous regime is characterized by weaker regularizing effects, giving rise to new mathematical phenomena, including non-uniqueness of weak solutions and intricate asymptotic behavior. The 2D HV-CNS framework allows for general pressure laws, encompasses both strong and weak solution theories, and has been the subject of intensive analysis in recent literature, notably in (Li et al., 2022, Liang et al., 11 Jan 2026), and (Li et al., 2019).

1. Mathematical Formulation and Hypo-Viscosity

The fundamental system for $$(\rho, u): [0,T] \times \mathbb{T}^2 \to (0,\infty) \times \mathbb{R}^2$$ (2-torus case) reads: $$\begin{aligned} &\partial_t \rho + \operatorname{div}(\rho u) = 0, \ &\partial_t (\rho u) + \operatorname{div}(\rho u \otimes u) + \nabla p(\rho) = -\nu (-\Delta)^\alpha u, \end{aligned}$$ where $p(\rho)$ is a $C^2$ pressure law, $\nu>0$ is viscosity, and $\alpha \in (0,1)$ defines the order of the fractional Laplacian. The operator $(-\Delta)^\alpha$ is taken spectrally: $$\mathcal{F}[(-\Delta)^\alpha f](k) = |k|^{2\alpha} \mathcal{F}[f](k),\quad k \in \mathbb{Z}^2 \setminus \{0\}.$$ In the absence of standard (Laplacian) bulk viscosity, lower-order terms may supplement the momentum equation, but the principal dissipation remains fractional and acts directly on velocity.

For barotropic flows, a common pressure law is $p(\rho) = \rho^\gamma$, $\gamma \geq 1$ (Liang et al., 11 Jan 2026). The quasi-linear nature of the system and the degeneration of hypo-viscous dissipation at high frequencies lead to delicate issues of well-posedness, regularity, and asymptotic analysis.

2. Initial Value Problem and Weak Solution Framework

The Cauchy problem is well-posed for a broad class of initial data:

• $\rho_0 \in L^\infty(\mathbb{T}^2)$ with $\rho_0(x)>0$ almost everywhere,
• $u_0 \in L^2(\mathbb{T}^2)$, equivalently $m_0 = \rho_0 u_0 \in L^2$.

Weak solutions $(\rho, m)$, $m = \rho u$, satisfy:

• $\rho \geq 0$, $\rho \in L_{t,x}^\infty$, $m \in L^2_{t,x}$,
• For all $\phi \in C_c^\infty([0,T) \times \mathbb{T}^2)$ and $$\psi \in C_c^\infty([0,T) \times \mathbb{T}^2;\mathbb{R}^2)$$: $$\int_{\mathbb{T}^2} \rho_0 \phi(0) dx + \int_0^T \int_{\mathbb{T}^2} (\rho\, \partial_t \phi + m \cdot \nabla \phi)\,dx\,dt = 0,$$

$$\int_{\mathbb{T}^2} m_0 \cdot \psi(0) dx + \int_0^T \int_{\mathbb{T}^2} [m \cdot \partial_t \psi + (m \otimes m/\rho) : \nabla \psi + p(\rho)\, \operatorname{div}\psi] dx dt =$$

$$- \nu \int_0^T \int_{\mathbb{T}^2} (-\Delta)^{\alpha/2}(m/\rho) \cdot (-\Delta)^{\alpha/2} \psi\, dx dt + \text{bulk terms}.$$

This definition is compatible with all $\alpha \in (0,1)$. For strong solutions, the analysis is typically conducted in Sobolev spaces $H^s(\mathbb{R}^2)$, $s>1$, with small perturbation from the constant state to preclude vacuum (Liang et al., 11 Jan 2026).

3. Existence, Regularity, and Optimal Decay of Strong Solutions

For subcritical exponents $\alpha \in [1/2, 1)$, global well-posedness and decay theory can be established for strong solutions under small initial perturbation:

• If $(\rho_0-1, u_0) \in H^s$, $s>1$, and $\|(\rho_0-1, u_0)\|_{H^s} \leq \delta$ (for sufficiently small $\delta$), there exists a unique global-in-time solution $(a,u) \in C([0,\infty); H^s)$ with $a = \rho-1$ and energy bound

$E_0(t) + \int_0^t D_0(\tau) d\tau \leq E_0(0)$

with explicit energy and dissipation functionals (cf. Section 5, (Liang et al., 11 Jan 2026)).

Time decay rates for solutions are sharp and governed by the order of fractional viscosity:

• For $(a_0, u_0) \in \dot{B}_{2,\infty}^{-1}$, the decay is

$$\|\Lambda^{s_1}(a,u)(t)\|_{L^2} \leq C (1+t)^{-(s_1+1)/(2\alpha)}$$

for $0 \leq s_1 \leq s$. This matches lower bounds under nonzero spatial mean, confirming the rates are optimal and coincide with the linearized semigroup behavior. When $\alpha \to 1$, these rates recover those of the classical compressible Navier–Stokes system; for $\alpha < 1$ decay is strictly slower, and no $L^2$ decay is possible in the inviscid limit $\alpha \to 0$ (Liang et al., 11 Jan 2026).

The analytic framework draws on Littlewood–Paley frequency decomposition, Fourier splitting methods (Schonbek-type), and commutator estimates (Kato–Ponce lemma).

4. Non-Uniqueness, Thresholds, and Convex Integration

For all $\alpha \in (0,1)$, the 2D HV-CNS admits infinitely many weak solutions for certain initial data, even with fixed $p \in [1,2]$ and Hölder regularity $s \in [0,1)$ such that $\alpha+s-1<0$: $$\exists\, (\rho_0>0, m_0) \in L^\infty \times L^2 \colon \; \text{CNS admits infinitely many } (\rho,m) \in C^0_t C^1_x \cap L^p_t C^s_x \text{ starting from } (\rho_0,m_0).$$ This constitutes the first non-uniqueness result for weak solutions to a viscous compressible fluid (Li et al., 2022).

In the incompressible hypo-viscous Navier–Stokes case (INS), $L^2([0,T];C^0)$ is the critical endpoint for well-posedness: non-uniqueness persists for every $\alpha<1$, and uniqueness holds for $\alpha=1$, thereby identifying $\alpha=1$ as the sharp threshold in $L^2_t C^0_x$ (see the Ladyzhenskaja–Prodi–Serrin criteria).

The non-uniqueness construction is achieved via convex integration, employing approximate solutions $(\rho_q, m_q)$ with small Reynolds stress $R_q$ and iteratively adding high-frequency, intermittent perturbations:

• Spatial building blocks: "Mikado flows" $W^{(k)}(x)$.
• Temporal intermittency and amplitude selection via geometric lemmas.
• The construction ensures $R_q \to 0$ in suitable spaces, so the sequence converges to a weak solution. Density and momentum correctors maintain the compressible structure (Li et al., 2022).

5. Energy Balance and Dissipation Structure

For smooth solutions, hypo-viscous CNS preserves the classical energy-dissipation identity: $$E(t) + \nu \int_0^t \|(-\Delta)^{\alpha/2}u(s)\|^2_{L^2} ds = E(0),\qquad E(t) = \int_{\mathbb{T}^2} \left(\frac12\rho |u|^2 + P(\rho)\right) dx,$$ where $P(\rho) = \rho \int_1^{\rho} \frac{p(r)}{r^2} dr$ is the internal energy.

In convex integration constructions for weak solutions, the energy balance is maintained only in the limit; the defect is controlled by the Reynolds stress through the iterative process, and all limiting solutions dissipate the same hypo-viscous energy: $$\frac{dE}{dt} + \nu \|(-\Delta)^{\alpha/2}u\|_{L^2}^2 = 0.$$ The existence of multiple solutions with the same energy dissipation shows that low-order fractional dissipation ($\alpha<1$) does not restore uniqueness, even in 2D (Li et al., 2022).

6. Vanishing Viscosity Limit and Rarefaction Waves

In regimes where physical viscosity coefficients vanish (the "hypo-viscous scaling"), solutions to the compressible Navier–Stokes system converge to rarefaction wave profiles of the inviscid Euler system. For the 2D setting with Riemann data in $x_1$ and genuinely nonlinear expansion,

$$\sup_{t \in [h,T]} \| (\rho^\varepsilon, u^\varepsilon)(t) - (\rho^r, u^r)(t)\|_{L^\infty} \leq C_{h,T} \varepsilon^{1/6} |\ln \varepsilon|,$$

with $(\rho^r, u^r)$ the planar 2-rarefaction wave and $\varepsilon\to0$ (Li et al., 2019). The technical framework for such convergence utilizes a composite ansatz (involving an approximate rarefaction profile and a hyperbolic correction) and hierarchies of energy estimates in Sobolev spaces. The optimal convergence rate is a consequence of the balance between weak viscosity and hyperbolic flux.

7. Analytical Techniques and Significance

The analysis of 2D HV-CNS integrates methods from harmonic analysis (Littlewood–Paley, commutators), nonlinear semigroup decay, and convex integration. The identification of sharp thresholds for uniqueness and decay, and the construction of non-unique, energy-dissipating solutions, highlight the fundamentally different role of fractional viscosity in compressible flows relative to the classical theory.

The results provide direct evidence that lowering the order of dissipation $(\alpha < 1)$ is insufficient to restore uniqueness or guarantee rapid convergence to equilibrium, even for small, regular data. The interplay between regularity, dissipative mechanisms, and nonlinearity in these equations remains a focal point for ongoing research on the boundary between hyperbolic and parabolic regimes (Li et al., 2022, Liang et al., 11 Jan 2026, Li et al., 2019).