---
title: '2D HV-CNS: Hypo-Viscous Compressible Navier–Stokes'
url: https://www.emergentmind.com/topics/2d-isentropic-hypo-viscous-compressible-navier-stokes-equations
type: topic
---

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

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 [2212.05844], [2601.06889], and [1902.11166].

## 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 $p(\rho) = \rho^\gamma$, $\gamma \geq 1$ [2601.06889]. 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 [2601.06889].

## 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, [2601.06889]).

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 $\alpha \to 0$ [2601.06889].

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 [2212.05844].

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 [2212.05844].

## 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 [2212.05844].

## 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 $\varepsilon\to0$
[1902.11166]. 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 [2212.05844], [2601.06889], [1902.11166].

Source: https://www.emergentmind.com/topics/2d-isentropic-hypo-viscous-compressible-navier-stokes-equations