Quantum Navier-Stokes Equations

• Quantum Navier-Stokes (QNS) equations are quantum hydrodynamic models that extend classical fluid dynamics by integrating Bohm potential and cold pressure for vacuum control.
• Analytical techniques like cold pressure regularization and effective velocity transforms ensure global existence and stability of finite-energy weak solutions.
• The vanishing quantum limit bridges quantum and classical regimes, demonstrating convergence of QNS models to classical Navier-Stokes dynamics.

Quantum Navier-Stokes (QNS) equations are quantum hydrodynamic models that extend classical Navier-Stokes theory by incorporating quantum mechanical effects, primarily through dispersive (Bohm) terms originating from the Madelung transform. These systems describe dissipative quantum fluids, such as superfluids, electron gases, or cold-atomic gases where kinetic and quantum pressure (non-local, dispersive) effects are significant. The leading mathematical structure is the compressible or incompressible Navier-Stokes system augmented by a third-order quantum term, typically of the form $\rho\nabla(\Delta\sqrt{\rho}/\sqrt{\rho})$, and in certain cases, additional mechanisms to control vacuum and degeneracy.

1. Mathematical Structure and Key Terms

The canonical barotropic compressible QNS system for density $\rho(t,x)>0$ and velocity $u(t,x)\in\mathbb R^d$ on a periodic domain $T^d$ (with $d\le3$) is

$$\begin{cases} \partial_t\rho + \nabla\cdot(\rho u) = 0, \ \partial_t(\rho u) + \nabla\cdot(\rho u \otimes u) + \nabla p(\rho) + \nabla p_s(\rho) = 2\nu \nabla\cdot(\rho D(u)) + \epsilon^2\rho \nabla\left(\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}\right), \end{cases}$$

where $D(u)=\frac12(\nabla u+\nabla u^T)$, $p(\rho)=\rho^\gamma$, $\gamma>1$, and $\nu>0$ is the viscosity coefficient. The scaled Planck constant $\epsilon>0$ quantifies quantum dispersion. A key addition for analytical purposes is the singular "cold" pressure $p_s(\rho)$, which penalizes low densities (vacuum): $$p_s'(\rho)= \begin{cases} c\rho^{-4k-1}, & 0<\rho\le1,\ k>1,\ \rho^{\gamma-1}, & \rho>1. \end{cases}$$ This ensures strict positivity of density ($\rho>0$ everywhere).

The quantum term $$\epsilon^2\rho\nabla(\Delta\sqrt{\rho}/\sqrt{\rho})$$—the Bohm potential—originates from the Madelung transformation of complex-valued Schrödinger-type evolution equations and represents quantum pressure or capillarity effects. In weak form, this term is commonly expressed as $\epsilon^2\nabla\cdot(\rho\nabla^2\log\rho)$.

2. Finite-Energy Weak Solutions and Fundamental Estimates

Weak solutions $(\rho,u)$ are defined via standard integral identities for mass and momentum:

• For all scalar tests $\varphi\in C_c^\infty$, $$\int_0^T\int_{T^d} (\rho\partial_t\varphi+\rho u\cdot\nabla\varphi)\,dx\,dt+\int_{T^d}\rho_0\varphi(0)=0$$.
• For all vector tests $\psi\in C_c^\infty$,

$$\int_0^T\int ( \rho u\cdot\partial_t\psi + \rho u\otimes u : \nabla\psi + (p(\rho)+p_s(\rho))\nabla\cdot\psi ) + 2\nu \int_0^T\int \rho D(u):\nabla\psi - \epsilon^2\int_0^T\int \rho\nabla^2\log\rho\cdot\psi = \int_{T^d}\rho_0u_0\cdot\psi(0).$$

Key a priori bounds, holding uniformly in the quantum parameter $\epsilon$, are:

• Energy inequality:

$$E(\rho,u)(t)+2\nu\int_0^t\int \rho|D(u)|^2 \leq E(\rho_0,u_0),$$

with $$E(\rho,u) = \int_{T^d} \left( \frac12\rho|u|^2 + H(\rho)+H_s(\rho)+2\nu^2|\nabla\sqrt{\rho}|^2 \right) dx$$, $H''(\rho)=p'(\rho)/\rho$, $H_s''(\rho)=p_s'(\rho)/\rho$.

• BD-entropy (effective velocity) estimate: for $w=u+\nu\nabla\log\rho$,

$$\int_0^T\int \rho|w|^2 + \nu|\nabla\sqrt{\rho}|^2 + \mathrm{const}\cdot\epsilon^2 \int_0^T\int |\Delta\sqrt{\rho}|^2 \leq C(E(\rho_0,u_0)).$$

The cold pressure term enables use of the standard weak-solution framework by prohibiting vacuum.

Function spaces for analysis include $L^\infty(0,T;L^\gamma)$ for $\rho$, $L^2(0,T;H^1)$ for $\sqrt{\rho}u$, and $L^\infty(0,T;L^2)$ for $\nabla\sqrt{\rho}$.

3. Analytical Techniques and Global Existence

The proof strategy, as established by Gisclon & Lacroix-Violet (Gisclon et al., 2014), proceeds as follows:

1. Cold Pressure Regularization: Singular $p_s(\rho)$ ensures positivity and analytic control near vacuum.
2. Effective Velocity: Transform $u \mapsto w = u+\nu\nabla\log\rho$ eliminates degeneracy and rewrites the system to expose parabolic structure.
3. Approximation Schemes: Use Galerkin methods in velocity, fixed points for the mass equation, regularizations (higher-order parabolic smoothing, truncation of highly nonlinearities) to construct smooth approximations.
4. Uniform Estimates: Derive uniform (in approximations) energy and BD-entropy bounds. Cold pressure enables uniform $L^\infty$ bounds for $\rho^{-k}$.
5. Compactness: Apply Lions–Feireisl–Mellet–Vasseur theoretical machinery to promote strong convergence of density and momentum, using Aubin–Simon compactness argument.
6. Limit Passage: Justify convergence in all nonlinear terms (pressure, cold pressure, viscosity, and Bohm term) via strong/weak convergences, ultimately yielding global existence of finite-energy weak solutions.

4. Vanishing Quantum Limit and Relation to Classical NS

A central result is the rigorous justification of the limit $\epsilon\to 0$, i.e., vanishing quantum effects. The uniformity of all a priori bounds in $\epsilon$—in particular, the control on $\epsilon^2\|\Delta\sqrt{\rho}\|_{L^2}^2$—permits strong convergence in $(\rho,u)$ as $\epsilon\to0$, while the quantum Bohm term drops out. One obtains, in the limit, a solution to the classical compressible Navier-Stokes system (including cold pressure for vacuum control): $$\partial_t\rho+\nabla\cdot(\rho u)=0,\ \partial_t(\rho u)+\nabla\cdot(\rho u\otimes u)+\nabla p(\rho)+\nabla p_s(\rho)=2\nu\nabla\cdot(\rho D(u)).$$

The cold pressure term continues to regularize the solution: vacuum remains excluded due to divergence of $p_s$ as $\rho\to 0$.

5. Physical and Mathematical Significance

The Quantum Navier-Stokes framework codifies interaction between dissipative (viscous), compressible fluid dynamics and quantum mechanical phenomena—quantum pressure regularizes density and counteracts concentration singularities. The analytical toolbox developed here enables existence theory and stability analysis for quantum fluids even with arbitrary adiabatic exponents $\gamma>1$ and in dimensions $d\leq3$, provided appropriate control near vacuum. Cold pressure regularization, effective velocity transforms, and the uniform estimates in $\epsilon$ are essential for treating the passage between quantum and classical regimes.

These results embed the QNS theory into the broader context of degenerate compressible fluid equations, create a unified framework for quantum–classical limits, and inform the mathematical investigation of quantum hydrodynamic models with density-dependent viscosities and dispersive corrections.

6. Connections to Broader Quantum Hydrodynamics

The QNS equations as developed by Gisclon & Lacroix-Violet (Gisclon et al., 2014) and related works form the analytical foundation for more elaborate models, including:

• Quantum Navier-Stokes-Poisson (QNSP) systems for charged quantum fluids or plasmas.
• Multi-dimensional generalizations, domain extensions (whole space, domains with boundary), and confining potentials.
• Hybrid quantum–classical numerical algorithms for simulating fluid flow in quantum hardware environments, including preconditioned quantum linear solvers (see, e.g., (Song et al., 2024)).
• Quantum hydrodynamics in materials (e.g., graphene) and low-dimensional settings, and transport coefficient computations in integrable or nearly integrable quantum gases.

The QNS equations establish the crucial analytical and physical bridge between nonlinear quantum mechanics (Schrödinger/Madelung dynamics) and viscous fluid hydrodynamics under the regime where quantum dispersion and hydrodynamic dissipation are both operative.