Quantum Navier-Stokes Equations

• Quantum Navier-Stokes equations are compressible fluid models that integrate viscous stresses and quantum dispersive effects via the Bohm potential and capillarity.
• They serve as hydrodynamic closures for quantum kinetic models, applicable in superfluids, quantum gases, and semiconductor devices.
• Advanced techniques like effective velocity reformulation and BD entropy ensure global solution existence and prevent vacuum formation under stochastic forcing.

Quantum Navier-Stokes Equations (QNS) represent a class of compressible fluid models incorporating both conventional viscous stresses and quantum dispersive effects, typically implemented via the Bohm potential or Korteweg-type capillarity. These equations serve as hydrodynamic closures for quantum kinetic models, with rigorous mathematical theory and substantial physical motivation in superfluids, quantum gases, semiconductor devices, and stochastic hydrodynamics. Recent advances have focused on global existence, regularity, stochastic forcing, effective velocity methods, and vanishing-dispersion limits in both deterministic and random environments.

1. Mathematical Structure of the One-Dimensional Stochastic QNS

The core one-dimensional stochastic quantum Navier–Stokes system, as established on the flat torus $\mathbb{T}=[0,1]$ with periodic boundary conditions, comprises:

• Fluid density $\rho(t,x) > 0$
• Velocity $u(t,x) \in \mathbb{R}$
• Pressure law $p(\rho) = \rho^\gamma$ for $\gamma > 1$
• Viscosity $\mu(\rho) = \rho^\alpha$, $\alpha \ge 0$
• Quantum capillarity: Bohm potential reduces in 1D to

$$K(\rho) = \rho \partial_x\Bigl(\frac{\partial_{xx}\sqrt{\rho}}{\sqrt{\rho}}\Bigr) = \rho \partial_x(\partial_{xx}\log\rho)$$

The system is stochastically forced via a cylindrical Wiener process $W(t) = \sum_{k \ge 1} e_k W_k(t)$ in a separable Hilbert space $\mathcal{U}$, leading to: $\begin{align} d\rho + \partial_x(\rho u)\,dt &= 0 \tag{1.1a} \ d(\rho u) + \partial_x(\rho u^2 + p(\rho))\,dt &= [4\gamma\mu\,\partial_x(\rho\,\partial_x u) + \rho\,\partial_x(\partial_{xx}\log\rho)]\,dt + G(\rho,\rho u)\,dW \tag{1.1b} \end{align}$ The noise coefficient $G(\rho, \rho u)$ satisfies

$$G_k(\cdot,0,0) = 0, \quad |G_k| \le g_k (\rho + |q|),\quad |\partial_{x,\rho,q}^m G_k| \le g_k, \quad \sum_k g_k^2 < \infty$$

ensuring requisite smoothness and growth conditions for stochastic analysis.

2. Functional Setting and Data Assumptions

The analysis is performed on a filtered probability space $$(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\ge 0}, \mathbb{P})$$. The initial data is given by an $\mathcal{F}_0$-measurable random variable

$$(\rho_0, u_0) : \Omega \to H^{s+1}(\mathbb{T}) \times H^{s}(\mathbb{T}), \quad s > 2.5$$

subject to the nondegeneracy and regularity conditions

$$0 < C^{-1} \le \rho_0(x) \le C < \infty, \quad \rho_0 \in H^{s+1}, \; u_0 \in H^s, \quad \mathbb{E}[ \|\rho_0\|_{H^{s+1}}^p + \|u_0\|_{H^s}^p ] < \infty, \; \forall p \ge 1$$

3. Local and Global Well-Posedness: Analytical Approach

Local well-posedness (Theorem 2.3) is demonstrated by the following steps:

• Nonlinearity regularization: Approximation via cutoff functions $\chi_R(\|(\rho, u)\|_{W^{2,\infty}})$ to render the nonlinearities globally Lipschitz.
• Galerkin discretization: Space discretization by Fourier truncation to $H_m$.
• Uniform high-order energy estimates: In $H^s$, uniform in Galerkin dimension $m$.
• Compactness and martingale solutions: Pathwise limit via Prokhorov–Skorokhod, pathwise uniqueness for $s > 5/2$ (ensured by $H^s \hookrightarrow W^{2,\infty}$).
• Gyöngy–Krylov Principle: Pathwise solutions obtained from martingale ones.

Globality (Theorem 2.4) leverages higher-order a priori bounds and, crucially, the Bresch–Desjardins (BD) entropy.

4. Bresch–Desjardins Entropy, Effective Velocity, and Regularity Mechanisms

Key to control and prevention of vacuum formation is the effective velocity

$v = u + \partial_x \log \rho$

with BD entropy functional

$$\mathcal{E}_{\rm BD}(t) = \int_{\mathbb{T}} \left\{ \tfrac12 \rho |v|^2 + H(\rho) \right\} dx, \quad H(\rho) = \frac{\rho^\gamma}{\gamma - 1}$$

The entropy inequality (informal, up to noise terms) is

$$\frac{d}{dt} \int (\tfrac12 \rho |v|^2 + H(\rho))^p dx + \int \rho |\partial_x v|^2 (\tfrac12 \rho |v|^2 + H)^{p-1}\,dx \leq \text{noise terms}$$

which implies:

• $L^2_{t,x}$ control over $\partial_x \log\rho$, $\partial_{xx} \log\rho$
• Uniform in time $L^q_{t,x}$ bounds on $\rho$, $1/\rho$
• High-order $H^s$ norm estimates by commutator technique

The local strong solution is continued globally because the norms $\|\rho,u\|_{W^{2,\infty}}$ are shown not to blow up in finite time, and a uniform positive lower bound for $\rho$ is established, preventing vacuum genesis.

5. Uniqueness, Continuation, and No-Vacuum Propagation

The absorption of all possible blow-up scenarios into stopping times $T_R$ (at which $\|(\rho,u)\|_{W^{2,\infty}}$ reaches threshold $R$) ensures that, under BD-entropy and a-priori energy, these stopping times tend to $+\infty$ almost surely, yielding global solutions with persistent positivity of density. This machinery is fundamentally enabled by the BD-entropy, which structurally rules out vacuum even in the presence of stochastic drivers.

6. Stochastic QNS: Physical Relevance and Mathematical Innovations

• The results supply, up to the time of publication, among the first global-in-time strong solutions for 1D stochastic compressible fluids with quantum (Korteweg) capillarity—where the density is guaranteed nonvanishing, and all derivatives are properly controlled for all $t \ge 0$.
• The stochastic QNS model is physically relevant for quantum hydrodynamics with random forcing, such as 1D superfluid flows, Bose-Einstein condensates, and related semiconductor/quantum device models.
• The interplay of deterministic energy and BD entropy bounds with stochastic forcing is nontrivial, as noise can be density-dependent and physically nonisotropic.
• The work establishes a robust probabilistic pathwise theory extending previous deterministic QNS results to the stochastic setting.

7. Methodological and Future Directions

Major analytic tools and possible research extensions include:

• Invariant Measures & Ergodicity: Building stochastic QNS as a dissipative system where rigorous long-time invariants and stochastic attractors can be constructed.
• Higher dimensions and BD structure: Extension to $d\ge 2$ is fundamentally obstructed by failure of BD entropy in higher dimensions. Achieving stochastic QNS regularity in multidimensional settings remains open.
• Noise Classes: The methodology currently accommodates multiplicative noise satisfying polynomial growth and regularity; extension to additive, rough, or transport noise is possible by adapting the cut-off/Galerkin/entropy approach.
• Generalizations: Broader pressure and viscosity laws, and additional dissipation mechanisms can be incorporated within the BD-entropy plus effective velocity framework.

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Summary Table: Core Elements of the 1D Stochastic QNS (Donatelli et al., 18 Jan 2024)

Mathematical Component	Statement/Role	Analytical Consequence
PDE System	See eqns (1.1a)-(1.1b); periodic domain $\mathbb{T}$	Framework for stochastic QNS analysis
Initial Law	$H^{s+1} \times H^s$, $s > 2.5$; density bounded, moments finite	Regularity and positivity of $\rho$
Noise Coefficient	Bounded, $C^s$, polynomial growth, $G(0,0) = 0$	Ensures well-posed stochastic integral
BD Entropy (effective velocity)	$\mathcal{E}_{\rm BD}(t)$, $v = u + \partial_x\log\rho$	Rules out vacuum, controls high norms
Energy/Entropy Inequality	High-order controls, $L^2_{t,x}$ on $\partial_x\log\rho$, etc.	No finite-time singularity
Pathwise Uniqueness	True for $s > 5/2$ (Sobolev embedding), a.s. in probability	Strong solution in $C([0,T]; H^s)$
Global Existence	A-priori bounds prevent blow-up; stopping times diverge	$T = \infty$ almost surely

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In sum, the stochastic quantum Navier–Stokes equations in 1D admit a robust global well-posedness theory for strong pathwise solutions, based on a delicate mixture of effective-velocity reformulation, Bresch–Desjardins entropy methods, and high-regularity energy estimates; these developments open the path for stochastic extensions of quantum hydrodynamics and for controlled analysis of random effects in quantum fluids (Donatelli et al., 18 Jan 2024).