Deterministic Fractional Navier–Stokes Equations

• Deterministic FNSE are a nonlocal generalization of classical fluid equations, replacing the standard Laplacian with a fractional Laplacian to model spatially nonlocal viscous dissipation.
• They use a probabilistic DSY cascade formulation to capture the interplay between nonlinearity, dissipation, and potential singularity formation, especially in scaling-supercritical regimes.
• The stochastic representation reveals phenomena like nonuniqueness and finite-time loss of integrability in scalar models, while symmetry considerations in 2D can ensure global solution persistence.

The deterministic Fractional Navier–Stokes Equations (FNSE) constitute a generalization of the classic Navier–Stokes equations in which the standard Laplacian (modeling local viscous dissipation) is replaced by a fractional Laplacian of order $\gamma \in (0,1)$, imposing nonlocal spatial dissipation that fundamentally alters the analytic and geometric properties of the solutions. Such equations are central in the mathematical analysis of turbulent flows, as they interpolate between the inviscid Euler model and the fully dissipative Navier–Stokes equations, and their paper illuminates the delicate balance of nonlinearity, dissipation, and potential singularity formation, especially in scaling-supercritical regimes.

1. Scaling, Criticality, and Parametric Regimes

The FNSE in $d$ dimensions for velocity $u(x, t)$ and pressure $p(x, t)$ with dissipation $(-\Delta)^\gamma$ take the form

$$u_t + (u \cdot \nabla)u + \nabla p + \nu (-\Delta)^\gamma u = 0, \qquad \nabla \cdot u = 0.$$

Criticality is dictated by the scaling of the energy: $$E[u](t) = \frac{1}{2} \int_{\mathbb{R}^d} |u(x, t)|^2 dx + \int_0^t \int_{\mathbb{R}^d} |(-\Delta)^{\gamma/2} u(x, s)|^2 dx ds,$$ which is invariant under

$$u_\lambda(x, t) = \lambda^{2\gamma - 1}u(\lambda x, \lambda^{2\gamma} t)$$

when $\gamma = (d+2)/4$. Thus,

• $\gamma < (d+2)/4$: scaling-supercritical regime,
• $\gamma = (d+2)/4$: critical,
• $\gamma > (d+2)/4$: subcritical.

A key result is that functional and geometric properties of DSY (doubly stochastic Yule) branching cascades associated with FNSE distinguish explosive from nonexplosive parametric regions in the $(d, \gamma)$-plane. The boundary between these regimes is set by the critical line $\gamma = (d+2)/4$, with complex behavior even arbitrarily close to this line (Dascaliuc et al., 13 Sep 2025).

2. DSY Cascade Representation and Connection to FNSE

The Fourier-transformed mild formulation of FNSE is recast probabilistically as an infinite branching process—the doubly stochastic Yule (DSY) cascade—on a binary tree. In the Fourier domain, introducing a majorizing kernel $h(\xi)$ ensures the integral equation

$$\chi(\xi, t) = \chi_0(\xi) e^{-|\xi|^{2\gamma} t} + \int_0^t |\xi|^{2\gamma} e^{-|\xi|^{2\gamma} s} \int_{\mathbb{R}^d} \chi(\eta, t-s) \odot_\xi \chi(\xi-\eta, t-s) H(\eta|\xi) d\eta ds,$$

where $\odot_\xi$ is a symmetrized product encoding incompressibility and nonlinearity, and $H$ is a probability measure on frequency splits.

The DSY cascade encodes the recursive evaluation of this mild formulation as a stochastic branching process: at each node (Fourier frequency), one draws exponentially distributed waiting times, selects branching frequencies according to $H$, and recurses. The key probabilistic quantity is the explosion time $S$—the total branching time along the genealogical path. If $S < \infty$ with positive probability, the process (and thus the solution representation) is explosive, otherwise nonexplosive.

Averaging the solution process $X(\xi, t)$ over the tree gives the solution to the FNSE wherever this expectation is integrable.

3. Nonuniqueness, Local Existence, and Majorization Principles

The stochastic representation reveals that even when classical mild solutions exist, multiple, distinct solutions ("nonuniqueness") can arise, especially when the DSY cascade is explosive. This occurs because, in the scalar PDE associated to the "majorization" of the FNSE (obtained by bounding $|\odot_\xi|$), the solution can lose integrability in finite time for sufficiently large initial data. The majorization principle takes the form

$|a \odot_\xi b| \leq \frac{1}{2}|a||b|$

and allows the paper of blow-up or loss of integrability in the scalar equation

$$\psi(\xi, t) = e^{-|\xi|^{2\gamma} t} \psi_0(\xi) + \int_0^t |\xi|^{2\gamma} e^{-|\xi|^{2\gamma}s} \int_{\mathbb{R}^d} \psi(\eta, t-s)\psi(\xi-\eta, t-s) H(\eta|\xi) d\eta ds.$$

• Local existence of solutions to the FNSE is established in pseudomeasure spaces $PM^a$ as long as the expectation of the cascade is integrable (Dascaliuc et al., 13 Sep 2025).
• Nonuniqueness arises when multiple "branching representations" of the stochastic solution exist, corresponding to different selection mechanisms for the stochastic cascade.

Importantly, even in the presence of deterministic well-posedness (e.g., subcritical regimes), the stochastic majorization reflects potential nonuniqueness and ill-posedness phenomena in the underlying deterministic FNSE.

4. Finite-Time Blow-Up in Associated Scalar Models

By analyzing the scalar Montgomery–Smith equation (the majorized model), one can show, using the DSY cascade structure, that for sufficiently large initial data, the expected value (and thus the $L^1$ norm in frequency) of the minimal solution process becomes infinite at a finite critical time $t_c$. This indicates finite-time blow-up of the stochastic representation, though it does not guarantee blow-up of the FNSE solution itself. This result is quantified via lower bounds on recursive moments of the DSY cascade; for each branching, combinatoric and geometric (angular) structure provides explicit time estimates for the loss of integrability. In higher-dimensional or less symmetric settings, such blow-up mechanisms may signal true singularity formation, whereas in special cases (e.g., 2D vortex flows), cancellations intervene.

5. Explicit Solution and Radial Symmetry in $d=2$

For $d=2$, the divergence-free constraint admits a simplification of the nonlinear term: $$f(\eta) \odot_\xi g(\xi-\eta) = \frac{i}{2}\operatorname{sign}(\xi\times\eta)[(f(\eta)\cdot e_{\eta^\perp})(g(\xi-\eta)\cdot e_{(\xi-\eta)^\perp})]\sin(\theta_{\xi, \eta}-\theta_{\xi, \xi-\eta}) e_{\xi^\perp}.$$ This allows a closed-form expression for the solution process $X(\xi, t)$ (restricted to non-explosion events), involving products over branch levels in the DSY tree: $$X(\xi, t) 1_{[S>t]} = (i/2)^{N(\xi,t)} \prod_{v \in \text{branch}} [\operatorname{sign}(W_v \times W_{v_1}) \sin(\theta_{W_v, W_{v_1}} - \theta_{W_v, W_{v_2}})] \prod_{v \in \text{leaves}} (\chi_0(W_v) \cdot e_{W_v^\perp}) e_{\xi^\perp}.$$ This explicit formula is central for establishing lower bounds leading to finite-time loss of integrability for large initial data, and for identifying symmetries responsible for cancellations in special cases.

In the radially symmetric (vortex) case in $d=2$, symmetries of the DSY tree induce cancellations in the product over angular factors. Upon correct "symmetry-adapted averaging", the solution persists globally and reduces to the standard linear fractional heat flow

$$\tilde{u}(\xi, t) = \tilde{u}_0(\xi) e^{-|\xi|^{2\gamma} t}$$

even beyond the time where the minimal cascade representation loses integrability.

6. Implications and Research Directions

The probabilistic representation of deterministic FNSE via DSY cascades provides a precise correspondence between explosion/non-explosion of the stochastic process and the analytic behavior of solutions (existence/nonuniqueness, loss of control over moments, and potential singularity). This framework yields:

• A phase diagram in the $(d, \gamma)$-plane distinguishing regions of deterministic local well-posedness from regimes where the stochastic representation predicts loss of integrability or nonuniqueness.
• Identification of nonuniqueness phenomena not only as a theoretical pathology but as a consequence of stochastically encoded branching structure inherent to the nonlinear FNSE.
• Fine-grained understanding of blow-up scenarios in model scalar PDEs (via majorization), and the possible continuation of solutions in symmetric or averaged contexts even after stochastic cascade integrability fails.
• A mechanism revealing that blow-up in the majorizing stochastic model does not necessarily imply deterministic blow-up for the mild or classical FNSE.

The methods illustrated provide a foundation for analyzing deterministic FNSE in supercritical regimes, understanding the fine structure of singularities, and rigorously characterizing ill-posedness and continuation phenomena in both scalar and vector contexts. This multifaceted approach connects deterministic PDE theory, harmonic analysis, and probability in the paper of nonlocal fluid dynamics (Dascaliuc et al., 13 Sep 2025).