Neural Networks and Existence Theory for the Navier-Stokes Equations

The study of fluid dynamics occupies a central role in applied mathematics and physics, withthe Navier-Stokes equations (NSE) modeling incompressible viscous flows. Despite theirsimple appearance, three-dimensional incompressible NSE are challenging both analyticallyand numerically. Classical methods include finite difference, finite volume, and spectralmethods. Recently, neural networks have emerged as a flexible framework for approximatingcomplex functional relationships. Physics-informed neural networks (PINNs) incorporate thegoverning PDEs directly into the loss function, allowing accurate approximations of velocityand pressure fields while leveraging physics to improve generalization.

In this bookthe 2PCF and 3PCF are considered.

The analysis of the Navier–Stokes equations and their derived forms is often complicatedby the potential lack of smoothness of solutions. In particular, when the velocity field uloses regularity for instance, if it develops only H¨older continuity of low order or singularstructures the nonlinear convective term(u · ∇)a in associated PDEs may become unbounded or even ill-defined. This fundamental obstruc-tion highlights the difficulty of reconciling the nonlinear transport structure of fluid equationswith weak solution concepts. A central question therefore arises: under what conditions cansuch convective terms be consistently defined, and what approximations or closures are avail-able when smoothness is absent?One proposed remedy is the introduction of algebraic closures, which replace the originalvelocity field by a composite field built from quadratic products of its components. Theprototypical example is the closureu = b = (uyuz ) i + (uxuz ) j + (uxuy) k,which enforces algebraic dependencies among the velocity components. This relation dras-tically reduces the effective degrees of freedom of the velocity vector: only two componentsremain independent, while the third is reconstructed via nonlinear constraints. In this way,the closure projects the dynamics onto a nonlinear surface in R3, thereby regularizing orconstraining certain convective interactions.

In the framework of Cannone and Karch, the forcing terms can be modeled as elements in singular spaces

(e.g., Besov-type or critical Morreyspaces), and the existence of solutions depends on delicate balance conditions between thesingularity of the forcing and the integrability/smoothness of the velocity field.If the velocity field is H¨older continuous with blowup in first or second derivatives, then:- The Laplacian ∇2 b and nonlinear term b · ∇b may become singular. - However, the closureremains valid in the weak or distributional sense as long as the products are in L1_loc or canbe interpreted via compensated compactness.

The closure model u = b is universal under the assumption of locally H¨oldercontinuous velocity components ui provided: - The singularities are such that the nonlinearterms and derivatives remain in suitable weak function spaces. - The forcing terms can bematched to these singular behaviors, as in the theory of Cannone and Karch (2022).