On existence of general solution of the Navier-Stokes equations for 3D non-stationary incompressible flow

A new presentation of general solution of Navier-Stokes equations is considered here. We consider equations of motion for 3-dimensional non-stationary incompressible flow. The field of flow velocity as well as the equation of momentum should be split to the sum of two components: an irrotational (curl-free) one, and a solenoidal (divergence-free) one. The obviously irrotational (curl-free) part of equation of momentum used for obtaining of the components of pressure gradient. As a term of such an equation, we used the irrotational (curl-free) vector field of flow velocity, which is given by the proper potential (besides, the continuity equation determines such a potential as a harmonic function). The other part of equation of momentum could also be split to the sum of 2 equations: - with zero curl for the field of flow velocity (viscous-free), and the proper Eq. with viscous effects but variable curl. A solenoidal Eq. with viscous effects is represented by the proper Heat equation for each component of flow velocity with variable curl. Non-viscous case is presented by the PDE-system of 3 linear differential equations (in regard to the time-parameter), depending on the components of solution of the above Heat Eq. for the components of flow velocity with variable curl. So, the existence of the general solution of Navier-Stokes equations is proved to be the question of existence of the proper solution for such a PDE-system of linear equations. Final solution is proved to be the sum of 2 components: - an irrotational (curl-free) one and a solenoidal (variable curl) components.

💡 Research Summary

The paper attempts to prove the existence of a general solution for the three‑dimensional, time‑dependent, incompressible Navier‑Stokes equations by decomposing the velocity field into an irrotational (curl‑free) component and a solenoidal (divergence‑free) component, following the Helmholtz decomposition. The irrotational part is represented as the gradient of a scalar potential φ. Using the continuity equation ∇·u = 0, the authors deduce that φ satisfies Laplace’s equation (∇²φ = 0), i.e., φ is a harmonic function. This component is then used to express the pressure gradient, assuming that the pressure can be obtained solely from the irrotational velocity and external forces.

The solenoidal component is further split into two sub‑equations. The first sub‑equation is described as “viscous‑free” and retains zero curl, while the second incorporates viscous effects and is said to have a “variable curl”. The authors claim that the viscous sub‑equation reduces to a heat‑type equation for each Cartesian component of the velocity: ∂u_i/∂t = ν∇²u_i, where ν is the kinematic viscosity. They argue that, because the heat equation has well‑known existence and uniqueness results for appropriate initial and boundary data, a solution for the solenoidal part always exists.

Having obtained a solution for the solenoidal (viscous) part, the authors assert that the irrotational part can be reconstructed from the harmonic potential φ, which is determined by the pressure gradient obtained earlier. Consequently, the full velocity field is given by the sum u = u_irrotational + u_solenoidal, and the pressure field is completely specified by the irrotational component. The paper concludes that the existence of a general Navier‑Stokes solution is equivalent to solving a linear system of partial differential equations derived from the above decomposition.

While the overall strategy—splitting the flow into potential and solenoidal parts—is mathematically sound in principle, several critical issues undermine the rigor of the claimed proof. First, the paper does not specify the boundary conditions required for the harmonic potential φ. In realistic fluid‑mechanical problems, boundary conditions couple pressure and velocity in a non‑trivial way, and without a clear specification it is impossible to guarantee a unique φ. Second, the assumption that viscous terms act only on the solenoidal component contradicts the standard Navier‑Stokes formulation, where the viscous Laplacian applies to the entire velocity field. Ignoring ν∇²φ in the irrotational part removes a potentially significant contribution, especially near solid boundaries where no‑slip conditions generate vorticity. Third, the treatment of the nonlinear convection term u·∇u is insufficient. The authors claim that this term becomes “obviously irrotational” after decomposition, yet the convection term generally contributes to both curl‑free and divergence‑free parts, and its proper projection onto each subspace requires careful analysis that is absent from the manuscript.

Moreover, the reduction of the solenoidal equation to a set of independent heat equations neglects the coupling introduced by the incompressibility constraint and the pressure field. In the full Navier‑Stokes system, pressure enforces the divergence‑free condition and couples the three velocity components; treating them as decoupled heat equations overlooks this essential interaction. Finally, the existence proof relies on the classical heat‑equation theory but does not address the compatibility of the initial data for the two sub‑systems, nor does it discuss regularity issues that are central to the Millennium Prize problem.

In summary, the paper presents an intriguing decomposition‑based viewpoint and correctly identifies that solving linear PDEs is simpler than tackling the full nonlinear Navier‑Stokes system. However, the omission of rigorous boundary‑condition treatment, the unjustified restriction of viscous effects, and the inadequate handling of the nonlinear convection term mean that the claim of having proved the existence of a general solution is not substantiated. Further work would need to incorporate a full projection of the Navier‑Stokes operator, explicit boundary and initial data, and a careful analysis of the coupling between pressure and velocity to make the argument mathematically convincing.