Existence and Smoothness of the Navier-Stokes equation using the Boundary Integral Method

Technical Summary: Existence and Smoothness of the Navier-Stokes Equation using the Boundary Integral Method

Problem Statement
The paper addresses the Millennium Prize problem concerning the existence and smoothness of solutions to the three-dimensional, unsteady, incompressible Navier-Stokes equations. Specifically, it considers a fluid filling all of space ($\mathbb{R}^3$) with no external body forces, starting from a smooth ($C^\infty$), divergence-free initial velocity field that decays like a Stokeslet at infinity. The central challenge is to prove that a smooth solution exists for all subsequent time $t \ge 0$ without finite-time blow-up (singularity formation), a problem where previous approaches using generalized functions (Ladyzhenskaya) or domain distributions of Stokeslets (Ramm) have faced difficulties regarding inertial blow-up and logarithmic unboundedness.

Methodology
The author proposes a novel boundary integral formulation based on a moving reference frame and a specific class of fundamental solutions called "nslets." The methodology relies on several key theoretical shifts:

1. Space-Time Boundary Integral Formulation: The problem is restated in a four-dimensional space-time domain ($\Sigma^+$) enclosed by a hypersurface consisting of an initial time volume, a far-field "spherinder" (spherical cylinder), and a vanishingly small boundary enclosing the fluid point.
2. Moving Reference Frame and Lagrangian-Eulerian Coincidence: A critical innovation is the alignment of the boundary integral calculation with a reference frame moving with the fluid velocity at the point of interest. In this limit, as the boundary encloses the fluid point, the Lagrangian material derivative and the Eulerian partial derivative coincide. Consequently, the non-linear quadratic term (the convective acceleration $u_j u_{i,j}$) vanishes in the boundary integral formulation because the non-linearity inherent in the Eulerian description is absorbed into the moving frame's material derivative.
3. Fundamental Solutions (Nslets): The solution is constructed using "nslets," which are fundamental solutions to the Navier-Stokes equations.
   • Near-Field (Eulerlet): Close to the origin, the nslet approximates an "eulerlet," a solution to the Euler equations where inertial terms dominate. The eulerlet contains a singularity line aligned with the fluid velocity (a streakline), ensuring the velocity direction is uniquely defined.
   • Far-Field (Stokeslet): Far from the origin, the nslet decays and approximates the unsteady Stokeslet (or Oseenlet in a moving frame).
4. Handling Non-Linearity via Divergence: The non-linear terms are treated as a divergence of a vector field directed along the radial spoke of the fundamental solution. By applying the divergence theorem, the non-linear domain integral is converted to a boundary integral. The paper argues that in the limit of the small boundary enclosing the fluid point, the singular contributions from the non-linear term cancel with the singular pressure contributions of the eulerlet, and the remaining terms vanish due to symmetry or exponential decay.

Key Contributions and Results

• Velocity Representation: The paper derives a boundary integral representation for the velocity field $u_k$ at time $T$ solely in terms of the initial velocity distribution $u_i(x', 0)$ and the nslet fundamental solution:
  $$u_k(x, T) = -\int_V u_i(x', 0) u_{ki}(x-x', T) dV'$$
  This representation is shown to be valid because the contributions from the far-field boundary vanish, and the near-field boundary yields the velocity itself.
• Proof of Existence and Smoothness:
  • The author demonstrates that for a sufficiently small time step $T$, the nslet approximates the eulerlet with a bounded error of order $O(\nu \sqrt{T})$.
  • By iterating this process over time steps, the paper argues that the velocity and its derivatives remain smooth ($C^\infty$) for all time.
  • The proof is conducted via ansatz: assuming smoothness to derive the integral representation, and then showing that the resulting representation indeed yields a smooth solution, thereby completing the proof.
• Zero Initial Velocity: The formulation naturally yields the null solution if the initial velocity is zero, satisfying the uniqueness condition for the homogeneous case.
• Chaotic Dynamics: The paper notes that because the singularity line of the fundamental solution (the eulerlet) must align with the local fluid velocity, the system constitutes a dynamical system where the fundamental solutions interact with the flow field. This mechanism allows for chaotic behavior, exemplified by the reduction of the formulation to the "blinking vortex" model in the high Reynolds number limit.

Significance and Claims
The paper claims to provide a proof of existence and smoothness for the 3D unsteady Navier-Stokes equations by circumventing the "inertial blow-up" singularity. This is achieved by shifting the analysis to a moving reference frame where the non-linear convective term disappears from the boundary integral formulation, allowing the use of standard linear boundary integral theory (Green's functions) developed by Oseen and Ladyzhenskaya.

The author asserts that this approach successfully models:

1. Low Reynolds number flows (validating against Oseen approximations).
2. High Reynolds number non-linear flows (validating against Blasius and triple-deck theories).
3. Chaotic flows (demonstrating reduction to blinking vortex dynamics).

The paper concludes that the resulting integral representation exists and is smooth for all time, provided the initial conditions are smooth and decay appropriately. The significance lies in offering a constructive method to prove smoothness by transforming the non-linear problem into a linear boundary integral problem through a specific choice of reference frame and fundamental solutions, while acknowledging that the modeling of fully developed turbulence remains a subject for future investigation.