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Analytic Solution of the N-Dimensional Incompressible Navier-Stokes Equations

Nathan Strange

TL;DR

The paper advances an analytic, recurrence-based framework for the $N$-dimensional incompressible Navier–Stokes equations by deriving derivative recurrences that generate arbitrary-order Taylor expansions under analytic data. It connects this framework to the Cauchy–Kovalevskaya theorem and prior symbolic Taylor approaches, producing a structured solution scheme for $u_j$ and $p$ that enforces the divergence-free constraint and yields a pressure Poisson relation. The main contributions include explicit recurrence relations for time derivatives of velocity, a derived pressure recurrence, and energy bounds that together address the Clay Math Millennium Problem variants, with analytic initial data guaranteeing existence and smoothness in the periodic setting and bounded-energy behavior in the unbounded case. The proposed method provides a rigorous, analytically tractable lens on NSE behavior, offering insights into convergence, stability, and potential avenues for analytic continuation, while also highlighting how such symbolic-series techniques can inform numerical practices and boundary-condition consistency checks.

Abstract

This paper presents an analytic solution of the incompressible Navier-Stokes equations as recurrence relations for the solution's derivatives, addressing the Clay Mathematics Institute's Millennium Prize problem on Navier-Stokes existence and smoothness.

Analytic Solution of the N-Dimensional Incompressible Navier-Stokes Equations

TL;DR

The paper advances an analytic, recurrence-based framework for the -dimensional incompressible Navier–Stokes equations by deriving derivative recurrences that generate arbitrary-order Taylor expansions under analytic data. It connects this framework to the Cauchy–Kovalevskaya theorem and prior symbolic Taylor approaches, producing a structured solution scheme for and that enforces the divergence-free constraint and yields a pressure Poisson relation. The main contributions include explicit recurrence relations for time derivatives of velocity, a derived pressure recurrence, and energy bounds that together address the Clay Math Millennium Problem variants, with analytic initial data guaranteeing existence and smoothness in the periodic setting and bounded-energy behavior in the unbounded case. The proposed method provides a rigorous, analytically tractable lens on NSE behavior, offering insights into convergence, stability, and potential avenues for analytic continuation, while also highlighting how such symbolic-series techniques can inform numerical practices and boundary-condition consistency checks.

Abstract

This paper presents an analytic solution of the incompressible Navier-Stokes equations as recurrence relations for the solution's derivatives, addressing the Clay Mathematics Institute's Millennium Prize problem on Navier-Stokes existence and smoothness.
Paper Structure (18 sections, 7 theorems, 64 equations)

This paper contains 18 sections, 7 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. The Clay Math Navier-Stokes Problem
  3. Preliminaries
  4. Multi-Index Notation
  5. Derivative Notation
  6. Taylor Sum Notation
  7. Real Analytic Functions
  8. Differentiation and Integration
  9. Multiplying Series
  10. Series Solution of the Navier-Stokes Equations
  11. Divergence-Free Constraint
  12. Pressure Poisson Equation
  13. Solving the Momentum Equations
  14. Total System Energy
  15. Bounds on Kinetic Energy
  16. ...and 3 more sections

Key Result

Proposition 2.2

Let $f \in C^\infty(U)$ for some $U \in \mathbb{R}^N$. The function $f$ is in fact in $C^\omega(U)$ if and only if, for each $x_1,\ldots,x_N \in U$, there is an open ball $V$, with $x_1,\ldots,x_N \in V \subseteq U$, and constants $C>0$ and $R>0$ such that the derivatives of $f$ satisfy:

Theorems & Definitions (15)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 5 more