# Computability of the Solutions to Navier-Stokes Equations via Recursive Approximation

## TL;DR

This paper investigates whether solutions to the 2D Navier-Stokes equations can be computed recursively, establishing a framework that makes the mild and strong solutions uniformly computable using a novel representation of divergence-free vector fields.

## Contribution

It constructs a natural encoding for divergence-free vector fields on a 2D domain and proves the computability of both mild and strong solutions to the Navier-Stokes equations within this framework.

## Key findings

- The representation makes the mild solution to the Stokes problem computable.
- The strong local solution to the Navier-Stokes initial value problem is also computable.
- The paper develops intricate estimates to establish these computability results.

## Abstract

As one of the seven open problems in the addendum to their 1989 book "Computability in Analysis and Physics", Pour-El and Richards proposed ``... the recursion theoretic study of particular nonlinear problems of classical importance. Examples are the Navier-Stokes equation, the KdV equation, and the complex of problems associated with Feigenbaum's constant.'' In this paper, we approach the question of whether the Navier-Stokes Equation admits recursive solutions in the sense of Weihrauch's Type-2 Theory of Effectivity. A natural encoding (``representation'') is constructed for the space of divergence-free vector fields on 2-dimensional open square $\Omega = (-1, 1)^2$. This representation is shown to render first the mild solution to the Stokes Dirichlet problem and then a strong local solution to the nonlinear inhomogeneous incompressible Navier-Stokes initial value problem uniformly computable. Based on classical approaches, the proofs make use of many subtle and intricate estimates which are developed in the paper for establishing the computability results.