Computability of the Solutions to Navier-Stokes Equations via Effective Approximation

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 Ω = (− 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.

Computer Science for Continuous Data

“…Next up on the to-do list is a complexity-theoretic classification of the (hyperbolic) linear Wave Equation, and of the non-linear Navier-Stokes Equation. Subject the Millennium Prize Problem, Navier-Stokes maintains regularity and its solutions remain in classical spaces of continuously differentiable functions with their established coding and computability and complexity theory [61]. But regarding the Wave Equation, its regularity theory is well-established to require Sobolev spaces for computability investigations [53,66]; and Subsect.…”

“…c) Picard's method for solving ODEs amounts to iterations according to Banach's Fixedpoint Theorem in a suitable space of smooth functions. d) Solutions to Navier-Stokes' nonlinear PDE are also mathematically shown to exist [18, §2] and being computable [61] by means of iterations in some space of integrable functions.…”