Constructive analysis of the Navier-Stokes equation

Abstract

A global time-discretized scheme for the Navier-Stokes equation system in its Leray projection form is defined. It is shown that the scheme converges to a bounded global classical solution for smooth data which have polynomial decay at infinity. Furthermore, the algorithm proposed is extended to the situation of initial-boundary value problems. Algorithms constructed in a different context (cf. [4, 10, 5, 9]) may be used within the proposed scheme in order to compute the solution of Leray's form of the Navier-Stokes system. The main idea for global existence is to define a control function dynamically and show explicitly that the scheme which solves a controlled Navier-Stokes type equation can control the modulus of velocity and the first derivatives of velocity to be bounded. The method described here can be extended to Navier-Stokes equations on compact manifolds which is done in a subsequent paper.

105 p., former version revised and simplified according to the ideas in arXiv:1205.4888v4 [math.AP]

References in corpus (4)

• On the multivariate Burgers equation and the incompressible Navier-Stokes equation (Part II)
• Global existence, regularity and a probabilistic scheme for a class of ultraparabolic Cauchy problems
• On local analytic expansions of densities in the context of (micro-)hypoelliptic and classes of semi-elliptic equations
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Cited by in corpus (3)

• On the multivariate Burgers equation and the incompressible Navier-Stokes equation (Part II)
• Some infinite matrix analysis, a Trotter product formula for dissipative operators, and an algorithm for the incompressible Navier-Stokes equation
• On a class of singular solutions to the incompressible 3-D Euler equation