On the multivariate Burgers equation and the incompressible Navier-Stokes equation (part III)

TL;DR

This paper develops simplified controlled schemes for the multivariate Burgers and incompressible Navier-Stokes equations, establishing local contraction, higher regularity, and convergence results to construct classical solutions.

Contribution

It introduces simplified controlled schemes that improve the analysis of regularity and convergence for these fluid dynamics equations, avoiding complex adjoint properties.

Findings

Local contraction in Hm-norms over time and space.

Preservation of higher order regularity with polynomial decay.

Constructive approach for local and global classical solutions.

Abstract

In this paper we first obtain local contraction results in a Hm-norm with respect to time and space for a local scheme. We show that a global controlled scheme preserves higher order regularity with respect to the spatial variables together with polynomial decay of order m of the data at each time step. Here, we simplify the controlled scheme considered in [1] and [2]. Especially, for the simplified controlled scheme we get an upper bound for the Leray projection term. Furthermore, the schemes discussed in [1] and [2] are simplified in the sense that the estimates are achieved without the use of some properties concerning the adjoint of a local fundamental solutions with variable drift terms. We note that the pointwise and absolute convergence of the local functional series and their first order time derivatives and their spatial derivatives leads to a constructive approach of local and global classical solutions.