On the multivariate Burgers equation and the incompressible Navier-Stokes equation (Part I)

TL;DR

This paper presents a constructive proof of global existence for the multivariate viscous Burgers equation using a time discretized scheme, which can aid in computational approaches for fluid dynamics equations.

Contribution

It introduces a novel semiexplicit perturbative expansion method with convergence guarantees for solving the multivariate Burgers equation on unbounded domains.

Findings

Proves global existence for the multivariate Burgers system.

Develops a convergent time discretization scheme.

Provides a foundation for computational fluid dynamics methods.

Abstract

We provide a constructive global existence proof for the multivariate viscous Burgers equation system defined on the whole space or on a domain isomorphic to the n-torus and with time horizon up to infinity and C^{\infty}- data (satisfying some growth conditions if the problem is posed on the whole space). The proof is by a time discretized semiexplicit perturbative expansion in transformed coordinates where the convergence is guaranteed by certain a priori estimates. The scheme is useful in order to define computation for related equation systems of fluid dynamics.