Exact linear representations of the nonlinear Cauchy problems and their smooth solutions

TL;DR

This paper presents conditions for representing nonlinear PDEs, including Navier-Stokes and Euler equations, as linear equations with smooth solutions, using formal Taylor series and the Borel-Whitney lemma.

Contribution

It introduces a method to obtain exact linear representations of nonlinear Cauchy problems with smooth solutions, expanding the understanding of nonlinear PDEs.

Findings

Established conditions for linear representation of nonlinear PDEs.

Applied the method to Navier-Stokes and Euler equations.

Proved existence of smooth solutions matching formal Taylor expansions.

Abstract

The paper establishes conditions under which there are exact linear representations of nonlinear partial differential equations (Cauchy problems). By introducing a certain linear operator $A$, it is shown that under these conditions there are three equivalent equations (one linear and two nonlinear), while the formal operator solution is common for all of them and is the expansion of the desired function $v(t,x)$ into a formal Taylor series. Using the Borel-Whitney lemma, which states that any smooth function $v(t,x)$ in a neighborhood of a point is defined by its formal Taylor series, we obtain the following statement. If all parameters of the operator $A$ are smooth functions, then there exists a smooth function $\tilde{v}(t,x)$, which for $t=0$ has the same power expansion as $v(t,x) $ and this function solves all equivalent equations. The Navier-Stokes and Euler equations are considered as a non-trivial examples of this approach.