On classes of globally smooth solutions to the Euler equations in several dimensions

TL;DR

This paper demonstrates that the Euler equations admit a class of globally smooth solutions with linear velocity profiles, and shows stability of these solutions under small Sobolev norm perturbations, with constructed examples.

Contribution

It introduces a new class of globally smooth solutions with linear velocity profiles and proves their stability under small perturbations in Sobolev spaces.

Findings

Existence of globally smooth solutions with linear velocity profiles.

Stability of these solutions under small Sobolev norm perturbations.

Constructed explicit examples of such solutions.

Abstract

It is shown that if the system of the Euler equations has a special global in time smooth solution with the linear profile of velocity, then another solutions with Cauchy data, close in the Sobolev norm to the initial data of the given solution, will be globally smooth as well. The examples of the solutions are constructed.