Least total curvature solutions to steady Euler system and monotone solutions to semilinear equations in a strip

TL;DR

This paper establishes the existence of least total curvature solutions to the steady Euler system in a strip and constructs monotone solutions to semilinear elliptic PDEs, advancing understanding of these complex fluid and PDE systems.

Contribution

It introduces a minimization approach to find monotone heteroclinic solutions and constructs positive, stable solutions with non-convex superlevel sets in a strip domain.

Findings

Existence of least total curvature solutions to the Euler system in a strip.

Construction of monotone heteroclinic solutions to semilinear elliptic PDEs.

Positive, stable solutions with non-convex superlevel sets in a strip domain.

Abstract

This paper focuses on establishing the existence of a class of steady solutions, termed least total curvature solutions, to the incompressible Euler system in a strip. The solutions obtained in this paper complement the least total curvature solutions already known. Our approach employs a minimization procedure to identify a monotone heteroclinic solution for a conveniently chosen semilinear elliptic PDE. This method also enables us to construct positive and monotone (and consequently stable) solutions to semilinear elliptic PDEs with non-convex superlevel sets in a strip domain. This can be regarded as a negative answer to a generalized problem raised in [27].