A construction of two different solutions to an elliptic system

TL;DR

This paper constructs two distinct solutions to a specialized elliptic system on the 2D torus, using finite-dimensional approximation and numerical analysis, revealing properties related to the linearized operator and its skew-symmetric part.

Contribution

It introduces a novel analytical approach to construct multiple solutions for an elliptic regularization of the 2D Burgers system, combining finite-dimensional analysis with numerical simulations.

Findings

Existence of two different solutions for the elliptic system

Analytical proof valid for large parameters and specific force

Numerical simulations support theoretical results

Abstract

The paper aims at constructing two different solutions to an elliptic system $$ u \cdot \nabla u + (-\Delta)^m u = \lambda F $$ defined on the two dimensional torus. It can be viewed as an elliptic regularization of the stationary Burgers 2D system. A motivation to consider the above system comes from an examination of unusual propetries of the linear operator $\lambda \sin y \partial_x w + (-\Delta)^{m} w$ arising from a linearization of the equation about the dominant part of $F$. We argue that the skew-symmetric part of the operator provides in some sense a smallness of norms of the linear operator inverse. Our analytical proof is valid for a particular force $F$ and for $\lambda > \lambda_0$, $m> m_0$ sufficiently large. The main steps of the proof concern finite dimension approximation of the system and concentrate on analysis of features of large matrices, which resembles standard numerical analysis. Our analytical results are illustrated by numerical simulations.