A construction of two different solutions to an elliptic system
Provides analytical construction of multiple solutions for a specific elliptic system, but the result is incremental and limited to a particular force and large parameters.
The paper constructs two distinct solutions to an elliptic system on a 2D torus, serving as an elliptic regularization of the stationary Burgers equation, valid for large parameters λ and m. The results are supported by numerical simulations.
The paper aims at constructing two different solutions to an elliptic system $$ u \cdot \nabla u + (-Δ)^m u = λF $$ 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 $λ\sin y \partial_x w + (-Δ)^{m} w$ 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 $λ> λ_0$, $m> m_0$ 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.