A construction of two different solutions to an elliptic system

作者：Cyranka Jacek^*; Mucha Piotr Boguslaw

来源：Journal of Mathematical Analysis and Applications, 2018, 465(1): 500-530.

DOI：10.1016/j.jmaa.2018.05.010

• Nonlinear elliptic problem
• 2D stationary Burgers equation
• Nonuniqucness
• Construction of two solutions
• Large matrices

The paper aims at constructing two different solutions to an elliptic system
u . del 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 properties of the linear operator lambda sin y partial derivative(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.

• 出版日期2018-9-1
• 单位rutgers