Abstract
In this article, we introduce a new approach to obtain the property of the dissipative structure for a system of differential equations. If the system has a viscosity or relaxation term which possesses symmetric property, Shizuta and Kawashima in 1985 introduced the suitable stability condition called in this article Classical Stability Condition for the corresponding eigenvalue problem of the system, and derived the detailed relation between the coefficient matrices of the system and the eigenvalues. However, there are some complicated physical models which possess a non-symmetric viscosity or relaxation term and we cannot apply Classical Stability Condition to these models. Under this situation, our purpose in this article is to extend Classical Stability Condition for complicated models and to make the relation between the coefficient matrices and the corresponding eigenvalues clear. Furthermore, we shall explain the new dissipative structure through the several concrete examples.
Highlights
We are interested in the profile of solutions for a system of differential equations
There are a lot of physical models which do not have enough properties to analyze the corresponding eigenvalue problem. (We will study several problems in Sections 3 and 4)
We focus on a general linear system with weak dissipation and try to construct the useful condition which induces the notable property of eigenvalues in this article
Summary
We are interested in the profile of solutions for a system of differential equations. Classical Kalman Rank Condition (CR): For each ω ∈ Sn−1 , the m2 × m Kalman matrix has rank m, that is Under this situation, the following theorem is obtained. Condition (S): There is a real compensating matrix S with the following properties: (SA0 ) T = SA0 and (SL)] + L] ≥ 0 on Rm , i (SA(ω ))[ = 0 They derived the sufficient condition which is a combination of Condition (K) and (S) to get the uniformly dissipativity of the type (1, 2), which is the regularity-loss type. Some physical models which possess the regularity-loss structure do not satisfy the stability condition in [1] (e.g., [16,17,18]).
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