Blow-up analysis for a periodic two-component μ-Hunter–Saxton system

Abstract

The two-component μ-Hunter–Saxton system is considered in the spatially periodic setting. Firstly, two wave-breaking criteria are derived by employing the transport equation theory and the localization analysis method. Secondly, a sufficient condition of the blow-up solutions is established by using the classic method. The results obtained in this paper are new and different from those in previous works.

Global Conservative Solutions of the Two-Component μ -Hunter–Saxton System

“…Hence, we can say 2-μHS system equation ( 4) lies in an intermediate between 2-CH and 2-HS systems. e Cauchy problem for equation (4) has been studied extensively in [24][25][26][27]. In addition, research studies have shown that this system is locally well-posed [26] for (u 0 , ρ 0 ) ∈ H s × H s− 1 , s > (3/2); besides, its global classical solutions [26] and finite-time blowup solutions [25,27] have also been found, and its geometric background has been comprehensively given by Escher in [24].…”

“…e Cauchy problem for equation (4) has been studied extensively in [24][25][26][27]. In addition, research studies have shown that this system is locally well-posed [26] for (u 0 , ρ 0 ) ∈ H s × H s− 1 , s > (3/2); besides, its global classical solutions [26] and finite-time blowup solutions [25,27] have also been found, and its geometric background has been comprehensively given by Escher in [24]. e global admissible weak solution of system equation (4) has been obtained in [28] by mollifying the initial date.…”