一类双分量可积系统在Besov空间的局部适定性与爆破

Local well-posedness and blow-up of a two-component system in Besov spaces

摘要: 主要研究一类双分量可积系统的Cauchy问题的局部适定性及其爆破现象.首先,利用Littlewood-Paley分解和输运方JP程理论,借助对数插值不等式和Osgood引理,证明了该系统在Besov空间B_p, 1^1 / p \times B_p, 1^1 / p(p∈1,2)中的局部适定性.其次,得到系统解在有限时刻发生爆破的判断准则.最后,基于以上爆破准则进一步推导出系统解发生爆破的充要条件.

Abstract: The local well-posedness and blow-up phenomena for the Cauchy problem of a two-component integrable system are investigated. By employing the Littlewood-Paley decomposition and transport equation theory, together with logarithmic interpolation inequalities and the Osgood lemma, the local well-posedness of the system is established in the Besov space B_p, 1^1 / p \times B_p, 1^1 / p for p∈1,2. Moreover, a criterion determining the occurrence of finite-time blow-up for solutions of the system is derived. Finally, based on this blow-up criterion, necessary and sufficient conditions for solution blow-up are further obtained.