一类经典的Boussinesq系统解的若干问题研究  

Study on Some Problems of Solutions for a Classical Boussinesq System

作　　者：李琪琪 周寿明 段俊 LI Qiqi;ZHOU Shouming;DUAN Jun(School of Mathematical Sciences,Chongqing Normal University;Economy and Management,Chongqing Normal University,Chongqing 401331,China)

机构地区：[1]重庆师范大学数学科学学院,重庆401331 [2]重庆师范大学经济与管理,重庆401331

出　　处：《重庆师范大学学报（自然科学版）》2021年第3期68-77,共10页Journal of Chongqing Normal University:Natural Science

基　　金：国家自然科学基金(No.11771063);重庆市教育委员会科学技术项目(No.KJQN202000518,No.KJZD-M201900501);重庆市社会科学规划项目(No.2019WT12)。

摘　　要：【目的】研究Boussinesq系统弱解的存在唯一性以及强解的爆破准则。【方法】通过拟抛正规化方法建立Boussinesq系统在Sobolev空间H^(s)(R)×H^(s-1)(R)(s≥1)中弱解的存在和唯一性,利用能量估计、输运方程理论等方法得到系统强解精确的爆破准则。【结果】首先得到了Boussinesq系统在Sobolev空间H^(s)(R)×H^(s-1)(R)(s>3/2)中强解的局部适定性,其次得到系统弱解在Sobolev空间H^(s)(R)×H^(s-1)(R)(s≥1)中的存在和唯一性,最后得到强解的爆破准则。【结论】Boussinesq系统强解的爆破只以波裂形式发生且仅与分量u的空间斜率有关而与分量ρ无关。[Purposes]The existence and uniqueness of weak solutions and blow up criteria of strong solutions for Boussinesq system were studied.[Methods]The existence and uniqueness of weak solution of Boussinesq system in Sobolev spaces H^(s)(R)×H^(s-1)(R)(s≥1)was established by using the quasi parabolic regularization method.The blow up criteria for strong solution of the system was obtained by means of energy estimation and transport equation theory.[Findings]Firstly,the local well-posedness of strong solution for Boussinesq system in Sobolev spaces H^(s)(R)×H^(s-1)(R)(s>3/2)wasobtained.Secondly,the existence and uniqueness of weak solution for Boussinesq system in Sobolev spaces H^(s)(R)×H^(s-1)(R)(s≥1)was obtained.Finally,the blow up criterion of strong solution was obtained.[Conclusions]The blow up of strong solution for the Boussinesq system only occurs in the form of wave breaking,which is related to the spatial slope of the component u but independent of the componentρ.

关 键 词：BOUSSINESQ系统 局部适定性 弱解 爆破准则 

分 类 号：O175[理学—数学]