Smooth Solution of Multi-dimensional Nonhomogeneous Conservation Law: Its Formula, and Necessary and Sufficient Blowup Criterion  

作　　者：Gao-wei CAO Hui KAN Wei XIANG Xiao-zhou YANG 

机构地区：[1]Wuhan Institute of Physics and Mathematics,Chinese Academy of Sciences,Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences,Wuhan 430071,China [2]City University of Hong Kong,Hong Kong,China

出　　处：《Acta Mathematicae Applicatae Sinica》2023年第1期17-27,共11页应用数学学报（英文版）

基　　金：The research of Gaowei Cao was supported in part by the NSFC(Grant 11701551 and Grant 11971024);the China Scholarship Council No.202004910200.The research of Hui Kan was supported in part by the NSFC(Grant 11801551);The research of Wei Xiang was supported in part by the Research Grants Council of the HKSAR,China(Project No.City U 11332916,Project No.City U 11304817 and Project No.City U 11303518);The research of X.Z.Yang was supported in part by the NSFC(Grant 11471332)。

摘　　要：In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term,where the initial data lies in W1,∞(Rn) ∩ C1(Rn). We obtain the solution formula for smooth solution, and then apply it to establish and prove the necessary and sufficient condition for the global existence of smooth solution. Moreover, if the smooth solution blows up at a finite time, the exact lifespan of the smooth solution can be obtained. In particular, when the source term vanishes, the corresponding theorem for the homogeneous case is obtained too. Finally, we give two examples as its applications, one for the global existence of the smooth solution and the other one for the blowup of the smooth solutions at any given positive time.

关 键 词：global smooth solution BLOWUP multi-dimensional conservation law solution formula 

分 类 号：O175[理学—数学]