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

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Abstract

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.

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Authors and Affiliations

  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

    Gao-wei Cao, Hui Kan & Xiao-zhou Yang

  2. City University of Hong Kong, Kowloon Tong, Hong Kong, China

    Wei Xiang

Authors
  1. Gao-wei Cao
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  2. Hui Kan
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  3. Wei Xiang
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  4. Xiao-zhou Yang
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Corresponding author

Correspondence to Xiao-zhou Yang.

Additional information

The research of Gaowei Cao was supported in part by the NSFC (Grant 11701551 and Grant 11971024), and 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. CityU 11332916, Project No. CityU 11304817 and Project No. CityU 11303518). The research of X.Z. Yang was supported in part by the NSFC (Grant 11471332).

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Cao, Gw., Kan, H., Xiang, W. et al. Smooth Solution of Multi-dimensional Nonhomogeneous Conservation Law: Its Formula, and Necessary and Sufficient Blowup Criterion. Acta Math. Appl. Sin. Engl. Ser. 39, 17–27 (2023). https://doi.org/10.1007/s10255-023-1036-9

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  • DOI: https://doi.org/10.1007/s10255-023-1036-9

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