Abstract.
For the Cauchy problem with a kind of non-smooth initial data for weakly linearly degenerate hyperbolic systems of conservation laws with the linear damping term, we prove the existence and uniqueness of global weakly discontinuous solution u = u(t, x) containing only n weak discontinuities with small amplitude on t ≥ 0, and this solution possesses a global structure similar to that of the similarity solution \(u = U(\frac{x}{t})\) of the corresponding homogeneous Riemann problem. As an application of our result, we obtain the existence and uniqueness of global weakly discontinuous solution, continuous and piecewise C 1 solution with discontinuous first order derivatives, of the flow equations of a model class of fluids with viscosity induced by fading memory.
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Zhi-Qiang Shao: Supported by the National Natural Science Foundation of China (Grant 70371025), the Scientific Research Foundation of the Ministry of Education of China (Grant 02JA790014) and Natural Science Foundation of Fujian Province(Grant 2001J009), the Science and Technology Developmental Foundation of Fuzhou University (Grant 2004-XQ-16).
Ya-Chun Li: Supported by the National Science Foundation of China(Grant 10571120) and Shanghai Natural Science Foundation (Grant 04ZR14090).
De-Xing Kong: Supported by the National Science Foundation of China(Grant 10371073), the Special Funds for Major State Basic Research Projects of China (Grant 2000077306), the Qi Ming Xing programme of Shanghai Government, and the Project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, the Ministry of Education of China.
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Shao, ZQ., Li, YC. & Kong, DX. Global weakly discontinuous solutions to quasilinear hyperbolic systems of conservation laws with damping with a kind of non-smooth initial data. Z. angew. Math. Phys. 59, 935–968 (2008). https://doi.org/10.1007/s00033-007-5096-0
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DOI: https://doi.org/10.1007/s00033-007-5096-0