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Global classical solutions of 3D compressible viscoelastic system near equilibrium

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Abstract

In this paper, we prove the global existence of general small solutions to compressible viscoelastic system. We remove the “initial state” assumption (\({\widetilde{\rho }}_0 \det F_0 =1\)) and the “div-curl” structure assumption compared with previous works. It then broadens the class of solutions to a great extent. More precisely, the initial density state would not be constant necessarily, and no more structure is needed for global well-posedness theory. It’s different from the elasticity system in which structure plays an important role. Since we can not obtain dissipation information for density and deformation tensor directly, we introduce a new effective flux. The core thought is regarding the wildest “nonlinear term” as “linear term”. Although the Sobolev norms of solution may increase now, we can still obtain the global existence for it.

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Acknowledgements

The author sincerely appreciates the helpful suggestion from Professor Zhen Lei. The author is supported by Shanghai Sailing Program (18YF1405500), Fundamental Research Funds for the Central Universities (222201814026), China Postdoctoral Science Foundation (2018M630406, 2019T120308) and NSFC (11801175).

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Authors and Affiliations

  1. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, People’s Republic of China

    Yi Zhu

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  1. Yi Zhu
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Correspondence to Yi Zhu.

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Communicated by F.-H. Lin.

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Zhu, Y. Global classical solutions of 3D compressible viscoelastic system near equilibrium. Calc. Var. 61, 21 (2022). https://doi.org/10.1007/s00526-021-02127-x

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