Global well-posedness and optimal large-time behavior of strong solutions to the non-isentropic particle-fluid flows

Abstract

In this paper, we study the three-dimensional non-isentropic compressible fluid–particle flows. The system involves coupling between the Vlasov–Fokker–Planck equation and the non-isentropic compressible Navier–Stokes equations through momentum and energy exchanges. For the initial data near the given equilibrium we prove the global well-posedness of strong solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. For the periodic domain the same global well-posedness result still holds while the convergence rate is exponential. New ideas and techniques are developed to establish the well-posedness and large-time behavior. For the global well-posedness our methods are based on the new macro–micro decomposition which involves less dependence on the spectrum of the linear Fokker–Plank operator and fine energy estimates; while the proofs of the optimal large-time behavior rely on the Fourier analysis of the linearized Cauchy problem and the energy-spectrum method, where we provide some new techniques to deal with the nonlinear terms.

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Acknowledgements

Y. Mu was partially supported by NSFC (Grant No. 11701268), Natural Science Foundation of Jiangsu Province of China (BK20171040) and Chinese Postdoctoral Science Foundation (2018M642277). D. Wang’s research was supported in part by the NSF Grants DMS-1613213 and DMS-1907519. The authors thank the referee for valuable comments and suggestions.

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Correspondence to Dehua Wang.

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Mu, Y., Wang, D. Global well-posedness and optimal large-time behavior of strong solutions to the non-isentropic particle-fluid flows. Calc. Var. 59, 110 (2020). https://doi.org/10.1007/s00526-020-01776-8

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Mathematics Subject Classification

  • 35Q30
  • 76D03
  • 76D05
  • 76D07