Optimal time-decay rates of the Keller–Segel system coupled to compressible Navier–Stokes equation in three dimensions

Boundary Value Problems

Rui Sun¹  Yuting Guo¹  Weiwei Wang¹ 

[1] College of Mathematics and Statistics, Fuzhou University;

关键词: Compressible chemotactic fluids;    Global existence;    Uniqueness;    Fourier theory;    Optimal time-decay rates;   

DOI  :  10.1186/s13661-022-01618-w

来源: DOAJ

【 摘 要 】

Abstract Recently, Hattori–Lagha established the global existence and asymptotic behavior of the solutions for a three-dimensional compressible chemotaxis system with chemoattractant and repellent (Hattori and Lagha in Discrete Contin. Dyn. Syst. 41(11):5141–5164, 2021). Motivated by Hattori–Lagha’s work, we further investigated the optimal time-decay rates of strong solutions with small perturbation to the three-dimensional Keller–Segel system coupled to the compressible Navier–Stokes equations, which models for the motion of swimming bacteria in a compressible viscous fluid. First, we reformulate the system into a perturbation form. Then we establish a prior estimates of solutions and prove the existence of the global-in-time solutions based on the local existence of unique solutions. Finally, we will establish the optimal time-decay rates of the nonhomogeneous system by the decomposition technique of both low and high frequencies of solutions as in (Wang and Wen in Sci. China Math., 2020, https://doi.org/10.1007/s11425-020-1779-7 ). Moreover, the decay rate is optimal since it agrees with the solutions of the linearized system.

【 授权许可】

Unknown