Hydrodynamic limit for the inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Su, Yunfei¹  Yao, Lei¹ 

[1] Northwest Univ, Sch Math, Xian 710127, Peoples R China

关键词: Hydrodynamic limit;    Inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations;    Relative entropy method;   

DOI  :  10.1016/j.jde.2019.12.027

来源: Elsevier

【 摘 要 】

In this paper, we study the hydrodynamic limit for the inhomogeneous incompressible NavierStokes/Vlasov-Fokker-Planck equations in a two or three dimensional bounded domain when the initial density is bounded away from zero. The proof relies on the relative entropy argument to obtain the strong convergence of macroscopic density of the particles n(epsilon) in L-infinity (0, T; L-1(Omega)), which extends the works of Goudon-Jabin-Vasseur [15] and Mellt-Vasseur [26] to inhomogeneous incompressible NavierStokes/Vlasov-Fokker-Planck equations. Precisely, the relative entropy estimates in [15] and [26] give the strong convergence of u(epsilon) and n(epsilon), rho(epsilon) and n(epsilon), respectively. However, we only obtain the strong convergence of n(epsilon) and u(epsilon) from the relative entropy estimate, and we use another way to obtain the strong convergence of rho(epsilon) via the convergence of u(epsilon). Furthermore, when the initial density may vanish, taking advantage of compactness result L-M hooked right arrow hooked right arrow H(-1)of Orlicz spaces in 2D, we obtain the convergence of no in L-infinity (0, T; H-1(Omega)), which is used to obtain the relative entropy estimate, thus we also show the hydrodynamic limit for 2D inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations when there is initial vacuum. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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