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<p>We consider the Cauchy problem for the three-dimensional pressureless Euler/Navier-Stokes system, which consists of the pressureless Euler equations for (<i>p, u</i>) coupled with the compressible Navier-Stokes equations for (<i>n, v</i>) through a drag force term <i>ρ</i>(<i>u</i> − <i>v</i>). We prove the global-in-time well-posedness and asymptotic behavior of solutions to this coupled system, where the initial data (<i>ρ</i><sub>0</sub>, <i>u</i><sub>0</sub>, <i>n</i><sub>0</sub>, <i>v</i><sub>0</sub>) satisfies (<i>ρ</i><sub>0</sub>, <i>u</i><sub>0</sub>, <i>n</i><sub>0</sub>, <i>v</i><sub>0</sub>)(<i>x</i>) → (0, 0, n<sub>*</sub>, 0) with a positive constant <i>n</i><sub>*</sub> &gt; 0 as ∣<i>x</i>∣ → ∞, i.e., the (initial) density <i>ρ</i> to the pressureless Euler flow contains the vacuum state at the far field. The pressureless structure and the appearance of the vacuum state cause the loss of time-decay properties of the density to the pressureless Euler equations and also lead to the invalidness of the dissipation effects of the source term. In order to overcome these difficulties and control the time growth of nonlinear terms, we make use of the combined time-weighted estimate and spectral analysis to establish the Lyapunov stability of the density <i>ρ</i> through the transport equation and obtain the optimal time-decay rates of the other flow part (<i>u, n, v</i>) in the Sobolev space, so as to show the global existence of strong solutions to the pressureless Euler/Navier-Stokes system.</p>

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Global existence and large-time behavior of solutions to the pressureless Euler/Navier-Stokes system

  • 期刊论文

  • 发表日期：

    2025年4月22日

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  • Hai-Liang Li,
  • Hou-Zhi Tang,
  • Yue Zhang

摘要

We consider the Cauchy problem for the three-dimensional pressureless Euler/Navier-Stokes system, which consists of the pressureless Euler equations for (p, u) coupled with the compressible Navier-Stokes equations for (n, v) through a drag force term ρ(u − v). We prove the global-in-time well-posedness and asymptotic behavior of solutions to this coupled system, where the initial data (ρ0, u0, n0, v0) satisfies (ρ0, u0, n0, v0)(x) → (0, 0, n*, 0) with a positive constant n* > 0 as ∣x∣ → ∞, i.e., the (initial) density ρ to the pressureless Euler flow contains the vacuum state at the far field. The pressureless structure and the appearance of the vacuum state cause the loss of time-decay properties of the density to the pressureless Euler equations and also lead to the invalidness of the dissipation effects of the source term. In order to overcome these difficulties and control the time growth of nonlinear terms, we make use of the combined time-weighted estimate and spectral analysis to establish the Lyapunov stability of the density ρ through the transport equation and obtain the optimal time-decay rates of the other flow part (u, n, v) in the Sobolev space, so as to show the global existence of strong solutions to the pressureless Euler/Navier-Stokes system.