Optimal Large-Time Behavior of the Two-Phase Fluid Model in the Whole Space

Wu, Guochun; Zhang, Yinghui; Zou, Lan

doi:10.1137/20m1331202

Cited by 35 publications

(37 citation statements)

References 17 publications

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“…for large t. Next, we apply interpolation inequality to obtain the lower bound of decay rate for the second order spatial derivative of large solution (cf. [54]). The decay estimate (3.24) together with interpolation inequality: Then…”

Section: Proofmentioning

confidence: 99%

The optimal decay rate of strong solution for the compressible Navier–Stokes equations with large initial data

Gao

Wei

Yao

2020

Physica D: Nonlinear Phenomena

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In recent paper [5], it is shown that the upper decay rate of global solution of compressible Navier-Stokes(CNS) equations converging to constant equilibrium state (1, 0) when the initial data is large and belongs to H 2 (R 3 ) ∩ L p (R 3 )(p ∈ [1, 2)). Thus, the first result in this paper is devoted to showing that the upper decay rate of the first order spatial derivative converging to zero in H 1 −norm is (1 + t)For the case of p = 1, the lower bound of decay rate for the global solution of CNS equations converging to constant equilibrium state (1, 0) in L 2 −norm is (1 + t) − 3 4 if the initial data satisfies some low frequency assumption additionally. In other words, the optimal decay rate for the global solution of CNS equations converging to constant equilibrium state in L 2 −norm is (1 + t) − 3 4 although the associated initial data is large.compressible Navier-Stokes equations. Matsumura and Nishida [15] first established the global existence with the small initial data in H 3 −framework. Later, Valli [29] and Kawashita [10] obtained the global existence with the small initial data in H 2 −framework. Recently, Huang, Li and Xin [8] proved the global existence and uniqueness of system (1.1) with the density containing vacuum in the condition that the initial energy is small. For further results about the well-posedness, we refer the readers to [4,11] and the references therein.The decay problem has been one of main interests in mathematical fluid dynamics, there are many interesting work has been obtained. The optimal decay rate of strong solution was addressed in whole space firstly by Matsumura and Nishida [16], and the optimal L p (p ≥ 2) decay rate is established by Ponce [19]. The authors obtained the optimal decay rate for Navier-Stokes system with an external potential force in series of papers [2,3,28]. By assuming the initial perturbation is bounded inḢ −s rather than L 1 , Guo and Wang [4] built the time decay rate for the solution of system (1.1) by using a general energy method. It should be emphasized that their method in [4] can be used to many other kinds of equations, such as Boltzmann equation, as well as some related fluid models. Many other results for the decay problem for the isentropic or non-isentropic Navier-Stokes equations, one can refer to [12,14,27,30] and the references therein.However, the most of above decay results are established under the condition that the initial data is a small perturbation of constant equilibrium state. A interesting question is what may happen about the large time behavior of global strong solution with general initial data. Very recently, He, Huang and Wang [5] proved global stability of large solution to the system (1.1). Let us give a short review of their work. By using some techniques about the blow-up criterion come from [7,8,25,26,31], and assuming the density is bounded uniformly in time in C α with α arbitrarily small, that is sup t≥0 ρ(t) C α ≤ M , they obtained uniform-in-time bounds for the global solution. This allows them to improve th...

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Section: Proofmentioning

confidence: 99%

The optimal decay rate of strong solution for the compressible Navier–Stokes equations with large initial data

Gao

Wei

Yao

2020

Physica D: Nonlinear Phenomena

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“…However, the decay rate for the highest order spatial derivative of global solution obtained in articles mentioned above is still not optimal. Recently, this tricky problem is addressed simultaneously in a series of articles [4,54,57] by using the spectrum analysis of the linearized system, and [13] by combining the energy estimates with the interpolation between negative and positive Sobolev spaces.…”

Section: Introductionmentioning

confidence: 99%

“…Due to the presence of potential force term ρ∇φ, we can not apply the time weighted method mentioned above to build the optimal decay rate for the third order spatial derivative of global solution directly. In order to overcome this difficult, motivated by [57], we establish some energy estimate for the quantity |ξ|≥η ∇ 2 u• ∇ 3 (ρ − ρ * )dξ, namely the higher frequency part, rather than ∇ 2 u • ∇ 3 (ρ − ρ * )dx. Here ∇ 2 u and ∇ 3 (ρ − ρ * ) stand for the Fourier part of ∇ 2 u and ∇ 3 (ρ − ρ * ), respectively.…”

Section: Introductionmentioning

confidence: 99%

Optimal decay of full compressible Navier-Stokes equations with potential force

Gao¹,

Li²,

Yao³

2022

Preprint

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In this paper, we aim to investigate the optimal decay rate for the higher order spatial derivative of global solution to the full compressible Navier-Stokes (CNS) equations with potential force in R 3 . We establish the optimal decay rate of the solution itself and its spatial derivatives (including the highest order spatial derivative) for global small solution of the full CNS equations with potential force. With the presence of potential force in the considered full CNS equations, the difficulty in the analysis comes from the appearance of non-trivial ststionary solutions. These decay rates are really optimal in the sense that it coincides with the rate of the solution of the linerized system. In addition, the proof is accomplished by virtue of time weighted energy estimate, spectral analysis, and high-low frequency decomposition.

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“…The first result was given in Chio [5] for the small initial perturbation in H l -space, where they obtained the global existence and the L 2 -decay rate for the periodic domain, and the global existence in the whole space. Later on, Wu et al [35] resolved the whole space problem and gave the optimal L 2 -decay rate of the solution and its higher-order spatial derivatives, where they used Hodge decomposition and the spectral analysis. Very recently, Tang and Zhang [30] reconsidered the Cauchy problem by the spectral analysis but without using the Hodge decomposition.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the global existence and L 2 -decay rate in H l with l ≥ 3 have been been given in [35], our main contribution here is the pointwise space-time estimates as above. To the best of our knowledge, it is the first result in this direction for two-phase models.…”

Section: Introductionmentioning

confidence: 99%

Pointwise space-time estimates of two-phase fluid model in dimension three

Zhou²

2021

Preprint

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In this paper, we investigate the pointwise space-time behavior of two-phase fluid model derived by Choi [5] [SIAM J. Math. Anal., 48(2016), pp. 3090-3122], which is the compressible damped Euler equations coupled with compressible Naiver-Stokes equations. Based on Green's function method together with frequency analysis and nonlinear coupling of different wave patterns, it shows that both of two densities and momentums obey the generalized Huygens' principle as the compressible Navier-Stokes equations [21], however, it is different from the compressible damped Euler equations [33]. The main contributions include seeking suitable combinations to avoid the singularity from the Hodge decomposition in the low frequency part of the Green's function, overcoming the difficulty of the non-conservation arising from the damped mechanism of the system, and developing the detailed description of the singularities in the high frequency part of the Green's function. Finally, as a byproduct, we extend L 2 -estimate in [35] [

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Optimal Large-Time Behavior of the Two-Phase Fluid Model in the Whole Space

Cited by 35 publications

References 17 publications

The optimal decay rate of strong solution for the compressible Navier–Stokes equations with large initial data

The optimal decay rate of strong solution for the compressible Navier–Stokes equations with large initial data

Optimal decay of full compressible Navier-Stokes equations with potential force

Pointwise space-time estimates of two-phase fluid model in dimension three

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