Decay estimates of solutions to the compressible Navier–Stokes–Maxwell system in $\mathbb{R}^3$

Tan, Zhong; Tong, Leilei

doi:10.4310/cms.2016.v14.n5.a1

Cited by 7 publications

(6 citation statements)

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“…Specifically, they combined energy estimates with the interpolation between negative and positive Sobolev norms and obtained the time decay rates for the isentropic compressible Navier-Stokes equations and Boltzmann equation. The new method developed in [5] has a wide range of applications recently, see [38][39][40][41][42][43][44]. It should be noticed that all of these decay results are established under the assumption that the initial data is a small perturbation of constant equilibrium state.…”

Section: Introductionmentioning

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: Introductionmentioning

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 this section, we will derive the evolution of the negative Besov norms of the solution. The negative Besov space also used in [18], however, we derive a different form of estimates for our low regularity assumption.…”

Section: Energy Evolution Of Negative Besov Normsmentioning

confidence: 99%

Optimal Time Decay of Navier-Stokes Equations With Low Regularity Initial Data

Jia¹,

Peng²

2015

Preprint

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In this paper, we study the optimal time decay rate of isentropic Navier-Stokes equations under the low regularity assumptions about initial data. In the previous works about optimal time decay rate, the initial data need to be small in H [N/2]+2 (R N ). Our work combined negative Besov space estimates and the conventional energy estimates in Besov space framework which is developed by R. Danchin. Though our methods, we can get optimal time decay rate with initial data just small in ḂN/2−1,N/2+1 ∩ ḂN/2−1,N/2 and belong to some negative Besov space(need not to be small). Finally, combining the recent results in [21] with our methods, we can only need the initial data to be small in homogeneous Besov space ḂN/2−2,N/2 ∩ ḂN/2−1 to get the optimal time decay rate in space L 2 .

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“…Motivated by the work of Refs. 22, 41, 42, in this paper, we will consider the Cauchy problem of the compressible quantum Navier–Stokes–Maxwell equations with the linear damping in

\mathbb {R}^3

, and prove the global existence and time decay rate of the classical solution. The system () is supplemented by the initial data as follows:

\begin{align} (\rho ,u,E,M)^T(x, t)|_{t=0}=(\rho _0, u _0, E_0, M_0)^T(x), \end{align}

and the compatible conditions will be

$$\begin{align*} {\rm div}E_0=1-\rho _0,\ \ {\rm div}M_0=0.

…”

Section: Introductionmentioning

confidence: 99%

Global existence and the time decay estimates of solutions to the compressible quantum Navier–Stokes–Maxwell system in R3$\mathbb {R}^3$

Tong,

Luo

2023

Stud Appl Math

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We consider the Cauchy problem of the compressible quantum Navier–Stokes–Maxwell equations with the linear damping in the isentropic case under the small perturbation of the constant equilibrium state in three dimensions. Based on the refined energy method, we establish the classical solution globally in time in Sobolev space. By the combination of the energy estimates with the interpolation between the positive Sobolev norms and the negative Sobolev norms with , we also obtain the algebraic decay rates of the classical solution. What is more, the –L2 types of the time decay rates of the solution are obtained without small assumption on the initial data in L1 norm.

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Decay estimates of solutions to the compressible Navier–Stokes–Maxwell system in $\mathbb{R}^3$

Cited by 7 publications

References 0 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 Time Decay of Navier-Stokes Equations With Low Regularity Initial Data

Global existence and the time decay estimates of solutions to the compressible quantum Navier–Stokes–Maxwell system in R3$\mathbb {R}^3$

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