Time Analyticity for the Navier-Stokes Equations with Nonlinear Damping

LI Ge-ping; ZHU Chao-sheng

doi:10.13718/j.cnki.xsxb.2021.03.019

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In this paper, the algebraic method has been used to deal with the Stokes-Ossen kernel. Then, we prove the time analyticity for the bounded mild solutions of the Navier-Stokes equations with nonlinear damping.

Corresponding author: ZHU Chao-sheng

Received Date: 28/06/2020

Available Online: 20/03/2021

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Abstract: In this paper, the algebraic method has been used to deal with the Stokes-Ossen kernel. Then, we prove the time analyticity for the bounded mild solutions of the Navier-Stokes equations with nonlinear damping.

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近年来，Navier-Stokes方程解析性的相关研究已经引起了广泛的关注^[1-4]. 空间解析性是一个局部性质，即在解的某一个领域内讨论其可导性，通常情况下它都是成立的. 然而时间解析性就很难说明了. 本文代数化地处理Stokes-Ossen核函数，就可以得到$\mathbb{R}$^d(d=2, 3)内带非线性阻尼项的Navier-Stokes方程

其有界温和解的时间解析性. 方程(1)中的阻尼项产生于水流运动的阻力，它可以描述一些物理现象，如多孔介质的流动、阻力、摩擦效应以及某些耗散机制^[5-7]. 这里未知函数u(x, t), p(x, t)分别表示不可压缩流体的速度和压强，β为满足 $\frac{7}{2} \leqslant \beta \leqslant 5$的常数.

本文主要结果如下：

定理1 若问题(1)的温和解u满足

则对任意整数n≥1有

其中N≥1为充分大的常数. 于是，u(x, t)对任意t∈(0, 1]是时间解析的.

本文中C₁, C₂, C₃代表特定的常数，c代表不定常数.

1. 时间解析性

引理1^[9]  对任意n≥1, 下述不等式成立：

其中c > 0为与n无关的常数.

引理2^[9]  设f, g为$\mathbb{R}$上的两个光滑函数，对任意n≥1, 下述关系式成立：

令Stokes-Ossen核为 $E(\boldsymbol{x}, t)=\widetilde{P} \mathit{\Gamma} (\boldsymbol{x}, t)$, 其中$\widetilde{P}$为$\mathbb{R}$^d内的Leray-Hopf投影， $\mathit{\Gamma}=(4 \pi t)^{-\frac{d}{2}} \mathrm{e}^{-\frac{|x|^{2}}{4 t}}$为热核. E为具有半群性质的齐次热方程的解，且 $E(\boldsymbol{x}, t)=t^{-\frac{d}{2}} E\left(\frac{\boldsymbol{x}}{\sqrt{t}}, 1\right)$, 其中E(·, 1)为$\mathbb{R}$^d内与 $\frac{C}{|\boldsymbol{x}|^{d}}$(x→∞) 衰减速率相同的光滑函数. 此外，(∂_tE)(x, 1)与 $\frac{C}{|\boldsymbol{x}|^{d+2}}(\boldsymbol{x} \rightarrow \infty)$衰减速率相同，(∇E)(x, 1)与 $\frac{c}{|\boldsymbol{\boldsymbol{x}}|^{d+1}}(x \rightarrow \infty)$衰减速率相同^[13]. 于是，对任意t > 0和整数k≥1有

其中C₂≥1为常数. 由Leibniz公式^[15]可得

其中C₃≥1为常数.

引理3  在定理1条件成立的前提下，对任意整数n≥1有

其中N≥1为充分大的常数.

证  因为u是问题(1)的温和解，所以对任意t∈(0, 1]有

其中*为空间卷积. 因为

所以

由(2)和(9)式，可得

其中N≥1为充分大的常数. 估计I₂之前，先将其作一个简单的变形，即

所以

由(2)和(4)式可得对任意整数k=1, 2, …, n-1有

以及k=n时有

其中N≥1为充分大的常数. 于是

由(2), (6), (10), (14), (15)式和引理1, 可得

其中N≥1为充分大的常数. 所以

与估计I₂同理，先将I₃作一个简单的变形，即

于是

由(2)和(4)式可得对任意整数k=1, 2, …, n-1有

以及k=n时有

其中N≥1为充分大的常数. 由(2), (5), (9), (17), (18)式和引理1可得

其中N≥1为充分大的常数，所以

将(13), (16)和(19)式代入(12)式，可得对任意t∈(0, 1]有

对(20)式利用Gronwall不等式，可得对任意t∈(0, 1]有

即

其中N≥1为充分大的常数. 综上，引理3证明完成.

定理1的证明  由(3)式可得对任意t∈(0, 1]和整数k=1, 2, …, n有

当k=n时有

再利用(11)式，可得

当k=n-1时有

再利用(21)式可得

由归纳法可得当k=1时有

即

其中N≥1为充分大的常数. 综上，定理1证明完成.

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[1]	BAE H, BISWAS A, TADMOR E. Analyticity and Decay Estimates of the Navier-Stokes Equations in Critical Besov Spaces[J]. Archive for Rational Mechanics and Analysis, 2012, 205(3): 963-991. doi: 10.1007/s00205-012-0532-5 CrossRef Google Scholar
[2]	BAE H. Analyticity of the Inhomogeneous Incompressible Navier-Stokes Equations[J]. Applied Mathematics Letters, 2018, 83: 200-206. doi: 10.1016/j.aml.2018.04.001 CrossRef Google Scholar
[3]	DONG H J, PAN X H. Time Analyticity for Inhomogeneous Parabolic Equations and the Navier-Stokes Equations in the Half Space[J]. Journal of Mathematical Fluid Mechanics, 2020, 22(4): 1-20. Google Scholar
[4]	SUN J Y, YANG M H, CUI S B. Existence and Analyticity of Mild Solutions for the 3D Rotating Navier-Stokes Equations[J]. Annali Di Matematica Pura Ed Applicata, 2016, 196(4): 1203-1229. doi: 10.1007/s10231-016-0613-4 CrossRef Google Scholar
[5]	GHISI M, GOBBINO M. Linear Wave Equations with Time-Dependent Propagation Speed and Strong Damping[J]. Journal of Differential Equations, 2016, 260(2): 1585-1621. doi: 10.1016/j.jde.2015.09.037 CrossRef Google Scholar
[6]	PAN X H. Global Existence of Solutions to 1-d Euler Equations with Time-Dependent Damping[J]. Nonlinear Analysis, 2016, 132: 327-336. Google Scholar
[7]	ZHU X S. Blowup of the Solutions for the IBVP of the Isentropic Euler Equations with Damping[J]. Journal of Mathematical Analysis and Applications, 2015, 432(2): 715-724. doi: 10.1016/j.jmaa.2015.06.072 CrossRef Google Scholar
[8]	CHARVE F, DANCHIN R, XU J. Gevrey Analyticity and Decay for the Compressible Navier-Stokes System with Capillarity[EB/OL]. (2018-05-04)[2020-05-20]. https://arxiv.org/abs/1805.01764. Google Scholar
[9]	DONG H J, ZHANG Q S. Time Analyticity for the Heat Equation and Navier-Stokes Equations[J]. Journal of Functional Analysis, 2020, 279(4): 108563-108577. doi: 10.1016/j.jfa.2020.108563 CrossRef Google Scholar
[10]	LEMARIE-RIEUSSET P G. The Navier-Stokes Problem in the 21st Century[M]. Boca Raton: Taylor & Francis, 2016. Google Scholar
[11]	PAZY A. Semigroups of Linear Operators and Applications to Partial Differential Equations[M]. New York: Springer-Verlag, 1983. Google Scholar
[12]	ROBINSON J C, PIERRE C. Infinite-Dimensional Dynamical Systems: an Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors[M]. Cambridge: Cambridge University Press, 2001. Google Scholar
[13]	SOLONNIKOV V A. Estimates for Solutions of Anon-Stationary Linearized System of Navier-Stokes Equations[J]. Trudy Matematicheskogo Instituta imeni V. A. Steklova, 1964(5): 213-317. Google Scholar
[14]	TEMAM R. Navier-Stokes Equations: Theory and Numerical Analysis[M]. New York: North-Holland PublishingCompany, 1977. Google Scholar
[15]	华东师范大学数学系. 数学分析-下册[M]. 4版. 北京: 高等教育出版社, 2010: 185-188. Google Scholar

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