Global Regularity for a Model of Inhomogeneous Three-dimensional Navier-Stokes Equations
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摘要: 考虑一个非均质三维Navier-Stokes方程模型,借助能量方法、Littlewood-Paley仿积分解技巧和Sobolev嵌入定理研究解的整体正则性. 用-D2u近似替代经典非均质Navier-Stokes方程中的耗散项Δu,得到一个新的Navier-Stokes方程模型,其中D是一个傅里叶乘子,其特征是m(ξ)=|ξ|5/4,对于任意小的正常数ε和δ,当初值(ρ0,u0)∈H3/2+ε×Hδ时,证明了该模型解的爆破准则和整体正则性.
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关键词:
- 非均质Navier-Stokes方程 /
- Littlewood-Paley仿积分解 /
- 整体正则性 /
- 爆破准则
Abstract: A model of inhomogeneous three-dimensional Navier-Stokes equations was studied in this paper. By using the energy method, Littlewood-Paley paraproduct decomposition techniques and Sobolev embedding theorem study of the global regularity of solutions were adopted. The dissipative term Δu in the classical inhomogeneous Navier-Stokes equations is replaced by -D2u and a new Navier-Stokes equations model was obtained, where D was a Fourier multiplier whose symbol is m(ξ)=|ξ|5/4. Blow-up criterion and global regularity of this model were proved for the initial data (ρ0,u0)∈H3/2+ε×Hδ, where ε and δ are arbitrary small positive constants. -
[1] KAZHIKOV A V.Resolution of boundary value problems for nonhomogeneous vicous fluids[J]. Dokl Akad Nauh, 1974, 216:1008-1010. [2] SIMON J.Sur les fluides visqueux incompressibles et non homogènes[J]. C R Acad Sci Paris, 1989, 309: 447-452. [3] SIMON J.Nonhomogeneous vicous incompressible fluids: existence of velocity, density, and pressure[J]. SIAM J Math Anal, 1990, 21: 1093-1117. [4] LIONS P L.Mathematical topics in fluid mechanics: vol 1 incompressible models, oxford lecture series in mathematics and its applications[M]. New York: The Clarendon Press, Oxford University Press, 1996: 237. [5] LADYZHENSKAYA O, SOLONNIKOV V A.Unique solvability of an initial and boundary value problem for vicous incompressible nonhomogeneous fluids[J].J Soviet Math, 1978(9): 697-749. [6] PADULA M.An existence theorem for non-homogeneous incompressible fluids[J]. Rend Circ Mat Palermo, 1982, 31: 119-124. [7] SALVI R.The equations of vicous incompressible nonhomogeneous fluids: on the existence and regularity[J]. J Austral Math Soc: Ser B, 1991, 33: 94-110. [8] CHOE H J, KIM H.Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids[J]. Partial Differential Equations, 2003, 28: 1183-1201. [9] TONG L N, YUAN H J.Classical solutions to Navier-Stokes equations for nonhomogeneous incompressible fluids with non-negetive densities[J]. J Math Anal Appl, 2010, 362: 476-504. [10] FUJITA H, KATO T.On the Navier-Stokes initial value problems I[J]. Arch Rational Mech Anal, 1964, 16: 269-315. [11] DANCHIN R.Density-dependent incompressible viscous fluids in critical spaces[J]. Proc Roy Soc Edinburgh Sect A, 2003, 133: 1311-1334. [12] DANCHIN R.Local and global well-posedness results for flows of inhomogeneous viscous fluids[J].Adv Differential Equations, 2004(9): 353-386. [13] TAO T.Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation[J].Anal PDE, 2009(2): 361-366. [14] 苗长兴, 吴家宏, 章志飞. Little-Paley理论及其在流体动力学方程中的应用[M]. 北京: 科学出版社, 2015: 54-56. [15] DANCHIN R.The inviscid limit for density-dependent incompressible fluids[J].Ann Fac Sci Toulouse Math, 2006(15): 637-688. [16] BAHOURI H, CHEMIN J Y, DANCHIN R.Fourier analysis and nonlinear partial differential equations[M]. Heidelberg: Springer, 2011: 343. -
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