高阶各向异性Cahn-Hilliard-Navier-Stokes系统的弱解

数学物理学报 ›› 2020, Vol. 40 ›› Issue (6): 1599-1611.

高阶各向异性Cahn-Hilliard-Navier-Stokes系统的弱解

罗娇(),漆前(),罗宏*()

四川师范大学数学科学学院成都 610066

收稿日期:2020-06-03 出版日期:2020-12-26 发布日期:2020-12-29
通讯作者: 罗宏 E-mail:656508503@qq.com;1374852064@qq.com;lhscnu@163.com
作者简介:罗娇, E-mail: 656508503@qq.com|漆前, E-mail: 1374852064@qq.com
基金资助:
国家自然科学基金(10771306);国家自然科学基金(11701399)

Weak Solutions to Higher-Order Anisotropic Cahn-Hilliard-Navier-Stokes Systems

Jiao Luo(),Qian Qi(),Hong Luo*()

School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066

Received:2020-06-03 Online:2020-12-26 Published:2020-12-29
Contact: Hong Luo E-mail:656508503@qq.com;1374852064@qq.com;lhscnu@163.com
Supported by:
the NSFC(10771306);the NSFC(11701399)

摘要/Abstract

摘要：

该文主要研究二维高阶各向异性的Cahn-Hilliard-Navier-Stokes系统的弱解.该系统由高阶各向异性的Cahn-Hilliard方程与不可压缩的Navier-Stokes方程耦合而成.首先引入泛函空间, 给出弱解的定义.其次, 给出了解的能量估计, 由Galerkin方法得到该系统的弱解, 最后得到解的唯一性.

关键词: 各向异性, Cahn-Hilliard-Navier-Stokes系统, 弱解, 存在性, 唯一性

Abstract:

In this paper, we are concerned with weak solutions to higher-order anisotropic Cahn-Hilliard-Navier-Stokes systems in a two-dimensional bounded domain. The system consists of an incompressible Navier-Stokes equation coupled with a higher-order anisotropic Cahn-Hilliard equation. First some functional spaces and the definition of weak solutions are introduced. Then energy estimates of the solutions are given, and weak solutions of the system are obtained by Galerkin approximation scheme. Furthermore, the uniqueness of the solution to the system is proved.

Key words: Anisotropic, Cahn-Hilliard-Navier-Stokes system, Weak solutions, Existence, Uniqueness

中图分类号:

O175.29

罗娇,漆前,罗宏. 高阶各向异性Cahn-Hilliard-Navier-Stokes系统的弱解[J]. 数学物理学报, 2020, 40(6): 1599-1611.

Jiao Luo,Qian Qi,Hong Luo. Weak Solutions to Higher-Order Anisotropic Cahn-Hilliard-Navier-Stokes Systems[J]. Acta mathematica scientia,Series A, 2020, 40(6): 1599-1611.

参考文献 26

1	Anderson D M , Mcfadden G B , Wheeler A A . Diffuse-interface methods in fluid mechanics. Annual Review of Fluid Mechanics, 1998, 30 (1): 139- 165
2	Gurtin M E , Polignone D , Vinals J . Two-phase binary fluids and immiscible fluids described by an order parameter. Mathematical Models and Methods in Applied Sciences, 1996, 6 (6): 815- 831
3	Jasnow D , Vinals J . Coarse-grained description of thermo-capillary flow. Physics of Fluids, 1996, 8 (3): 660- 669
4	Lamorgese A G , Molin D , Mauri R . Phase field approach to multiphase flow modeling. Milan Journal of Mathematics, 2011, 79 (2): 597- 642
5	Ables H , Feireisl E . On a diffuse interface model for a two-phase flow of compressible viscous fluids. Indiana University Mathematics Journal, 2008, 57 (2): 659- 698
6	K im , Junseok . Phase-field models for multi-component fluid flows. Communications in Computational Physics, 2012, 12 (3): 613- 661
7	Liu H , Sengul T , Wang S , Zhang P . Dynamic transitions and pattern formations for a Cahn-Hilliard model with long-range repulsive interactions. Communications in Mathematical Sciences, 2015, 13 (5): 1289- 1315
8	Song L , Zhang Y , Ma T . Global attractor of the Cahn-Hilliard equation in H^k spaces. Journal of Mathematical Analysis and Applications, 2009, 355: 53- 62
9	You B . Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Communications on Pure and Applied Analysis, 2019, 18 (5): 2283- 2298
10	Gal C G , Grasselli M . Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2010, 27 (1): 401- 436
11	Cao C S , Gal C G . Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility. Nonlinearity, 2012, 25 (11): 3211- 3234
12	Cherfils L , Miranville A , Peng S R . Higher-order models in phase separation. Journal of Applied Analysis and Computation, 2017, 7 (1): 39- 56
13	Mininni R M , Miranville A , Romanelli S . Higher-order Cahn-Hilliard equations with dynamic boundary conditions. Journal of Mathematical Analysis and Applications, 2017, 449 (2): 1321- 1339
14	Cherfils L , Gatti S , Miranville A . Asymptotic behavior of higher-order Navier-Stokes-Cahn-Hilliard systems. Mathematical Methods in the Applied Sciences, 2018, 41 (12): 4476- 4794
15	Pan J J , Xing C , Luo H . Uniform regularity of the weak solution to higher-order Navier-Stokes-Cahn-Hilliard systems. Journal of Mathematical Analysis and Applications, 2020, 486 (2): 123925
16	Esenturk E , Caginalp G . Anisotropic phase field equations of arbitrary order. Discrete and Continuous Dynamical Systems, 2010, 4 (2): 311- 350
17	Cherfils L , Miranville A , Peng S R . Higher-order anisotropic models in phase separation. Advances in Nonlinear Analysis, 2019, 8 (1): 278- 302
18	Cherfils L , Miranville A , Peng S R , Zhang W . Higher-order generalized Cahn-Hilliard equations. Electronic Journal of Qualitative Theory of Differential Equations, 2017, 9: 1- 22
19	Peng S R , Zhu H Y . Well-posedness for modified higher-order anisotropic Cahn-Hilliard equations. Asymptotic Analysis, 2019, 111 (3/4): 201- 215
20	Cahn J W , Hilliard J E . Free energy of nonuniform system, Ⅰ. interfacial free energy. J Chemical Physis, 1958, 28 (2): 258- 267
21	Agmon S , Douglis A , Nirenberg L . Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions Ⅱ. Communications on Pure and Applied Mathematics, 1964, 17 (1): 35- 92
22	Cholewa J W , Dlotko T . Global Attractors in Abstract Parabolic Problems. Cambridge: Cambridge University Press, 2000
23	Giorgi C , Grasselli M , Pata V . Uniform attractors for a phase-field Model with memory and quadratic nonlinearity. Indiana University Mathematics Journal, 1999, 48 (4): 1395- 1444
24	Boyer F . Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptotic Analysis, 1999, 20 (4): 175- 212
25	Pal S , Haloi R . On solution to the Navier-Stokes equations with Navier slip boundary condition for three dimensional incompressible fluid. Acta Mathematica Scientia, 2019, 39B (6): 1628- 1638
26	李凯, 杨晗, 王凡. 三维带有衰减项的不可压缩磁流体力学方程组弱解与强解的研究. 数学物理学报, 2019, 39A (3): 518- 528
	Li K , Yang H , Wang F . Study on weak solution and strong solution of incompressible MHD equations with damping in three-dimensional systems. Acta Mathematica Scientia, 2019, 39A (3): 518- 528

[1]	西宣宣,侯咪咪,周先锋. 非自治Caputo分数阶发展方程弱解的适定性与不变集[J]. 数学物理学报, 2021, 41(1): 149-165.
[2]	王娟,原子霞. 具有对数敏感度和混合边界的一维趋化模型解的整体存在性和收敛性[J]. 数学物理学报, 2020, 40(6): 1646-1669.
[3]	佟玉霞,王薪茹,谷建涛. Orlicz空间中A-调和方程很弱解得L^ϕ估计[J]. 数学物理学报, 2020, 40(6): 1461-1480.
[4]	李海侠. 一类具有毒素的非均匀chemostat模型正解的存在性和唯一性[J]. 数学物理学报, 2020, 40(5): 1175-1185.
[5]	黄小梦,张贻民. 一类修正Gross-Pitaevskii方程基态解的存在性[J]. 数学物理学报, 2020, 40(4): 850-856.
[6]	钱红丽,黄小涛. 一类p-Laplacian方程解的存在性与对称性研究[J]. 数学物理学报, 2020, 40(3): 725-734.
[7]	张雅楠,闫硕,佟玉霞. 自然增长条件下的非齐次A-调和方程弱解的梯度估计[J]. 数学物理学报, 2020, 40(2): 379-394.
[8]	许诗敏,王春花. Kirchhoff方程单峰解的局部唯一性[J]. 数学物理学报, 2020, 40(2): 432-440.
[9]	张利民,徐海燕,金春花. 具渗流扩散和间接信号产生的Prey-Taxis模型解的整体存在性与稳定性[J]. 数学物理学报, 2019, 39(6): 1381-1404.
[10]	吴杰,林红霞. 关于抛物-抛物Keller-Segel类模型的全局解和渐近性[J]. 数学物理学报, 2019, 39(5): 1102-1114.
[11]	钟澎洪,杨干山,马璇. 双曲空间上的Landau-Lifshitz-Gilbert方程解的全局存在性与自相似爆破解[J]. 数学物理学报, 2019, 39(3): 461-474.
[12]	李凯,杨晗,王凡. 三维带有衰减项的不可压缩磁流体力学方程组弱解与强解的研究[J]. 数学物理学报, 2019, 39(3): 518-528.
[13]	计婷,胡良根,曾晶. 拟线性椭圆系统非径向爆破解的非存在性[J]. 数学物理学报, 2019, 39(2): 307-315.
[14]	周凯,金路. 以较低截断重数分担超平面的亚纯映射的唯一性问题[J]. 数学物理学报, 2019, 39(1): 1-14.
[15]	胡弟弟,汪璇. 记忆型无阻尼抽象发展方程的强时间依赖全局吸引子[J]. 数学物理学报, 2019, 39(1): 81-94.