On the Field-Induced Transport of Magnetic Nanoparticles in Incompressible Flow: Existence of Global Solutions
On the Field-Induced Transport of Magnetic Nanoparticles in Incompressible Flow: Existence of Global Solutions
复制标题
不可压缩流中磁性纳米颗粒的场致输运：全局解的存在性
DOI：
--
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发表时间：
2020
影响因子：
1.3
通讯作者：
P. Weiss
中科院分区：
文献类型：
--
作者：
G. Grün;P. Weiss
We prove global-in-time existence of weak solutions to a pde-model for the motion of dilute superparamagnetic nanoparticles in fluids influenced by quasi-stationary magnetic fields. This model has recently been derived in Grün and Weiß(On the field-induced transport of magnetic nanoparticles in incompressible flow: modeling and numerics, Mathematical Models and Methods in the Applied Sciences, in press). It couples evolution equations for particle density and magnetization to the hydrodynamic and magnetostatic equations. Suggested by physical arguments, we consider no-flux-type boundary conditions for the magnetization equation which entails H(div,curl)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$H({ ext {div}},{ ext {curl}})$$end{document}-regularity for magnetization and magnetic field. By a subtle approximation procedure, we nevertheless succeed to give a meaning to the Kelvin force (m·∇)hdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$(mathbf {m}cdot
abla )mathbf {h}$$end{document} and to establish existence of solutions in the sense of distributions in two space dimensions. For the three-dimensional case, we suggest two regularizations of the system which each guarantee existence of solutions, too.
We prove global-in-time existence of weak solutions to a pde-model for the motion of dilute superparamagnetic nanoparticles in fluids influenced by quasi-stationary magnetic fields. This model has recently been derived in Grün and Weiß(On the field-induced transport of magnetic nanoparticles in incompressible flow: modeling and numerics, Mathematical Models and Methods in the Applied Sciences, in press). It couples evolution equations for particle density and magnetization to the hydrodynamic and magnetostatic equations. Suggested by physical arguments, we consider no-flux-type boundary conditions for the magnetization equation which entails H(div,curl)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$H({ ext {div}},{ ext {curl}})$$end{document}-regularity for magnetization and magnetic field. By a subtle approximation procedure, we nevertheless succeed to give a meaning to the Kelvin force (m·∇)hdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$(mathbf {m}cdot
abla )mathbf {h}$$end{document} and to establish existence of solutions in the sense of distributions in two space dimensions. For the three-dimensional case, we suggest two regularizations of the system which each guarantee existence of solutions, too.