趋化NS系统弱解的全局存在性（英文）
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摘要
本文主要研究趋化NS系统在2维的有界光滑领域Ω?R~2中.本文利用Galerkin方法证明了不可压缩的NS系统弱解的存在性.其次,利用一系列检验程序,证明了带有初边值条件的趋化NS系统弱解的局部存在性,进一步得到该系统弱解的全局存在性.
In this paper, the chemotaxis NS system is considered in two-dimensional bounded domain Ω ? R~2 with smooth boundary. It proves the existence of weak solution of the incompressible NS system by using Galerkin method in two-dimensional. Moreover,it obtains that the system with given initial data and the corresponding initial-boundary value problem possesses a global weak solution by combining standard testing procedures with regularity estimate.
In this paper, the chemotaxis NS system is considered in two-dimensional bounded domain Ω ? R~2 with smooth boundary. It proves the existence of weak solution of the incompressible NS system by using Galerkin method in two-dimensional. Moreover,it obtains that the system with given initial data and the corresponding initial-boundary value problem possesses a global weak solution by combining standard testing procedures with regularity estimate.
引文
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[7] CAO X. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces[J]. Discrete and Continuous Dynamical Systems, 2015, 35:1891-1904.
[8] LIU D. Boundedness in a chemotaxis system with nonlinear signal production[J]. Applied Mathematics, 2016, 31:379-388.
[9] SOHR H. The navier-stokes equations. An elementary functional analytic approach[J]. Birkhauser Advanced Texts Basler Lehrbucher, 2001, 22:3917-3935.
[10] MA T. Theories and Methods in Partial Differential Equations[M]. Beijing:Science Press(in Chinese),2011.
[11] ESPEJO E, SUZUKI T. Reaction terms avoiding aggregation in slow fluids[J]. Nonlinear Analysis Real World Applications, 2015, 21:110-126.
[12] ZHENG J S. Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion[J]. Journal of Differential Equations, 2017, 5:1-28.
[13] WANG Y, XIANG Z. Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation[J]. Journal of Differential Equations, 2015, 259:7578-7609.
[14] CAO X, LANKEIT J. Global classical small-data solutions for a three-dimensional chemotaxis NavierStokes system involving matrix-valued sensitivities[J]. Calculus of Variations and Partial Differential Equations, 2016, 55:1-32.
[15] EVANS L C. Partial Differential Equations[M]. Providence, Rhode Island:American Mathematical Society, 1998.
[16] BLACK T. Sublinear signal production in a two-dimensional Keller-Segel-Stokes system[J]. Nonlinear Analysis Real World Applications, 2016, 31:593-609.
[17] LANKEIT J, LI Y. Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion[J]. Nonlinearity, 2016, 29:1564-1595.
[18] HENRY D. Geometric Theory of Semilinear Parabolic Equations[M]. New York/Berlin/Heidelberg:Springer-Verlag, 1981.
[2] HILLEN T, PAINTER K J. A user's guide to PDE models for chemotaxis[J]. Journal Of Mathematical Biology, 2009, 58:183-217.
[3] BEIIOMO N, BELLOUQUID A. On multiscale models of pedestrian crowds from mesoscopic to macroscopic[J]. Communications in Mathematical Sciences, 2015, 13:1649-1664.
[4] KELLER E F, SEGEL L A. Initiation of slide mold aggregation viewed as an instability[J]. Journal of Theoretical Biology, 1970, 26:399-415.
[5] AIDA M, TSUJIKAWA T, EFENDIEV M. et al. Lower estimate of the attractor dimension for a chemotaxis growth system[J]. Journal of the London Mathematical Society, 2006, 74:453-474.
[6] BELLOMO N, BELLOUQUID A, NIETO J. et al. Complexity and mathematical tools towards the modelling multicellular growing system in biology[J]. Mathematical and Computer Modelling, 2010,51:441-451.
[7] CAO X. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces[J]. Discrete and Continuous Dynamical Systems, 2015, 35:1891-1904.
[8] LIU D. Boundedness in a chemotaxis system with nonlinear signal production[J]. Applied Mathematics, 2016, 31:379-388.
[9] SOHR H. The navier-stokes equations. An elementary functional analytic approach[J]. Birkhauser Advanced Texts Basler Lehrbucher, 2001, 22:3917-3935.
[10] MA T. Theories and Methods in Partial Differential Equations[M]. Beijing:Science Press(in Chinese),2011.
[11] ESPEJO E, SUZUKI T. Reaction terms avoiding aggregation in slow fluids[J]. Nonlinear Analysis Real World Applications, 2015, 21:110-126.
[12] ZHENG J S. Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion[J]. Journal of Differential Equations, 2017, 5:1-28.
[13] WANG Y, XIANG Z. Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation[J]. Journal of Differential Equations, 2015, 259:7578-7609.
[14] CAO X, LANKEIT J. Global classical small-data solutions for a three-dimensional chemotaxis NavierStokes system involving matrix-valued sensitivities[J]. Calculus of Variations and Partial Differential Equations, 2016, 55:1-32.
[15] EVANS L C. Partial Differential Equations[M]. Providence, Rhode Island:American Mathematical Society, 1998.
[16] BLACK T. Sublinear signal production in a two-dimensional Keller-Segel-Stokes system[J]. Nonlinear Analysis Real World Applications, 2016, 31:593-609.
[17] LANKEIT J, LI Y. Boundedness in a chemotaxis-haptotaxis model with nonlinear diffusion[J]. Nonlinearity, 2016, 29:1564-1595.
[18] HENRY D. Geometric Theory of Semilinear Parabolic Equations[M]. New York/Berlin/Heidelberg:Springer-Verlag, 1981.