一个关于不可压液晶系统的适当弱解的局部Serrin型的正则性条件
|
摘要
本文研究液晶流中简化版的Ericksen-Leslie方程.对于三维不可压液晶系统,本文证明了它的适当弱解在Neustupa (2014)条件下的部分正则性结果.
In this paper, we study the simplified system of the original Ericksen-Leslie equations for the flow of liquid crystals. For the three-dimensional incompressible liquid crystal system, we prove partial regularity results of its suitable weak solutions under Neustupa's(2014) criteria.
In this paper, we study the simplified system of the original Ericksen-Leslie equations for the flow of liquid crystals. For the three-dimensional incompressible liquid crystal system, we prove partial regularity results of its suitable weak solutions under Neustupa's(2014) criteria.
引文
1 de Gennes P G.The Physics of Liquid Crystals.Oxford:Oxford University Press,1974
2 Ericksen J L.Hydrostatic theory of liquid crystals.Arch Ration Mech Anal,1962,9:371-378
3 Leslie F M.Some constitutive equations for liquid crystals.Arch Ration Mech Anal,1968,28:265-283
4 Lin F,Lin J,Wang C.Liquid crystal flows in two dimensions.Arch Ration Mech Anal,2010,197:297-336
5 Lin F,Wang C.Global existence of weak solutions of the nematic liquid crystal flow in dimension three.Commun Pure Appl Anal,2016,69:1532-1571
6 Lin F H,Liu C.Nonparabolic dissipative systems modeling the flow of liquid crystals.Commun Pure Appl Anal,1995,48:501-537
7 Lin F H,Liu C.Partial regularity of the dynamic system modeling the flow of liquid crystals.Discrete Contin Dyn Syst,1996,2:1-22
8 Liu Q,Cui S B.Regularity of solutions to 3-D nematic liquid crystal flows.Electron J Differential Equations,2010,2010:1-5
9 Liu X G,Min J Z,Wang K,et al.Serrin’s regularity results for the incompressible liquid crystals system.Discrete Contin Dyn Syst,2016,36:5579-5594
10 Serrin J.On the interior regularity of weak solutions of the Navier-Stokes equations.Arch Ration Mech Anal,1962,9:187-195
11 Struwe M.On partial regularity results for the navier-stokes equations.Commun Pure Appl Anal,1988,41:437-458
12 Liu Q,Zhao J,Cui S.A regularity criterion for the three-dimensional nematic liquid crystal flow in terms of one directional derivative of the velocity.J Math Phys,2011,52:033102
13 Gala S,Liu Q,Ragusa M A.Logarithmically improved regularity criterion for the nematic liquid crystal flows in˙B-1∞,∞space.Comput Math Appl,2013,65:1738-1745
14 Guillen-Gonzalez F,Rodriguez-Bellido M A,Rojas-Medar M A.Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model.Math Nachr,2009,282:846-867
15 Qian C Y.Remarks on the regularity criterion for the nematic liquid crystal flows in R3.Appl Math Comput,2016,274:679-689
16 Scheffer V.Partial regularity of solutions to the Navier-Stokes equations.Pacific J Math,1976,66:535-552
17 Scheffer V.Hausdorff measure and the Navier-Stokes equations.Comm Math Phys,1977,55:97-112
18 Caffarelli L,Kohn R,Nirenberg L.Partial regularity of suitable weak solutions of the Navier-Stokes equations.Comm Pure Appl Math,1982,35:771-831
19 Kuˇcera P,Skalak Z.A note on the generalized energy inequality in the Navier-Stokes equations.Appl Math,2003,48:537-545
20 Ladyzhenskaya O A,Seregin G A.On partial regularity of suitable weak solutions to the three-dimensional NavierStokes equations.J Math Fluid Mech,1999,1:356-387
21 Lin F H.A new proof of the Caffarelli-Kohn-Nirenberg theorem.Comm Pure Appl Math,1988,51:241-257
22 Vasseur A F.A new proof of partial regularity of solutions to Navier-Stokes equations.NoDEA Nonlinear Differential Equations Appl,2007,14:753-785
23 Wolf J.A new criterion for partial regularity of suitable weak solutions to the Navier-Stokes equations.In:Advances in Mathematical Fluid Mechanics.Berlin-Heidelberg:Springer,2010,613-630
24 Seregin G A.On the local regularity of suitable weak solutions of the Navier-Stokes equations.Uspekhi Mat Nauk,2007,62:149-168
25 Neˇcas J,Neustupa J.New conditions for local regularity of a suitable weak solution to the Navier-Stokes equation.JMath Fluid Mech,2002,4:237-256
26 Seregin G.On smoothness of L3,∞-solutions to the Navier-Stokes equations up to boundary.Math Ann,2005,332:219-238
27 Seregin G,Sverak V.On smoothness of suitable weak solutions to the Navier-Stokes equations.J Math Sci,2005,130:4884-4892
28 Takahashi S.On interior regularity criteria for weak solutions of the Navier-Stokes equations.Manuscripta Math,1990,69:237-254
29 Neustupa J.A removable singularity in a suitable weak solution to the Navier-Stokes equations.Nonlinearity,2012,25:1695-1708
30 Neustupa J.A refinement of the local serrin-type regularity criterion for a suitable weak solution to the Navier-Stokes equations.Arch Ration Mech Anal,2014,214:525-544
31 Gilbarg D,Trudinger N S.Elliptic Partial Differential Equations of Second Order.Berlin:Springer-Verlag,2001
32 Borchers W,Sohr H.On the equations rotv=g and divu=f with zero boundary conditions.Hokkaido Math J,1990,19:67-87
33 Saks R S.Spectral problems for curl and Stokes operators(in Russian).Dokl Akad Nauk,2007,416:446-450;translation in Dokl Math,2007,76:724-728
34 Lai B,Ma W.On the interior regularity criteria for liquid crystal flows.Nonlinear Anal Real World Appl,2018,40:1-13
35 Gustafson S,Kang K,Tsai T P.Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations.Comm Math Phys,2007,273:161-176
2 Ericksen J L.Hydrostatic theory of liquid crystals.Arch Ration Mech Anal,1962,9:371-378
3 Leslie F M.Some constitutive equations for liquid crystals.Arch Ration Mech Anal,1968,28:265-283
4 Lin F,Lin J,Wang C.Liquid crystal flows in two dimensions.Arch Ration Mech Anal,2010,197:297-336
5 Lin F,Wang C.Global existence of weak solutions of the nematic liquid crystal flow in dimension three.Commun Pure Appl Anal,2016,69:1532-1571
6 Lin F H,Liu C.Nonparabolic dissipative systems modeling the flow of liquid crystals.Commun Pure Appl Anal,1995,48:501-537
7 Lin F H,Liu C.Partial regularity of the dynamic system modeling the flow of liquid crystals.Discrete Contin Dyn Syst,1996,2:1-22
8 Liu Q,Cui S B.Regularity of solutions to 3-D nematic liquid crystal flows.Electron J Differential Equations,2010,2010:1-5
9 Liu X G,Min J Z,Wang K,et al.Serrin’s regularity results for the incompressible liquid crystals system.Discrete Contin Dyn Syst,2016,36:5579-5594
10 Serrin J.On the interior regularity of weak solutions of the Navier-Stokes equations.Arch Ration Mech Anal,1962,9:187-195
11 Struwe M.On partial regularity results for the navier-stokes equations.Commun Pure Appl Anal,1988,41:437-458
12 Liu Q,Zhao J,Cui S.A regularity criterion for the three-dimensional nematic liquid crystal flow in terms of one directional derivative of the velocity.J Math Phys,2011,52:033102
13 Gala S,Liu Q,Ragusa M A.Logarithmically improved regularity criterion for the nematic liquid crystal flows in˙B-1∞,∞space.Comput Math Appl,2013,65:1738-1745
14 Guillen-Gonzalez F,Rodriguez-Bellido M A,Rojas-Medar M A.Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model.Math Nachr,2009,282:846-867
15 Qian C Y.Remarks on the regularity criterion for the nematic liquid crystal flows in R3.Appl Math Comput,2016,274:679-689
16 Scheffer V.Partial regularity of solutions to the Navier-Stokes equations.Pacific J Math,1976,66:535-552
17 Scheffer V.Hausdorff measure and the Navier-Stokes equations.Comm Math Phys,1977,55:97-112
18 Caffarelli L,Kohn R,Nirenberg L.Partial regularity of suitable weak solutions of the Navier-Stokes equations.Comm Pure Appl Math,1982,35:771-831
19 Kuˇcera P,Skalak Z.A note on the generalized energy inequality in the Navier-Stokes equations.Appl Math,2003,48:537-545
20 Ladyzhenskaya O A,Seregin G A.On partial regularity of suitable weak solutions to the three-dimensional NavierStokes equations.J Math Fluid Mech,1999,1:356-387
21 Lin F H.A new proof of the Caffarelli-Kohn-Nirenberg theorem.Comm Pure Appl Math,1988,51:241-257
22 Vasseur A F.A new proof of partial regularity of solutions to Navier-Stokes equations.NoDEA Nonlinear Differential Equations Appl,2007,14:753-785
23 Wolf J.A new criterion for partial regularity of suitable weak solutions to the Navier-Stokes equations.In:Advances in Mathematical Fluid Mechanics.Berlin-Heidelberg:Springer,2010,613-630
24 Seregin G A.On the local regularity of suitable weak solutions of the Navier-Stokes equations.Uspekhi Mat Nauk,2007,62:149-168
25 Neˇcas J,Neustupa J.New conditions for local regularity of a suitable weak solution to the Navier-Stokes equation.JMath Fluid Mech,2002,4:237-256
26 Seregin G.On smoothness of L3,∞-solutions to the Navier-Stokes equations up to boundary.Math Ann,2005,332:219-238
27 Seregin G,Sverak V.On smoothness of suitable weak solutions to the Navier-Stokes equations.J Math Sci,2005,130:4884-4892
28 Takahashi S.On interior regularity criteria for weak solutions of the Navier-Stokes equations.Manuscripta Math,1990,69:237-254
29 Neustupa J.A removable singularity in a suitable weak solution to the Navier-Stokes equations.Nonlinearity,2012,25:1695-1708
30 Neustupa J.A refinement of the local serrin-type regularity criterion for a suitable weak solution to the Navier-Stokes equations.Arch Ration Mech Anal,2014,214:525-544
31 Gilbarg D,Trudinger N S.Elliptic Partial Differential Equations of Second Order.Berlin:Springer-Verlag,2001
32 Borchers W,Sohr H.On the equations rotv=g and divu=f with zero boundary conditions.Hokkaido Math J,1990,19:67-87
33 Saks R S.Spectral problems for curl and Stokes operators(in Russian).Dokl Akad Nauk,2007,416:446-450;translation in Dokl Math,2007,76:724-728
34 Lai B,Ma W.On the interior regularity criteria for liquid crystal flows.Nonlinear Anal Real World Appl,2018,40:1-13
35 Gustafson S,Kang K,Tsai T P.Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations.Comm Math Phys,2007,273:161-176