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Global regularity to the 2D inhomogeneous liquid crystal flows with large initial data and vacuum. (English) Zbl 1505.35074

Summary: We study the 2D incompressible nematic liquid crystal equations in a smooth bounded domain, where the velocity \(u\) and macroscopic molecular orientation \(d\) admit the Dirichlet and Neumann boundary condition, respectively. Under a geometric condition for the initial orientation field, we establish the global existence of strong solutions with large initial data. In particular, the initial density can be allowed to vanish. Furthermore, this result extends the corresponding results of J. Li [Methods Appl. Anal. 22, No. 2, 201–220 (2015; Zbl 1322.35117)] and X. Li [Discrete Contin. Dyn. Syst. 37, No. 9, 4907–4922 (2017; Zbl 1364.35242)] to the Neumann boundary condition and removes the smallness conditions on the initial data.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
76A15 Liquid crystals
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics

References:

[1] H. Brézis and S. Wainger, “A note on limiting cases of Sobolev embeddings and convolution inequalities”, Comm. Partial Differential Equations 5:7 (1980), 773-789. · Zbl 0437.35071 · doi:10.1080/03605308008820154
[2] S. Ding, J. Huang, and F. Xia, “Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum”, Filomat 27:7 (2013), 1247-1257. · Zbl 1324.76014 · doi:10.2298/FIL1307247D
[3] J. L. Ericksen, “Hydrostatic theory of liquid crystals”, Arch. Rational Mech. Anal. 9 (1962), 371-378. · Zbl 0105.23403 · doi:10.1007/BF00253358
[4] G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations: steady-state problems, 2nd ed., Springer, New York, 2011. · Zbl 1245.35002 · doi:10.1007/978-0-387-09620-9
[5] J. Gao, Q. Tao, and Z. Yao, “Strong solutions to the density-dependent incompressible nematic liquid crystal flows”, J. Differential Equations 260:4 (2016), 3691-3748. · Zbl 1333.35193 · doi:10.1016/j.jde.2015.10.047
[6] X. Huang and Y. Wang, “Global strong solution to the 2D nonhomogeneous incompressible MHD system”, J. Differential Equations 254:2 (2013), 511-527. · Zbl 1253.35121 · doi:10.1016/j.jde.2012.08.029
[7] F. M. Leslie, “Some constitutive equations for liquid crystals”, Arch. Rational Mech. Anal. 28:4 (1968), 265-283. · Zbl 0159.57101 · doi:10.1007/BF00251810
[8] J. Li, “Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions”, Nonlinear Anal. 99 (2014), 80-94. · Zbl 1285.35083 · doi:10.1016/j.na.2013.12.023
[9] J. Li, “Global strong solutions to the inhomogeneous incompressible nematic liquid crystal flow”, Methods Appl. Anal. 22:2 (2015), 201-220. · Zbl 1322.35117 · doi:10.4310/MAA.2015.v22.n2.a4
[10] J. Li, “Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density”, J. Differential Equations 263:10 (2017), 6512-6536. · Zbl 1370.76026 · doi:10.1016/j.jde.2017.07.021
[11] X. Li, “Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two”, Discrete Contin. Dyn. Syst. 37:9 (2017), 4907-4922. · Zbl 1364.35242 · doi:10.3934/dcds.2017211
[12] L. Li, Q. Liu, and X. Zhong, “Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum”, Nonlinearity 30:11 (2017), 4062-4088. · Zbl 1386.76016 · doi:10.1088/1361-6544/aa8426
[13] S. Liu and J. Zhang, “Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density”, Discrete Contin. Dyn. Syst. Ser. B 21:8 (2016), 2631-2648. · Zbl 1354.35107 · doi:10.3934/dcdsb.2016065
[14] Q. Liu, S. Liu, W. Tan, and X. Zhong, “Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows”, J. Differential Equations 261:11 (2016), 6521-6569. · Zbl 1364.76015 · doi:10.1016/j.jde.2016.08.044
[15] B. Lü and S. Song, “On local strong solutions to the three-dimensional nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum”, Nonlinear Anal. Real World Appl. 46 (2019), 58-81. · Zbl 1412.35225 · doi:10.1016/j.nonrwa.2018.09.001
[16] B. Lü, X. Shi, and X. Zhong, “Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum”, Nonlinearity 31:6 (2018), 2617-2632. · Zbl 1391.35332 · doi:10.1088/1361-6544/aab31f
[17] L. Nirenberg, “On elliptic partial differential equations”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115-162. · Zbl 0088.07601
[18] J. Simon, “Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure”, SIAM J. Math. Anal. 21:5 (1990), 1093-1117. · Zbl 0702.76039 · doi:10.1137/0521061
[19] H. Wen and S. Ding, “Solutions of incompressible hydrodynamic flow of liquid crystals”, Nonlinear Anal. Real World Appl. 12:3 (2011), 1510-1531. · Zbl 1402.76021 · doi:10.1016/j.nonrwa.2010.10.010
[20] H. Yu and P. Zhang, “Global regularity to the 3D incompressible nematic liquid crystal flows with vacuum”, Nonlinear Anal. 174 (2018), 209-222. · Zbl 1391.76048 · doi:10.1016/j.na.2018.04.022
[21] X. Zhong, “Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations”, Discrete Contin. Dyn. Syst. Ser. B 26:7 (2021), 3563-3578 · Zbl 1467.76075 · doi:10.3934/dcdsb.2020246
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