On the global existence of classical solutions for compressible nematic liquid crystal flows with vacuum

Yang Liu1,2
1College of Mathematics, Changchun Normal University, Changchun, People’s Republic of China
2Department of Mathematics, Nanjing University, Nanjing, People’s Republic of China

Tóm tắt

This paper deals with the Cauchy problem of compressible nematic liquid crystal flows in the whole space $$\mathbb {R}^3$$. We show that if, in addition, the conservation law of the total mass is satisfied (i.e., $$\rho _0\in L^1$$), then the global existence theorem with small density and $$L^3$$-norm of the gradient of $$d_0$$ holds for any $$\gamma >1$$. It is worth mentioning that the initial total energy can be arbitrarily large and the initial vacuum is allowed. Thus, the result obtained particularly extends the one due to Li et al. (J Math Fluid Mech 20:2105–2145, 2018), where the global well-posedness of classical solutions with small energy was proved.

Tài liệu tham khảo

Ding, S., Lin, J., Wang, C., Wen, H.: Compressible hydrodynamic flow of liquid crystals in 1D. Discrete Contin. Dyn. Syst. 32, 539–563 (2012) Ding, S., Wang, C., Wen, H.: Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete Contin. Dyn. Syst. Ser. B 15, 357–371 (2011) de Gennes, P.: The Physics of Liquid Crystals. Oxford University Press, Oxford (1974) Huang, T., Wang, C., Wen, H.: Strong solutions of the compressible nematic liquid crystal flow. J. Differ. Equ. 252, 2222–2265 (2012) Huang, T., Wang, C., Wen, H.: Blow up criterion for compressible nematic liquid crystal flows in dimension three. Arch. Ration. Mech. Anal. 204, 285–311 (2012) Huang, X., Wang, Y.: A Serrin criterion for compressible nematic liquid crystal flows. Math. Methods Appl. Sci. 36, 363–1375 (2013) Jiang, F., Jiang, S., Wang, D.: On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain. J. Funct. Anal. 265, 3369–3397 (2013) Jiang, F., Jiang, S., Wang, D.: Global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions. Arch. Ration. Mech. Anal. 214, 403–451 (2014) Leslie, F.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968) Li, J., Xu, Z., Zhang, J.: Global existence of classical solutions with large oscillations and vacuum to the three dimensional compressible nematic liquid crystal flows. J. Math. Fluid Mech. 20, 2105–2145 (2018) Lin, F.: Nonlinear theory of defects in nematic liquid crystals: phase transition and flow phenomena. Commun. Pure Appl. Math. 42, 789–814 (1998) Lin, F., Liu, C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48, 501–537 (1995) Lin, F., Liu, C.: Partial regularity of the nonlinear dissipative system modeling the flow of liquid crystals. Discrete Contin. Dyn. Syst. 2, 1–23 (1996) Lin, F., Wang, C.: Recent developments of analysis for hydrodynamic flow of nematic liquid crystals. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372(2029), 20130361 (2014) Lin, J., Lai, S., Wang, C.: Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three. SIAM J. Math. Anal. 47, 2952–2983 (2014) Liu, Y., Zheng, S., Li, H., Liu, S.: Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete Contin. Dyn. Syst. 37, 3921–3938 (2017) Liu, Y.: Large time asymptotic behavior of classical solutions to the 3D compressible nematic liquid crystal flows with vacuum. Acta Appl. Math. 146, 57–66 (2016) Nirenberg, L.: On elliptic partial differential equations. Ann. Sc. Norm. Super. Pisa 13, 115–162 (1959) Si, X., Zhang, J., Zhao, J.: Global classical solutions of compressible isentropic Navier–Stokes equations with small density. Nonlinear Anal. Real World Appl. 42, 53–70 (2018) Wang, T.: Global existence and large time behavior of strong solutions to the 2-D compressible nematic liquid crystal flows with vacuum. J. Math. Fluid Mech. 18, 539–569 (2016) Wu, G., Tan, Z.: Global low-energy weak solution and large-time behavior for the compressible flow of liquid crystals. J. Differ. Equ. 264, 6603–6632 (2018) Zlotnik, A.: Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations. Differ. Equ. 36, 701–716 (2000)