Abstract
The main purpose of this manuscript is to study the well-posedness and decay estimates for strong solutions to the Cauchy problem of 3D smectic-A liquid crystals equations. First, applying Banach fixed point theorem, we prove the local existence and uniqueness of strong solutions. Then, by establishing some nontrivial estimates with energy method and a standard continuity argument, we prove that there exists a unique global strong solution provided that the initial data are sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solutions.
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References
Abels, H.: Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Commun. Math. Phys. 289, 45–73 (2009)
Abels, H.: On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal. 194, 463–506 (2009)
Abels, H., Depner, D., Garcke, H.: On an incompressible Navier–Stokes/Cahn–Hilliard system with degenerate mobility. Ann. Inst. H. Poincaré Anal. Non Lineairé 30, 1175–1190 (2013)
Calderer, C., Liu, C.: Mathematical developments in the study of smectic A liquid crystals. Int. J. Eng. Sci. 38, 1113–1128 (2000)
Climent-Ezquerra, B., Guillén-González, F.: Global in-time solutions and time-periodicity for a semectic-A liquid crystal model. Commun. Pure Appl. Anal. 9, 1473–1493 (2010)
Climent-Ezquerra, B., Guillén-González, F.: On a double penalized smectic-A model. Discrete Contin. Dyn. Syst. Ser. A 32, 4171–4182 (2012)
Climent-Ezquerra, B., Guillén-González, F.: A review of mathematical analysis of nematic and smectic-A liquid crystal models. Eur. J. Appl. Math. 25, 133–153 (2014)
Climent-Ezquerra, B., Guillén-González, F.: Convergence to equilibrium for smectic-A liquid crystals in 3D domains without constraints for the viscosity. Nonlinear Anal. 102, 208–219 (2014)
Contreras, A., Garcia-Azpeitia, C., Garcia-Cervera, C.J., Joo, S.: The onset of layer undulations in smectic A liquid crystals due to a strong magnetic field. Nonlinearity 29, 2474–2496 (2016)
De Gennes, P.: Viscous flows in smectic-A liquid crystals. Phys. Fluids 17, 1645 (1974)
De Gennes, P., Prost, J.: The Physics of Liquid Crystals. Oxford Publications, London (1993)
Eleuteri, M., Rocca, E., Schimperna, G.: Existence of solutions to a two-dimensional model for nonisothermal two-phase flows of incompressible fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 1431–1454 (2016)
Ericksen, J.: Continuum theory of nematic liquid crystals. Res. Mechanica 21, 381–392 (1961)
Frigeri, S., Grasselli, M., Rocca, E.: A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility. Nonlinearity 28, 1257–1293 (2015)
Gal, C.G., Grasselli, M., Miranville, A.: Cahn–Hilliard–Navier-Stokes system with moving contact lines. Calc. Var. Partial Differ. Equ. 55, 1–47 (2016)
Gal, C.G., Grasselli, M., Wu, H.: Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities. Arch. Ration. Mech. Anal. 234(1), 1–56 (2019)
Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education Inc, Prentice-Hall (2004)
Guo, Y., Wang, Y.: Decay of dissipative equations and negative Sobolev spaces. Commun. Partial Differ. Equ. 37, 2165–2208 (2012)
Jiang, Z.: Asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term. Nonlinear Anal. 75(13), 5002–5009 (2012)
Jiang, Z., Fan, J.: Time decay rate for two 3D magnetohydrodynamics-\(\alpha \) models. Math. Methods Appl. Sci. 37(6), 838–845 (2014)
Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Commun. Pure Appl. Math. 41, 891–907 (1988)
Lam, K.F., Wu, H.: Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis. Eur. J. Appl. Math. 29, 595–644 (2018)
Leslie, F.: Theory of flow phenomena in liquid crystals. Adv. Liq. Cryst. 4, 1–81 (1979)
Liu, A., Liu, C.: Global attractor for a smectic-A liquid crystal model in 2D. Boll. Unione Mat. Ital. 11, 581–594 (2018)
Liu, C.: Dynamic theory for incompressible smectic liquid crystals: existence and regularity. Discrete Contin. Dyn. Syst. 6, 591–608 (2000)
Martin, P., Parodi, P., Pershan, P.: Unified hydrodynamic theory for crystals, liquid crystals, and normal fluids. Phys. Rev. A 6, 2401 (1972)
Nirenberg, L.: On elliptic partial differential equations. Annali della Scuola Normale Superiore di Pisa 13, 115–162 (1959)
Schonbek, M.E.: \(L^2\) decay for weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 88(2), 209–222 (1985)
Schonbek, M.E.: Large time behaviour of solutions to the Navier–Stokes equations. Commun. Partial Differ. Equ. 11(7), 733–763 (1986)
Segatti, A., Wu, H.: Finite dimensional reduction and convergence to equilibrium for incompressible smectic-A liquid crystal flows. SIAM J. Math. Anal. 43, 2445–2481 (2011)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Stewart, I.W., Vynnycky, M., McKee, S., Tomé, M.F.: Boundary layers in pressure-driven flow in smectic A liquid crystals. SIAM J. Appl. Math. 75, 1817–1851 (2015)
Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. North-Holland Publishing Co., Amsterdam (1977)
Wang, Y.: Decay of the Navier–Stokes–Poisson equations. J. Differ. Equ. 253, 273–297 (2012)
Weinan, E.: Nonlinear continuum theory of smectic-A liquid crystals. Arch. Ration. Mech. Anal. 137, 159–175 (1997)
Wu, H.: Well-posedness of a diffuse-interface model for two-phase incompressible flows with thermo-induced Marangoni effect. Eur. J. Appl. Math. 28(3), 380–434 (2017)
Wiegner, M.: Decay results for weak solutions of the Navier–Stokes equations on \(\mathbb{R}^n\). J. Lond. Math. Soc. 35, 303–313 (1987)
Ye, Z., Zhao, X.: Global well-posedness of the generalized magnetohydrodynamic equations. Z. Angew. Math. Phys. 69, 126 (2018)
Zhao, X.: On the Cauchy problem of a sixth-order Cahn–Hilliard equation arising in oil–water–surfactant mixtures. Asymptotic Anal. https://doi.org/10.3233/ASY-201616
Zhou, Y.: A remark on the decay of solutions to the 3-D Navier–Stokes equations. Math. Methods Appl. Sci. 30, 1223–1229 (2007)
Zhou, Y.: Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows. Nonlinearity 21(9), 2061–2071 (2008)
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XZ is supported by the Fundamental Research Funds for the Central Universities (Grant No. N2005031).
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Zhao, X., Zhou, Y. On well-posedness and large time behavior for smectic-A liquid crystals equations in \(\mathbb {R}^3\). Z. Angew. Math. Phys. 71, 179 (2020). https://doi.org/10.1007/s00033-020-01407-4
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DOI: https://doi.org/10.1007/s00033-020-01407-4