Previous |  Up |  Next


Stokes operator; spatially periodic problem; maximal $L^p$ regularity; nematic liquid crystal flow; quasilinear parabolic equations
We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal $L^p$-regularity of the periodic Laplace and Stokes operators and a local-in-time existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group $G:=\mathbb R^{n-1}\times \mathbb R / L \mathbb Z$ to obtain an $\mathcal {R}$-bound for the resolvent estimate. Then, Weis' theorem connecting $\mathcal {R}$-boundedness of the resolvent with maximal $L^p$ regularity of a sectorial operator applies.
[1] Alfsen, E. M.: A simplified constructive proof of the existence and uniqueness of Haar measure. Math. Scand. 12 (1963), 106-116. DOI 10.7146/math.scand.a-10675 | MR 0158022
[2] Amann, H.: Quasilinear evolution equations and parabolic systems. Trans. Am. Math. Soc. 293 (1986), 191-227. DOI 10.1090/S0002-9947-1986-0814920-4 | MR 0814920 | Zbl 0635.47056
[3] Bourgain, J.: Vector-valued singular integrals and the {$H^1$}-BMO duality. Probability Theory and Harmonic Analysis. Papers from the Mini-Conf. on Probability and Harmonic Analysis, Cleveland, 1983 Pure Appl. Math. 98 Marcel Dekker, New York (1986), 1-19 W. A. Woyczy{ń}ski. MR 0830227
[4] Bruhat, F.: Distributions sur un groupe localement compact et applications à l'étude des représentations des groupes {$\wp $}-adiques. Bull. Soc. Math. Fr. 89 French (1961), 43-75. DOI 10.24033/bsmf.1559 | MR 0140941
[5] Burkholder, D. L.: A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. Conf. on Harmonic Analysis in Honor of Antoni Zygmund, 1, Chicago, Ill., 1981 The Wadsworth Math. Ser. Wadsworth, Belmont (1983), 270-286 W. Beckner et al. MR 0730072
[6] Cartan, H.: Sur la mesure de Haar. C. R. Acad. Sci., Paris 211 French (1940), 759-762. MR 0005742
[7] Clément, P., Li, S.: Abstract parabolic quasilinear equations and application to a groundwater flow problem. Adv. Math. Sci. Appl. 3 (1993/1994), 17-32. MR 1287921
[8] Denk, R., Hieber, M., Prüss, J.: {$\scr R$}-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166 (2003), 114 pages. MR 2006641
[9] Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics 43 Cambridge Univ. Press, Cambridge (1995). MR 1342297 | Zbl 0855.47016
[10] Ericksen, J. L., Kinderlehrer, D.: Theory and Applications of Liquid Crystals. The IMA Volumes in Mathematics and Its Applications Vol. 5, Papers from the IMA workshop, Minneapolis Institute for Mathematics and Its Applications, University of Minnesota, Springer, New York (1987). MR 0900827
[11] Farwig, R., Ri, M.-H.: Resolvent estimates and maximal regularity in weighted {$L^q$}-spaces of the Stokes operator in an infinite cylinder. J. Math. Fluid Mech. 10 (2008), 352-387. DOI 10.1007/s00021-006-0235-5 | MR 2430805 | Zbl 1162.76322
[12] Haar, A.: Der Massbegriff in der Theorie der kontinuierlichen Gruppen. Ann. Math. (2) 34 German (1933), 147-169. DOI 10.2307/1968346 | MR 1503103 | Zbl 0006.10103
[13] Hieber, M., Nesensohn, M., Prü{ß}, J., Schade, K.: Dynamics of nematic liquid crystal flows. The quasilinear approach. (2014), 11 pages ArXiv:1302.4596 [math.AP]. MR 3465380
[14] Kunstmann, P. C., Weis, L.: Maximal {$L_p$}-regularity for parabolic equations, Fourier multiplier theorems and {$H^\infty$}-functional calculus. Functional Analytic Methods for Evolution Equations. Autumn School on Evolution Equations and Semigroups, Levico Terme, Trento, Italy, 2001 Lecture Notes in Mathematics 1855 Springer, Berlin (2004), 65-311 M. Iannelli, et al. DOI 10.1007/978-3-540-44653-8_2 | MR 2108959 | Zbl 1097.47041
[15] Lin, F.-H.: Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Commun. Pure Appl. Math. 42 (1989), 789-814. DOI 10.1002/cpa.3160420605 | MR 1003435
[16] Lin, F.-H., Liu, C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Commun. Pure Appl. Math. 48 (1995), 501-537. DOI 10.1002/cpa.3160480503 | MR 1329830
[17] Lunardi, A.: Interpolation Theory. Appunti. Scuola Normale Superiore di Pisa 9. Lecture Notes. Scuola Normale Superiore di Pisa Edizioni della Normale, Pisa (2009). MR 2523200 | Zbl 1171.41001
[18] Francia, J. L. Rubio de, Ruiz, F. J., Torrea, J. L.: Calderón-Zygmund theory for operator-valued kernels. Adv. Math. 62 (1986), 7-48. DOI 10.1016/0001-8708(86)90086-1 | MR 0859252
[19] Rudin, W.: Fourier Analysis on Groups. Interscience Tracts in Pure and Applied Mathematics, Vol. 12 Interscience Publishers, John Wiley, New York (1962). MR 0152834 | Zbl 0107.09603
[20] Sauer, J.: Weighted resolvent estimates for the spatially periodic Stokes equations. Ann. Univ. Ferrara Sez. VII Sci. Mat. 61 (2015), 333-354 DOI 10.1007/s11565-014-0221-4. DOI 10.1007/s11565-014-0221-4 | MR 3421709 | Zbl 1330.35342
[21] Sauer, J.: An extrapolation theorem in non-Euclidean geometries and its application to partial differential equations. (to appear) in J. Elliptic Parabol. Equ.
[22] Wang, C.: Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data. Arch. Ration. Mech. Anal. 200 (2011), 1-19. DOI 10.1007/s00205-010-0343-5 | MR 2781584 | Zbl 1285.35085
[23] Weil, A.: L'intégration Dans les Groupes Topologiques et Ses Applications. Actualités Scientifiques et Industrielles 869 Hermann & Cie., Paris French (1940). MR 0005741
[24] Weis, L.: A new approach to maximal {$L_p$}-regularity. Evolution Equations and Their Applications in Physical and Life Sciences. Proc. Bad Herrenalb Conf., Karlsruhe, 1999 Lect. Notes in Pure and Appl. Math. 215 Marcel Dekker, New York (2001), 195-214 G. Lumer et al. MR 1818002 | Zbl 0981.35030
[25] Zimmermann, F.: On vector-valued Fourier multiplier theorems. Stud. Math. 93 (1989), 201-222. DOI 10.4064/sm-93-3-201-222 | MR 1030488
Partner of
EuDML logo