On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals

Abstract

For any n-dimensional compact Riemannian manifold (M, g) without boundary and another compact Riemannian manifold (N, h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0, T),W ^1,n). For the hydrodynamic flow (u, d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ L _t ^∞ L _x ² ∩ L _t ² H _x ¹, ▿ P ∈ L _t ^4/3 L _x ^4/3, and ▿d ∈ L _t ^∞ L _x ² ∩ L _t ² H _x ²; or (ii) for n = 3, u ∈ L _t ^∞ L _x ² ∩ L _t ² H _x ¹ ∩ C ([0, T), L ⁿ), P ∈ L _t ^n/2 L _x ^n/2, and ▿d ∈ L _t ² L _x ² ∩ C ([0, T), L ⁿ). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.

Keywords

Hydrodynamic flow / Harmonic maps / Nematic liquid crystals / Uniqueness

References

[1] 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

[2] Chang K.. Heat flow and boundary value problem for harmonic maps. Annales de l’institut Henri Poincaré Analyse nonlinairé, 1989, 6(5): 363-395

[3] Chang K., Ding W., Ye R.. Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Differential Geom., 1992, 36: 507-515

[4] Chen Y., Ding W.. Blow-up and global existence for heat flows of harmonic maps. Invent. Math., 1990, 99(3): 567-578

[5] Chen Y., Lin F.. Evolution of harmonic maps with Dirichlet boundary conditions. Comm. Anal. Geom., 1993, 1(3–4): 327-346

[6] Chen Y., Struwe M.. Existence and partial regularity results for the heat flow for harmonic maps. Math. Z., 1989, 201(1): 83-103

[7] Ericksen J. L.. Hydrostatic theory of liquid crystal. Arch. Rational Mech. Anal., 1962, 9: 371-378

[8] Escauriaza L., Serëgin G., Sverak V.. L _3,∞-solutions of Navier-Stokes equations and backward uniqueness (in Russian). Uspekhi Mat. Nauk, 2003, 58(2): 3-44 Translation in Russian Math. Surveys, 58(2), 2003, 211–250

[9] Freire A.. Uniqueness for the harmonic map flow from surfaces to general targets. Comment Math. Helvetici, 1995, 70(1): 310-338

[10] Giga Y.. Solutions for semilinear parabolic equations in L ^p and regularity of weak solutions of the Navier-Stokes system. J. Diff. Eqns., 1986, 61: 186-212

[11] Huang T., Wang C. Y.. Notes on the regularity of harmonic map systems. Proc. Amer. Math. Soc., 2010, 138: 2015-2023

[12] Kato T.. Strong L ^p solutions of the Navier-Stokes equations in ℝ^m, with applications to weak solutions. Math. Z., 1984, 187: 471-480

[13] Leray J.. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math., 1934, 63: 193-248

[14] Leslie F. M.. Some constitutive equations for liquid crystals. Arch. Rational Mech. Anal., 1968, 28: 265-283

[15] Lin F. H.. Nonlinear theory of defects in nematic liquid crystal: phase transition and flow phenomena. Comm. Pure Appl. Math., 1989, 42: 789-814

[16] Lin F. H.. A new proof of the Caffarelli-Kohn-Nirenberg Theorem. Comm. Pure Appl. Math., 1998, LI: 0241-0257

[17] Lin F. H., Liu C.. Nonparabolic dissipative systems modeling the flow of liquid crystals. Comm. Pure Appl. Math., 1995, XLVIII: 501-537

[18] Lin F. H., Liu C.. Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals. Dis. Cont. Dyn. Sys., 1996, 2: 1-23

[19] Lin F. H., Lin J. Y., Wang C. Y.. Liquid crystal flows in two dimensions. Arch. Rational Mech. Anal., 2010, 197(1): 297-336

[20] Seregin G.. On the number of singular points of weak solutions to the Navier-Stokes equations. Comm. Pure Appl. Math., 2001, LIV: 1019-1028

[21] Struwe M.. On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helvetici, 1985, 60: 558-581

[22] Wang C. Y.. A remark on harmonic map flows from surfaces. Diff. Int. Eqs., 1999, 12(2): 161-166