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On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals. (English) Zbl 1208.35002
Summary: 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 $$n = 2$$, $$u \in L_t^{\infty } L_x^{2} \cap L_t^{2} H_x^{1}$$, $$\nabla P \in L_t^{\frac{4}{3}} L_x^{\frac{4}{3}}$$, and $$\nabla d \in L_t^{\infty } L_x^{2} \cap L_t^{2} H_x^{2}$$; or (ii) for $$n = 3$$, $$u \in L_t^{\infty} L_x^{2} \cap L_t^{2} H_x^{1} \cap C ([0, T), L^n)$$, $$P \in L_t^{\frac{n}{2}} L_x^{\frac{n}{2}}$$, and $$\nabla d \in L_t^{2} L_x^{2} \cap C ([0, T), L^n)$$. This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.

##### MSC:
 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35K55 Nonlinear parabolic equations 58J35 Heat and other parabolic equation methods for PDEs on manifolds 35D30 Weak solutions to PDEs
##### Keywords:
compact Riemannian manifold
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##### References:
 [1] Caffarelli, L., Kohn, R. and Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35, 1982, 771–831. · Zbl 0509.35067 [2] Chang, K., Heat flow and boundary value problem for harmonic maps, Annales de l’institut Henri Poincaré Analyse nonlinairé, 6(5), 1989, 363–395. [3] Chang, K., Ding, W. and Ye, R., Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom., 36, 1992, 507–515. · Zbl 0765.53026 [4] Chen, Y. and Ding, W., Blow-up and global existence for heat flows of harmonic maps, Invent. Math., 99(3), 1990, 567–578. · Zbl 0674.58019 [5] Chen, Y. and Lin, F., Evolution of harmonic maps with Dirichlet boundary conditions, Comm. Anal. Geom., 1(3–4), 1993, 327–346. · Zbl 0845.35049 [6] Chen, Y. and Struwe, M., Existence and partial regularity results for the heat flow for harmonic maps, Math. Z., 201(1), 1989, 83–103. · Zbl 0652.58024 [7] Ericksen, J. L., Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9, 1962, 371–378. · Zbl 0105.23403 [8] Escauriaza, L., Serëgin, G. and Sverak, V., L 3,solutions of Navier-Stokes equations and backward uniqueness (in Russian), Uspekhi Mat. Nauk, 58(2), 2003, 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, 70(1), 1995, 310–338. · Zbl 0831.58018 [10] Giga, Y., Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system, J. Diff. Eqns., 61, 1986, 186–212. · Zbl 0577.35058 [11] Huang, T. and Wang, C. Y., Notes on the regularity of harmonic map systems, Proc. Amer. Math. Soc., 138, 2010, 2015–2023. · Zbl 1191.35118 [12] Kato, T., Strong L p solutions of the Navier-Stokes equations in $$\mathbb{R}$$m, with applications to weak solutions, Math. Z., 187, 1984, 471–480. · Zbl 0545.35073 [13] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 1934, 193–248. · JFM 60.0726.05 [14] Leslie, F. M., Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28, 1968, 265–283. · Zbl 0159.57101 [15] Lin, F. H., Nonlinear theory of defects in nematic liquid crystal: phase transition and flow phenomena, Comm. Pure Appl. Math., 42, 1989, 789–814. · Zbl 0703.35173 [16] Lin, F. H., A new proof of the Caffarelli-Kohn-Nirenberg Theorem, Comm. Pure Appl. Math., LI, 1998, 0241–0257. · Zbl 0958.35102 [17] Lin, F. H. and Liu, C. Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., XLVIII, 1995, 501–537. · Zbl 0842.35084 [18] Lin, F. H. and Liu, C., Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Dis. Cont. Dyn. Sys., 2, 1996, 1–23. [19] Lin, F. H., Lin, J. Y. and Wang, C. Y., Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197(1), 2010, 297–336. · Zbl 1346.76011 [20] Seregin, G., On the number of singular points of weak solutions to the Navier-Stokes equations, Comm. Pure Appl. Math., LIV, 2001, 1019–1028. · Zbl 1030.35133 [21] Struwe, M., On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helvetici, 60, 1985, 558–581. · Zbl 0595.58013 [22] Wang, C. Y., A remark on harmonic map flows from surfaces, Diff. Int. Eqs., 12(2), 1999, 161–166. · Zbl 1008.58014
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