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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
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