Abstract
We investigate the well-posedness of (1) the heat flow of harmonic maps from \({\mathbb R^n}\) to a compact Riemannian manifold N without boundary for initial data in BMO; and (2) the hydrodynamic flow (u, d) of nematic liquid crystals on \({\mathbb R^n}\) for initial data in BMO−1 × BMO.
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Chen Y., Ding W.: Blow-up and global existence for heat flows of harmonic maps. Invent. Math. 99(3), 567–578 (1990)
Chang K., Ding W., Ye R.: Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Differ. Geom. 36(2), 507–515 (1992)
Coron J., Ghidaglia J.: Explosion en temps fini pour le flot des applications harmoniques. C. R. Acad. Sci. Paris Ser I 308, 339–344 (1989)
Chen Y., Lin F.: Evolution of harmonic maps with Dirichlet boundary conditions. Comm. Anal. Geom. 1(3–4), 327–346 (1993)
Chen Y., Struwe M.: Existence and partial regularity for heat flow for harmonic maps. Math. Z. 201, 83–103 (1989)
Eells J., Sampson J.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)
Ericksen J.L.: Hydrostatic theory of liquid crystal. Arch. Ration. Mech. Anal. 9, 371–378 (1962)
de Gennes, P.G.: The Physics of Liquid Crystals. Oxford, 1974
Hildebrandt S., Kaul H., Widman K.: An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138(1–2), 1–16 (1977)
Koch, H., Lamm, T.: Geometric flows with rough initial data. arXiv: 0902.1488v1, 2009
Koch H., Tataru D.: Well-posedness for the Navier–Stokes equations. Adv. Math. 157(1), 22–35 (2001)
Leslie F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968)
Lin F., Liu C.: Nonparabolic Dissipative Systems Modeling the Flow of Liquid Crystals. CPAM XLVIII, 501–537 (1995)
Lin F., Liu C.: Partial Regularity of The Dynamic System Modeling The Flow of Liquid Crystals. DCDS 2(1), 1–22 (1998)
Lin, F., Lin, J.Y., Wang, C.: Liquid crystal flows in two dimensions. Arch. Ration. Mech. Anal. (in press)
Lin, F., Wang, C.: The analysis of harmonic maps and their heat flows. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. xii+267 pp. ISBN: 978-981-277-952-6
Stein, E.: Harmonic Analysis, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, 1993
Struwe M.: On the evolution of harmonic maps of Riemannian surfaces. Comment. Math. Helv. 60, 558–581 (1985)
Wang C.: Heat flow of harmonic maps whose gradients belong to \({L^n_xL^\infty_t}\). Arch. Ration. Mech. Anal. 188(2), 351–369 (2008)
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Wang, C. Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data. Arch Rational Mech Anal 200, 1–19 (2011). https://doi.org/10.1007/s00205-010-0343-5
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DOI: https://doi.org/10.1007/s00205-010-0343-5