×

zbMATH — the first resource for mathematics

Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data. (English) Zbl 1285.35085
Summary: 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\(^{ - 1} \times \) BMO.

MSC:
35Q35 PDEs in connection with fluid mechanics
35K45 Initial value problems for second-order parabolic systems
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K15 Initial value problems for second-order parabolic equations
35K59 Quasilinear parabolic equations
58E20 Harmonic maps, etc.
76A15 Liquid crystals
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Chen Y., Ding W.: Blow-up and global existence for heat flows of harmonic maps. Invent. Math. 99(3), 567–578 (1990) · Zbl 0674.58019
[2] 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) · Zbl 0765.53026
[3] 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) · Zbl 0679.58017
[4] Chen Y., Lin F.: Evolution of harmonic maps with Dirichlet boundary conditions. Comm. Anal. Geom. 1(3–4), 327–346 (1993) · Zbl 0845.35049
[5] Chen Y., Struwe M.: Existence and partial regularity for heat flow for harmonic maps. Math. Z. 201, 83–103 (1989) · Zbl 0652.58024
[6] Eells J., Sampson J.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964) · Zbl 0122.40102
[7] Ericksen J.L.: Hydrostatic theory of liquid crystal. Arch. Ration. Mech. Anal. 9, 371–378 (1962) · Zbl 0105.23403
[8] de Gennes, P.G.: The Physics of Liquid Crystals. Oxford, 1974 · Zbl 0295.76005
[9] Hildebrandt S., Kaul H., Widman K.: An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138(1–2), 1–16 (1977) · Zbl 0356.53015
[10] Koch, H., Lamm, T.: Geometric flows with rough initial data. arXiv: 0902.1488v1, 2009 · Zbl 1252.35159
[11] Koch H., Tataru D.: Well-posedness for the Navier–Stokes equations. Adv. Math. 157(1), 22–35 (2001) · Zbl 0972.35084
[12] Leslie F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968) · Zbl 0159.57101
[13] Lin F., Liu C.: Nonparabolic Dissipative Systems Modeling the Flow of Liquid Crystals. CPAM XLVIII, 501–537 (1995) · Zbl 0842.35084
[14] Lin F., Liu C.: Partial Regularity of The Dynamic System Modeling The Flow of Liquid Crystals. DCDS 2(1), 1–22 (1998) · Zbl 0948.35098
[15] Lin, F., Lin, J.Y., Wang, C.: Liquid crystal flows in two dimensions. Arch. Ration. Mech. Anal. (in press) · Zbl 1346.76011
[16] 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
[17] Stein, E.: Harmonic Analysis, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, 1993 · Zbl 0821.42001
[18] Struwe M.: On the evolution of harmonic maps of Riemannian surfaces. Comment. Math. Helv. 60, 558–581 (1985) · Zbl 0595.58013
[19] Wang C.: Heat flow of harmonic maps whose gradients belong to $${L\^n_xL\^\(\backslash\)infty_t}$$ . Arch. Ration. Mech. Anal. 188(2), 351–369 (2008) · Zbl 1156.35052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.