Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the blow-up of heat flow for conformal $3$-harmonic maps


Authors: Chao-Nien Chen, L. F. Cheung, Y. S. Choi and C. K. Law
Journal: Trans. Amer. Math. Soc. 354 (2002), 5087-5110
MSC (2000): Primary 35K50, 35K65, 58E20
Published electronically: July 16, 2002
MathSciNet review: 1926851
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Using a comparison theorem, Chang, Ding, and Ye (1992) proved a finite time derivative blow-up for the heat flow of harmonic maps from $D^2$ (a unit ball in ${\mathbf R}^2$) to $S^2$ (a unit sphere in ${\mathbf R}^3$) under certain initial and boundary conditions. We generalize this result to the case of $3$-harmonic map heat flow from $D^3$ to $S^3$. In contrast to the previous case, our governing parabolic equation is quasilinear and degenerate. Technical issues such as the development of a new comparison theorem have to be resolved.


References [Enhancements On Off] (What's this?)

  • 1. K.C. Chang and W.Y. Ding, A result on the global existence for heat flows of harmonic maps from $D^{2}$ into $S^{2}$, in ``Nematics'', J.M. Coron et al ed., Kluwer Academic Publishers (1990) 37-48. MR 94c:58048
  • 2. K.C. Chang, W.Y. Ding and R.G. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Diff. Geometry, 36 (1992) 507-515. MR 93h:58043
  • 3. Y. Chen, M.C. Hong, N. Hungerbühler, Heat flow for p-harmonic maps with values into spheres, Math. Zeit. 215, 25-35, (1994). MR 94k:58145
  • 4. C.N. Chen, L.F. Cheung, Y.S. Choi and C.K. Law, Integrability of rotationally symmetric $n$-harmonic maps, preprint.
  • 5. C.N. Chen, L.F. Cheung, Y.S. Choi and C.K. Law, Asymptotic behaviours of rotationally symmetric harmonic maps and their heat flow, preprint.
  • 6. L.F. Cheung, C.K. Law, M.C. Leung and J.B. McLeod, Entire solutions of quasilinear differential equations corresponding to $p$-harmonic maps, Nonlinear Anal. T.M.A., 31 (1998) 701-715. MR 99g:58033
  • 7. R. Coron, J.-M. Ghidaglia, Explosion en temps fini pour le flot des applications harmoniques, C.R. Acad. Sci. Paris Sér. I 308, 339-344 (1989). MR 90g:58026
  • 8. W.Y. Ding, Lecture Notes on the Heat Flow of Harmonic Maps, NCTS, (1997).
  • 9. L.C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rat. Mech. Anal. 116, 101-113, (1991). MR 93m:5826
  • 10. J. Eells, J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86, 109-169 (1964) MR 29:1603
  • 11. M. Fuchs, The blow-up of $p$-harmonic maps, Manuscripta Math. 89-94, (1993). MR 94h:58060
  • 12. M. Giaquinta, E. Giusti, On the regularity of minima of variational integrals, Acta Math. 148, 31-40 (1982). MR 84b:58034
  • 13. D. Gilbarg and N. S. Trudinger; Elliptic partial differential equations of second order, 2nd edition, Springer-Verlag, 1983. MR 86c:35035
  • 14. J.F. Grotowski, Heat flow for harmonic maps, in ``Nematics'', J.M. Coron et al ed., Kluwer Academic Publishers (1990) 129-140. MR 98f:58042
  • 15. J.F. Grotowski, Finite time blow-up for the harmonic map heat flow, Calculus of Variations, 1 (1993) 231-236. MR 94k:58034
  • 16. Richard Hamilton, Harmonic Maps of Manifolds with Boundary, (1975), Lecture Notes in Mathematics, vol. 471, Springer-Verlag. MR 58:2872
  • 17. R. Hardt, D. Kinderlehrer, F.H. Lin, Mappings minimizing the $L^{p}$-norm of the gradient, Comm. Pure Appl. Math. 11, 555-588 (1987). MR 88k:58026
  • 18. F. Hélein, Regularité des applications faiblement harmoniques entre une surface et une varieté Riemannienne, C.R. Acad. Sci. Paris, Sér. I Math. 312, 591-596 (1991). MR 92e:58055
  • 19. S. Hildebrandt, H. Kaul, K.O. Widman, An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math., 138, 1-16, (1977). MR 55:6478
  • 20. N. Hungerbühler, $m$-harmonic flow, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24, no. 4, 593-632 (1997). MR 99c:58046
  • 21. N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Graduate Studies in Math., Vol. 12, (1996), American Mathematical Society, Providence, RI. MR 97i:35001
  • 22. S. Luckhaus, Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold, Indiana Univ. Math. J. 37, 346-367, (1988). MR 89m:58043
  • 23. O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'ceva, Linear and Quasilinear Equations of Parabolic type, Translation of Mathematical Monographs, 23, (1967), American Mathematical Society, Providence, RI. MR 39:3159h
  • 24. N. Nakauchi, S. Takakuwa, A remark on $p$-harmonic maps, Nonlinear Anal., 25, 169-195, (1995). MR 96e:58046
  • 25. M. Rigoli, M. Salvatori and M. Vignati, Volume growth and $p-$subharmonic functions on complete manifolds, Math. Z. 227, no. 3, 357-375, (1998). MR 99h:58177
  • 26. Wilhelm Schlag, Schauder and $L^{p}$ estimates for parabolic systems via Campanato spaces, Comm. Partial Differential Equations, 21 (1996), pp. 1141-1175. MR 97k:35108
  • 27. P. Strzelecki, Regularity of $p$-harmonic maps from the $p$-dimensional ball into a sphere, Manuscripta Math. 82, 407-415, (1994). MR 95b:58047
  • 28. J. Sacks, K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. 113, 1-24, (1981). MR 82f:58035
  • 29. R. Schoen, K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom., 17, 307-335, (1982). MR 84b:58037a; Correction, J. Diff. Geom., 18, 329, (1983). MR 84b:58037b
  • 30. M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60, 558-581 (1981). MR 87e:5856

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35K50, 35K65, 58E20

Retrieve articles in all journals with MSC (2000): 35K50, 35K65, 58E20


Additional Information

Chao-Nien Chen
Affiliation: Department of Mathematics, National Changhua University of Education, Changhua, Taiwan, Republic of China
Email: chenc@math.ncue.edu.tw

L. F. Cheung
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong
Email: matheclf@maun01.ma.polyu.edu.hk

Y. S. Choi
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: choi@math.uconn.edu

C. K. Law
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 80424, Republic of China
Email: law@math.nsysu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03054-4
PII: S 0002-9947(02)03054-4
Keywords: $p$-harmonic maps, heat flow, blow up, maximum principle
Received by editor(s): June 25, 2001
Received by editor(s) in revised form: December 21, 2001
Published electronically: July 16, 2002
Dedicated: Dedicated to Shui-Nee Chow on the occasion of his 60th birthday
Article copyright: © Copyright 2002 American Mathematical Society