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Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy


Author: Dung Le
Journal: Trans. Amer. Math. Soc. 365 (2013), 2723-2753
MSC (2010): Primary 35K65, 35B65
DOI: https://doi.org/10.1090/S0002-9947-2012-05720-5
Published electronically: August 27, 2012
MathSciNet review: 3020113
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Abstract: This paper introduces a new technique, using the so-called nonlinear heat approximation and BMO preserving homotopy, to investigate regularity properties of BMO weak solutions of strongly coupled nonlinear parabolic systems consisting of more than one equation defined on a domain of any dimension.


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

  • 1. H. Amann.
    Dynamic theory of quasilinear parabolic systems. III. Global existence.
    Math. Z., 202(1989), pp. 219-250. MR 1013086 (90i:35125)
  • 2. F. Duzaar and G. Mingione.
    Second order parabolic systems, optimal regularity, and singular sets of solutions.
    Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 6, pp. 705-751. MR 2172857 (2008h:35139)
  • 3. M. Giaquinta and M. Struwe.
    On the partial regularity of weak solutions of nonlinear parabolic systems.
    Math. Z. Vol. 179 (1982), pp. 437-451. MR 652852 (83f:35062)
  • 4. E. Giusti.
    Direct Methods in the Calculus of Variations.
    World Scientific, 2003. MR 1962933 (2004g:49003)
  • 5. O. John and J. Stara.
    Some (new) counterexamples of parabolic systems.
    Comment. Math. Univ. Carolin., 36 (1995), pp. 503-510. MR 1364491 (96j:35027)
  • 6. O. John and J. Stara.
    On the regularity of weak solutions to parabolic systems in two spatial dimensions.
    Comm. P.D.E., 27(1998), pp. 1159-1170. MR 1642595 (99k:35079)
  • 7. A. Koshelev.
    Regularity of solutions for some quasilinear parabolic systems.
    Math. Nachr., 162(1993), pp. 59-88. MR 1239576 (94h:35092)
  • 8. E. Kalita.
    On the Hölder continuity of solutions of nonlinear parabolic systems.
    Comment. Math. Univ. Carolin., 35,4(1994), pp. 675-680. MR 1321237 (96b:35097)
  • 9. K. H. W. Küfner.
    Invariant regions for quasilinear reaction-diffusion systems and applications to a two population model.
    NoDEA, 3(1996), pp. 421-444. MR 1418589 (97m:35135)
  • 10. O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'tseva.
    Linear and Quasilinear Equations of Parabolic Type.
    AMS Transl. Monographs, vol. 23, 1967. MR 0241822 (39:3159b)
  • 11. D. Le.
    Partial regularity of solutions to a class of strongly coupled degenerate parabolic systems.
    Discrete Contin. Dyn. Syst. 2005, suppl., pp. 576-586. MR 2192716 (2006h:35113)
  • 12. D. Le.
    Global existence for a class of strongly coupled parabolic systems.
    Ann. Mat. Pura Appl. (4) 185 (2006), no. 1, pp. 133-154. MR 2179585 (2006i:35190)
  • 13. D. Le.
    Global Existence Results for Near Triangular Nonlinear Parabolic Systems (submitted).
  • 14. D. Le and T. Nguyen.
    Everywhere regularity of solutions to a class of strongly coupled degenerate parabolic systems.
    Comm. Partial Differential Equations 31 (2006), no. 1-3, pp. 307-324. MR 2209756 (2007b:35161)
  • 15. D. Le and T. Nguyen.
    Global existence for a class of triangular parabolic systems on domains of arbitrary dimension.
    Proc. Amer. Math. Soc. 133 (2005), no. 7, pp. 1985-1992. MR 2137864 (2005k:35215)
  • 16. G. M. Lieberman.
    Second Order Parabolic Differential Equations.
    World Scientific, 1998. MR 1465184 (98k:35003)
  • 17. J. Necas and V. Sverak.
    On regularity of solutions of nonlinear parabolic systems.
    Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 18 (1)(1991), pp. 1-11. MR 1118218 (92d:35058)
  • 18. N. Shigesada, K. Kawasaki and E. Teramoto.
    Spatial segregation of interacting species.
    J. Theoretical Biology, 79 (1979), pp. 83-99. MR 540951 (80e:92038)
  • 19. J. Simon.
    Compact sets in the space $ L^p(0,T ;B)$,
    Ann. Mat. Pura Appl. 146 (4) (1987) pp. 65-96. MR 916688 (89c:46055)
  • 20. M. Wiegner.
    Global solutions to a class of strongly coupled parabolic systems.
    Math. Ann., 292 (1992), pp. 711-727. MR 1157322 (92m:35136)

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Additional Information

Dung Le
Affiliation: Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, Texas 78249
Email: dle@math.utsa.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05720-5
Keywords: Parabolic systems, Hölder regularity, BMO weak solutions.
Received by editor(s): June 10, 2011
Received by editor(s) in revised form: September 26, 2011
Published electronically: August 27, 2012
Additional Notes: The author was partially supported by NSF grant DMS0707229.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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