Bulletin of the Korean Mathematical Society (대한수학회보)

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REGULARITY CRITERIA FOR THE p-HARMONIC AND OSTWALD-DE WAELE FLOWS

  • Fan, Jishan (Department of Applied Mathematics Nanjing Forestry University) ;
  • Nakamura, Gen (Department of Mathematics Inha University) ;
  • Zhou, Yong (School of Mathematics Shanghai University of Finance and Economics, Nonlinear Analysis and Applied Mathematics (NAAM) Research Group King Abdulaziz University)
  • Received : 2014.04.25
  • Published : 2015.03.31
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Abstract

This paper considers regularity for the p-harmonic and Ostwald-de Waele flows. Some Serrin's type regularity criteria are established for 1 < p < 2.

Keywords

References

  1. J. Azzam and J. Bedrossian, Bounded mean oscillation and the uniqueness of active scalar equations, arXiv: 1108.2735 v2[math. AP] 3 Nov 2012.
  2. H.-O. Bae, H. J. Choe, and D. W. Kim, Regularity and singularity of weak solutions to Ostwald-De Waele flows, J. Korean Math. Soc. 37 (2000), no. 6, 957-975.
  3. H.-O. Bae, K. Kang, J. Lee, and J. Wolf, Regularity for Ostwald-de Waele type shear thickening fluids, Nonlinear Differ. Equ. Appl. 2014(in press).
  4. L. C. Berselli, L. Diening, and M. Ruzicka, Existence of strong solutions for incom- pressible fluids with shear dependent viscosities, J. Math. Fluid Mech. 12 (2010), no. 1, 101-132. https://doi.org/10.1007/s00021-008-0277-y
  5. M. Bertsch, R. dal Passo, and R. van der Hout, Nonuniqueness for the heat flow of harmonic maps on the disk, Arch. Ration. Mech. Anal. 161 (2002), no. 2, 93-112. https://doi.org/10.1007/s002050100171
  6. M. Bertsch, R. dal Passo, and A. Pisante, Point singularities and nonuniqueness for the heat flow for harmonic maps, Comm. Partial Differential Equations 28 (2003), no. 5-6, 1135-1160. https://doi.org/10.1081/PDE-120021189
  7. K.-C. Chang, W.-Y. Ding, and R. Ye, Finite time blow-up of heat flow of harmonic maps from surface, J. Differential Geom. 36 (1992), no. 2, 507-515.
  8. Y. Chen, M-C. Hong, and N. Hungerbuhler, Heat flow of p-harmonic maps with values into spheres, Math. Z. 215 (1994), no. 1, 25-35. https://doi.org/10.1007/BF02571698
  9. L. Diening and M. Ruzicka, Strong solutions for generalized Newtonian fluids, J. Math. Fluid Mech. 7 (2005), no. 3, 413-450. https://doi.org/10.1007/s00021-004-0124-8
  10. L. Diening, M. Ruzicka, and J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluid, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 1, 1-46.
  11. J. Fan and T. Ozawa, Logarithmically improved regularity criteria for Navier-Stokes and related equations, Math. Methods Appl. Sci. 32 (2009), no. 17, 2309-2318. https://doi.org/10.1002/mma.1140
  12. A. Fardoun and R. Regbaoui, Heat flow for p-harmonic maps with small initial data, Calc. Var. Partial Differential Equations 16 (2003), no. 1, 1-16. https://doi.org/10.1007/s005260100138
  13. C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137-193. https://doi.org/10.1007/BF02392215
  14. N. Hungerbuhler, Global weak solutions of the p-harmonic flow into homogeneous space, Indiana Univ. Math. J. 45 (1996), no. 1, 275-288.
  15. R. G. Iagar and S. Moll, Rotationally symmetric p-harmonic flows from $D^2$ to $S^2$ : local well-posedness and finite time blow-up, arXiv:1305.6552v1[math.AP], 2013.
  16. O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon and Breach, New York, 1969.
  17. J. Malek, J. Necas, M. Rokyta, and M. Ruzicka, Weak and Measure-valued Solutions to Evolutionary PDEs, Chapman & Hall, 1996.
  18. M. Misawa, On the p-harmonic flow into spheres in the singular case, Nonlinear Anal. 50 (2002), no. 4, 485-494. https://doi.org/10.1016/S0362-546X(01)00755-6
  19. T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal. 34 (2003), no. 6, 1318-1330. https://doi.org/10.1137/S0036141001395868
  20. M. Pokorny, Cauchy problem for the non-Newtonian viscous incompressible fluid, Appl. Math. 41 (1996), no. 3, 169-201.
  21. J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mech. 9 (2007), no. 1, 104-138. https://doi.org/10.1007/s00021-006-0219-5

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