Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SPATIAL BEHAVIOR OF SOLUTION FOR THE STOKES FLOW EQUATION

  • Liu, Yan (Department of Applied Mathematics Guangdong University of Finance) ;
  • Liao, Wenhui (Department of Applied Mathematics Guangdong University of Finance) ;
  • Lin, Changhao (School of Mathematical Sciences South China Normal University)
  • Received : 2009.09.07
  • Published : 2011.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, the equation of the transient Stokes flow of an incompressible viscous fluid is studied. Growth and decay estimates are established associating some appropriate cross sectional line and area integral measures. The method of the proof is based on a first-order differential inequality leading to an alternative of Phragm$\'{e}$n-Lindell$\"{o} $f type in terms of an area measure of the amplitude in question. In the case of decay, we also indicate how to bound the total energy.

Keywords

References

  1. J. N. Flavin, On Knowles' version of Saint-Venant's principle in two-dimensional elastostatics, Arch. Rational Mech. Anal. 53 (1973/74), 366-375.
  2. J. N. Flavin, R. J. Knops, and L. E. Payne, Asymptotic behaviour of solutions to semilinear elliptic equations on the half-cylinder, Z. Angew. Math. Phys. 43 (1992), no. 3,405-421. https://doi.org/10.1007/BF00946237
  3. C. O. Horgan, Recent developments concerning Saint-Venant's principle: an update, AMR 42 (1989), no. 11, part 1, 295-303.
  4. C. O. Horgan, Recent development concerning Saint-Venant's principle: a second update, Applied Mechanics Reviews 49 (1996), 101-111. https://doi.org/10.1115/1.3101961
  5. C. O. Horgan, Decay estimates for the biharmonic equation with applications to Saint-Venant principles in plane elasticity and Stokes flows, Quart. Appl. Math. 47 (1989), no. 1, 147-157. https://doi.org/10.1090/qam/987903
  6. C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant's principle, Adv. in Appl. Mech. 23 (1983), 179-269. https://doi.org/10.1016/S0065-2156(08)70244-8
  7. C. O. Horgan and L. E. Payne, Phragmen-Lindelof type results for harmonic functions with nonlinear boundary conditions, Arch. Rational Mech. Anal. 122 (1993), no. 2, 123-144. https://doi.org/10.1007/BF00378164
  8. J. K. Knowles, On the spatial decay of solutions of the heat equation, Z. Angew. Math. Phys. 22 (1971), 1050-1056. https://doi.org/10.1007/BF01590873
  9. J. K. Knowles, An energy estimate for the biharmonic equation and its application to Saint-Venant's principle in plane elastostatics, Indian J. Pure Appl. Math. 14 (1983), no. 7, 791-805.
  10. C. Lin, Spatial decay estimates and energy bounds for the Stokes flow equation, Stability and Appl. Anal. of Continuous Media 2 (1992), 249-264.
  11. C. Lin and L. E. Payne, Phragmen-Lindelof type results for second order quasilinear parabolic equations in $R^2$, Z. Angew. Math. Phys. 45 (1994), no. 2, 294-311. https://doi.org/10.1007/BF00943507
  12. L. E. Payne and P. W. Schaefer, Some Phragmen-Lindelof type results for the biharmonic equation, Z. Angew. Math. Phys. 45 (1994), no. 3, 414-432. https://doi.org/10.1007/BF00945929
  13. J. C. Song, Improved decay estimates in time-dependent Stokes flow, J. Math. Anal. Appl. 288 (2003), no. 2, 505-517. https://doi.org/10.1016/j.jmaa.2003.09.007