SPATIAL BEHAVIOR OF SOLUTION FOR THE STOKES FLOW EQUATION

Title & Authors
SPATIAL BEHAVIOR OF SOLUTION FOR THE STOKES FLOW EQUATION
Liu, Yan; Liao, Wenhui; Lin, Changhao;

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$\small{\`{e}}$n-Lindell$\small{\"{o} }$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
spatial behavior;transient Stokes flow;Saint-Venant`s principle;biharmonic equation;stream function;
Language
English
Cited by
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.

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.

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.

6.
C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant's principle, Adv. in Appl. Mech. 23 (1983), 179-269.

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.

8.
J. K. Knowles, On the spatial decay of solutions of the heat equation, Z. Angew. Math. Phys. 22 (1971), 1050-1056.

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.

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.

13.
J. C. Song, Improved decay estimates in time-dependent Stokes flow, J. Math. Anal. Appl. 288 (2003), no. 2, 505-517.