Continuous dependence results for a problem in penetrative convection

Ames, Karen A.; Payne, Lawrence E.

doi:10.1090/qam/1486548

Authors: Karen A. Ames and Lawrence E. Payne
Journal: Quart. Appl. Math. 55 (1997), 769-790
MSC: Primary 35Q35; Secondary 35B30, 76D05, 76R05
DOI: https://doi.org/10.1090/qam/1486548
MathSciNet review: MR1486548
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Abstract: Continuous dependence inequalities are derived for a system of equations that models penetrative convection in a thermally conducting viscous fluid with a linear buoyancy law. Both the forward-in-time problem and the improperly posed backward-in-time problem are analyzed. These results indicate that solutions depend continuously on a parameter in the boundary data.

Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
K. A. Ames and L. E. Payne, Continuous dependence results for solutions of the Navier-Stokes equations backward in time, Nonlinear Anal. 23 (1994), no. 1, 103–113. MR 1288501, DOI https://doi.org/10.1016/0362-546X%2894%2990254-2
Catherine Bandle, Isoperimetric inequalities and applications, Monographs and Studies in Mathematics, vol. 7, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980. MR 572958
F. Franchi and B. Straughan, Continuous dependence on the body force for solutions to the Navier-Stokes equations and on the heat supply in a model for double diffusive porous convection, J. Math. Anal. Appl. 172 (1993), no. 1, 117–129. MR 1199499, DOI https://doi.org/10.1006/jmaa.1993.1011
R. J. Knops and L. E. Payne, On the stability of solutions of the Navier-Stokes equations backward in time, Arch. Rational Mech. Anal. 29 (1968), 331–335. MR 226222, DOI https://doi.org/10.1007/BF00283897
L. E. Payne, Improperly posed problems in partial differential equations, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. Regional Conference Series in Applied Mathematics, No. 22. MR 0463736
L. E. Payne and H. F. Weinberger, Lower bounds for vibration frequencies of elastically supported membranes and plates, J. Soc. Indust. Appl. Math. 5 (1957), 171–182. MR 92431
James Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962) Univ. of Wisconsin Press, Madison, Wis., 1963, pp. 69–98. MR 0150444

J. C. Song, Continuous dependence on the initial data, viscosity, and body force for the Navier-Stokes equations, SAACM 2, 265–279 (1993)

René P. Sperb, Untere und obere Schranken für den tiefsten Eigenwert der elastisch gestützten Membran, Z. Angew. Math. Phys. 23 (1972), 231–244 (German, with English summary). MR 312800, DOI https://doi.org/10.1007/BF01593087
Brian Straughan, Mathematical aspects of penetrative convection, Pitman Research Notes in Mathematics Series, vol. 288, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1267954
W. Velte, Über ein Stabilitätskriterium der Hydrodynamik, Arch. Rational Mech. Anal. 9 (1962), 9–20 (German). MR 155501, DOI https://doi.org/10.1007/BF00253330