TY - JOUR

T1 - Stability of idgher norms in terms of energy-stability for the Boussinesq-equations

T2 - Remarks on the asymptotic behaviour of convection-roll-type solutions

AU - Kagei, Yoshiyuki

AU - Von Wahl, Wolf

PY - 1994/1/1

Y1 - 1994/1/1

N2 - In an infinite layer heated from below we perturb a steady viscous incomprssible fluid flow. The steady flow is assumed to be stable in the sense of Lyapunov with respect to the norm supt≥0 E1/2 (t), where E(t) is the kinetic energy of the perturbation at timet. Energy-stability may also set in at later times. Then the steady flow turns out to be stable in the sense ofLyapunov also with respect to L2-norms of higher order derivatives. In terms of these we are able to indicate absorbing sets for the disturbances if their initial values are small. The smallness required is discussed. As a by-product we obtain that (nonlinear) instability of any steady flow with respect to higher order norms of the disturbances, as considered in [1] by Galdi and Padula for perturbations of the motionless state in various situations, implies (nonlinear) instability with respect to kinetic energy, at least. Finally we consider some aspects of the time evolution of two-dimensional solutions of the Boussinesq-equations. Two-dimensional means that the velocity-field and the temperature do not depend on one of the plane space-variables. We speak of convection-roll-type solutions.

AB - In an infinite layer heated from below we perturb a steady viscous incomprssible fluid flow. The steady flow is assumed to be stable in the sense of Lyapunov with respect to the norm supt≥0 E1/2 (t), where E(t) is the kinetic energy of the perturbation at timet. Energy-stability may also set in at later times. Then the steady flow turns out to be stable in the sense ofLyapunov also with respect to L2-norms of higher order derivatives. In terms of these we are able to indicate absorbing sets for the disturbances if their initial values are small. The smallness required is discussed. As a by-product we obtain that (nonlinear) instability of any steady flow with respect to higher order norms of the disturbances, as considered in [1] by Galdi and Padula for perturbations of the motionless state in various situations, implies (nonlinear) instability with respect to kinetic energy, at least. Finally we consider some aspects of the time evolution of two-dimensional solutions of the Boussinesq-equations. Two-dimensional means that the velocity-field and the temperature do not depend on one of the plane space-variables. We speak of convection-roll-type solutions.

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M3 - Article

AN - SCOPUS:84972517067

SN - 0893-4983

VL - 7

SP - 921

EP - 948

JO - Differential and Integral Equations

JF - Differential and Integral Equations

IS - 3-4

ER -