We consider a horizontal rectangular piece of layer of viscous inhomogeneous fluid, bounded below by a rigid plane and above by a free surface with lateral periodicity conditions. In the presence of gravity and surface tension, the fluid admits many nonhomogeneous rest states each of which corresponds to a given vertical density distribution. We study some stability properties of a rest state with plane free boundary Γ , corresponding to the given (large potential) gravitational force, given volume V , and given mass M, in the class of motions deriving by perturbations of initial data into the same volume V and having the same total mass M. We assume the existence of global smooth nonsteady flows and study the control in time for the L2 norms of perturbations to velocity, density, and height, in terms of their values at initial time. In the class of linear basic density profiles, we solve the problem of stability in the mean. The model does not admit a decay in time for the density, nevertheless, we prove a decay to zero for the L2 norm of the velocity gradient along a sequence of times. Furthermore, if we do not perturb the initial density then we can prove that more regular norms of perturbations to velocity, density, and height are bounded for all times. Finally, for the homogeneous basic density distribution, the asymptotic decay to zero of these more regular norms of perturbations to velocity, density, and height takes place.
Free boundary problem for a layer of inhomogeneous fluid
PADULA, Mariarosaria
2004
Abstract
We consider a horizontal rectangular piece of layer of viscous inhomogeneous fluid, bounded below by a rigid plane and above by a free surface with lateral periodicity conditions. In the presence of gravity and surface tension, the fluid admits many nonhomogeneous rest states each of which corresponds to a given vertical density distribution. We study some stability properties of a rest state with plane free boundary Γ , corresponding to the given (large potential) gravitational force, given volume V , and given mass M, in the class of motions deriving by perturbations of initial data into the same volume V and having the same total mass M. We assume the existence of global smooth nonsteady flows and study the control in time for the L2 norms of perturbations to velocity, density, and height, in terms of their values at initial time. In the class of linear basic density profiles, we solve the problem of stability in the mean. The model does not admit a decay in time for the density, nevertheless, we prove a decay to zero for the L2 norm of the velocity gradient along a sequence of times. Furthermore, if we do not perturb the initial density then we can prove that more regular norms of perturbations to velocity, density, and height are bounded for all times. Finally, for the homogeneous basic density distribution, the asymptotic decay to zero of these more regular norms of perturbations to velocity, density, and height takes place.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
