Loss of control of motions from initial data for pending capillary liquid

Massari, Umberto; Padula, Mariarosaria; Shimizu, Senjo

doi:10.1090/S0033-569X-2011-01226-X

Authors: Umberto Massari, Mariarosaria Padula and Senjo Shimizu
Journal: Quart. Appl. Math. 69 (2011), 569-601
MSC (2000): Primary 35Q30, 76D05
DOI: https://doi.org/10.1090/S0033-569X-2011-01226-X
Published electronically: May 9, 2011
MathSciNet review: 2850746
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Abstract:

First, the problem of stability of an equilibrium figure $F_*$ for an abstract system is reduced to the sign of the difference between the energy of the perturbed motion at initial time, and that of $F_*$. All control conditions are only sufficient conditions to ensure nonlinear stability.

Second, employing the local character of the nonlinear stability, some nonlinear instability theorems are proven by a direct method.

Third, the definition of loss of control from initial data for motions $F$ is introduced. A class of equilibrium figures $F_*$ is constructed such that: $F_*$ is nonlinearly stable; the motions, corresponding to initial data sufficiently far from $F_*$, cannot be controlled by their initial data for all time. A lower bound is computed for the norms of initial data above which the loss of control from initial data occurs.

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