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Loss of control of motions from initial data for pending capillary liquid. (English) Zbl 1236.35209
A capillary liquid drop $$F$$ is pending below a rigid surface $$S$$ with air at rest all around. Its equilibrium position is denoted by $$F_*$$. Then the problem of stability of an equilibrium $$F_*$$ 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.
Further, employing the local character of the nonlinear stability, some nonlinear instability theorems are proven by a direct method.
Finally 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.
##### MSC:
 35R35 Free boundary problems for PDEs 76D45 Capillarity (surface tension) for incompressible viscous fluids 35Q35 PDEs in connection with fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids
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