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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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