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• W2024703146 abstract "First, <bold>the problem of stability</bold> of an equilibrium figure <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F Subscript asterisk> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> <mml:annotation encoding=application/x-tex>F_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for an abstract system <bold>is reduced to the sign of </bold>the difference between the energy of the perturbed motion at initial time, and that of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F Subscript asterisk> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> <mml:annotation encoding=application/x-tex>F_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding=application/x-tex>F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is introduced. A class of equilibrium figures <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F Subscript asterisk> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> <mml:annotation encoding=application/x-tex>F_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is constructed such that: <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F Subscript asterisk> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> <mml:annotation encoding=application/x-tex>F_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is nonlinearly stable; the motions, corresponding to initial data sufficiently far from <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F Subscript asterisk> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> <mml:annotation encoding=application/x-tex>F_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 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." @default.
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• W2024703146 date "2011-05-09" @default.
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• W2024703146 title "Loss of control of motions from initial data for pending capillary liquid" @default.
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