Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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


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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Additional Information

Umberto Massari
Affiliation: Department of Mathematics, University of Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy
Email: umberto.massari@unife.it

Mariarosaria Padula
Affiliation: Department of Mathematics, University of Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy
Email: pad@unife.it

Senjo Shimizu
Affiliation: Faculty of Science, Shizuoka University, Ohya 836, Shizuoka 422-8529, Japan
Email: ssshimi@ipc.shizuoka.ac.jp

DOI: https://doi.org/10.1090/S0033-569X-2011-01226-X
Keywords: Nonlinear instability, Rayleigh-Taylor instability, horizontal layer flow, free boundary problem, Navier-Stokes equation, surface tension, gravity force
Received by editor(s): March 30, 2010
Published electronically: May 9, 2011
Article copyright: © Copyright 2011 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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