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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

References:

[1] Helmut Abels, The initial-value problem for the Navier-Stokes equations with a free surface in \?^{\?}-Sobolev spaces, Adv. Differential Equations 10 (2005), no. 1, 45 – 64. · Zbl 1105.35072
[2] G. Allain, Small-time existence for the Navier-Stokes equations with a free surface, Appl. Math. Optim. 16 (1987), no. 1, 37 – 50. · Zbl 0655.76021 · doi:10.1007/BF01442184
[3] J. T. Beale, Large-time regularity of viscous surface waves, Arch. Rational Mech. Anal. 84 (1983/84), no. 4, 307 – 352. · Zbl 0545.76029 · doi:10.1007/BF00250586
[4] J. Thomas Beale and Takaaki Nishida, Large-time behavior of viscous surface waves, Recent topics in nonlinear PDE, II (Sendai, 1984) North-Holland Math. Stud., vol. 128, North-Holland, Amsterdam, 1985, pp. 1 – 14. · Zbl 0642.76048 · doi:10.1016/S0304-0208(08)72355-7
[5] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, Dover Pub. Inc., New York, 1981. · Zbl 0142.44103
[6] P. G. Drazin and William Hill Reid, Hydrodynamic stability, Cambridge University Press, Cambridge-New York, 1981. Cambridge Monographs on Mechanics and Applied Mathematics. · Zbl 0449.76027
[7] Robert Finn, On the equations of capillarity, J. Math. Fluid Mech. 3 (2001), no. 2, 139 – 151. · Zbl 0999.76025 · doi:10.1007/PL00000966
[8] E. Frolova and M. Padula, Free boundary problem for a layer of inhomogeneous fluid, Eur. J. Mech. B Fluids 23 (2004), no. 4, 665 – 679. · Zbl 1060.76542 · doi:10.1016/j.euromechflu.2004.01.001
[9] Giovanni P. Galdi and Mariarosaria Padula, A new approach to energy theory in the stability of fluid motion, Arch. Rational Mech. Anal. 110 (1990), no. 3, 187 – 286. · Zbl 0719.76035 · doi:10.1007/BF00375129
[10] Bum Ja Jin and Mariarosaria Padula, In a horizontal layer with free upper surface, Commun. Pure Appl. Anal. 1 (2002), no. 3, 379 – 415. · Zbl 1132.35502 · doi:10.3934/cpaa.2002.1.379
[11] Bum Ja Jin and Mariarosaria Padula, Steady flows of compressible fluids in a rigid container with upper free boundary, Math. Ann. 329 (2004), no. 4, 723 – 770. · Zbl 1149.35398
[12] Bernard Helffer and Olivier Lafitte, Asymptotic methods for the eigenvalues of the Rayleigh equation for the linearized Rayleigh-Taylor instability, Asymptot. Anal. 33 (2003), no. 3-4, 189 – 235. · Zbl 1065.34084
[13] G. Iooss and M. Rossi, Nonlinear evolution of the two-dimensional Rayleigh-Taylor flow, European J. Mech. B Fluids 8 (1989), no. 1, 1 – 22. · Zbl 0677.76106
[14] Takaaki Nishida, Yoshiaki Teramoto, and Hideaki Yoshihara, Global in time behavior of viscous surface waves: horizontally periodic motion, J. Math. Kyoto Univ. 44 (2004), no. 2, 271 – 323. · Zbl 1095.35028
[15] Takaaki Nishida, Yoshiaki Teramoto, and Hideaki Yoshihara, Hopf bifurcation in viscous incompressible flow down an inclined plane, J. Math. Fluid Mech. 7 (2005), no. 1, 29 – 71. · Zbl 1065.35053 · doi:10.1007/s00021-004-0104-z
[16] Takaaki Nishida, Yoshiaki Teramoto, and Htay Aung Win, Navier-Stokes flow down an inclined plane: downward periodic motion, J. Math. Kyoto Univ. 33 (1993), no. 3, 787 – 801. · Zbl 0815.76027
[17] M. Padula, Free work and control of equilibrium configurations, Ann. Univ. Ferrara Sez. VII (N.S.) 49 (2003), 375 – 396 (English, with English and Italian summaries). · Zbl 1232.74005
[18] Mariarosaria Padula, On direct Lyapunov method in continuum theories, Nonlinear problems in mathematical physics and related topics, I, Int. Math. Ser. (N. Y.), vol. 1, Kluwer/Plenum, New York, 2002, pp. 289 – 302. · Zbl 1046.35089 · doi:10.1007/978-1-4615-0777-2_17
[19] M. Padula, Free work identity and nonlinear instability in fluids with free boundaries, Recent advances in elliptic and parabolic problems, World Sci. Publ., Hackensack, NJ, 2005, pp. 203 – 216. · Zbl 1140.76339 · doi:10.1142/9789812702050_0015
[20] M. Padula and V. A. Solonnikov, On Rayleigh-Taylor stability, Ann. Univ. Ferrara Sez. VII (N.S.) 46 (2000), 307 – 336 (English, with English and Italian summaries). Navier-Stokes equations and related nonlinear problems (Ferrara, 1999). · Zbl 1001.76039
[21] Mariarosaria Padula and Vsevolod A. Solonnikov, A simple proof of the linear instability of rotating liquid drops, Ann. Univ. Ferrara Sez. VII Sci. Mat. 54 (2008), no. 1, 107 – 122. · Zbl 1248.76059 · doi:10.1007/s11565-008-0042-4
[22] Jan Prüss and Gieri Simonett, Analysis of the boundary symbol for the two-phase Navier-Stokes equations with surface tension, Nonlocal and abstract parabolic equations and their applications, Banach Center Publ., vol. 86, Polish Acad. Sci. Inst. Math., Warsaw, 2009, pp. 265 – 285. · Zbl 1167.35555 · doi:10.4064/bc86-0-17
[23] M. Padula and V. A. Solonnikov, Existence of non-steady flows of an incompressible, viscous drop of fluid in a frame rotating with finite angular velocity, Elliptic and parabolic problems (Rolduc/Gaeta, 2001) World Sci. Publ., River Edge, NJ, 2002, pp. 180 – 203. · Zbl 1039.35018 · doi:10.1142/9789812777201_0019
[24] V. A. Solonnikov, On the stability of axisymmetric equilibrium figures of a rotating viscous incompressible fluid, Algebra i Analiz 16 (2004), no. 2, 120 – 153 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 2, 377 – 400.
[25] V. A. Solonnikov, On the stability of nonsymmetric equilibrium figures of a rotating viscous incompressible liquid, Interfaces Free Bound. 6 (2004), no. 4, 461 – 492. · Zbl 1068.35114 · doi:10.4171/IFB/110
[26] V. A. Solonnikov, On instability of axially symmetric equilibrium figures of rotating viscous incompressible liquid, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 318 (2004), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 36 [35], 277 – 297, 313 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 136 (2006), no. 2, 3812 – 3825. · Zbl 1075.35042 · doi:10.1007/s10958-006-0202-y
[27] V. A. Solonnikov, On instability of equilibrium figures of rotating viscous incompressible liquid, J. Math. Sci. (N.Y.) 128 (2005), no. 5, 3241 – 3262. Problems in mathematical analysis. No. 30. · Zbl 1099.76022 · doi:10.1007/s10958-005-0267-z
[28] V. A. Solonnikov, Letter to the editor: Erratum to: ”On a linear problem connected with equilibrium figures of a uniformly rotating viscous liquid” [J. Math. Sci. (N. Y.) 132 (2006), no. 4, 502 – 521; MR2197341], J. Math. Sci. (N.Y.) 135 (2006), no. 6, 3522 – 3528. Problems in mathematical analysis. No. 32. · doi:10.1007/s10958-006-0176-9
[29] Y. Shibata and S. Shimizu, Local solvability of free surface problems for the Navier-Stokes equations with surface tension, Preprint (2008). · Zbl 1209.35158
[30] Donna Lynn Gates Sylvester, Large time existence of small viscous surface waves without surface tension, Comm. Partial Differential Equations 15 (1990), no. 6, 823 – 903. · Zbl 0731.35081 · doi:10.1080/03605309908820709
[31] Atusi Tani, Small-time existence for the three-dimensional Navier-Stokes equations for an incompressible fluid with a free surface, Arch. Rational Mech. Anal. 133 (1996), no. 4, 299 – 331. · Zbl 0857.76026 · doi:10.1007/BF00375146
[32] Atusi Tani and Naoto Tanaka, Large-time existence of surface waves in incompressible viscous fluids with or without surface tension, Arch. Rational Mech. Anal. 130 (1995), no. 4, 303 – 314. · Zbl 0844.76025 · doi:10.1007/BF00375142
[33] Yoshiaki Teramoto, The initial value problem for viscous incompressible flow down an inclined plane, Hiroshima Math. J. 15 (1985), no. 3, 619 – 643. · Zbl 0625.76030
[34] Yoshiaki Teramoto, On the Navier-Stokes flow down an inclined plane, J. Math. Kyoto Univ. 32 (1992), no. 3, 593 – 619. · Zbl 0784.76021
[35] J. A. Whitehead and M. M. Chen, Thermal instability and convection of a thin fluid layer bounded by a stably stratified region, J. Fluid Mech. 40 (1970), 549-576.
[36] M. Padula, Asymptotic stability of steady compressible flows, Lecture Notes in Mathematics (2011), to appear. · Zbl 1227.76003
[37] M. Padula and V. A. Solonnikov, On the local solvability of free boundary problem for the Navier-Stokes equations, J. Math. Sci. 170 (4), 2010, 522. · Zbl 1357.35247
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