Geodesic
Unconditional nonlinear exponential stability in the Bénard problem for a mixture: necessary and sufficient conditions
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 9 (1998) no. 3, pp. 221-236
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The Lyapunov direct method is applied to study nonlinear exponential stability of a basic motionless state to imposed linear temperature and concentration fields of a binary fluid mixture heated and salted from below, in the Oberbeck-Boussinesq scheme. Stress-free and rigid surfaces are considered and absence of Hopf bifurcation is assumed. We prove the coincidence of the linear and (unconditional) nonlinear critical stability limits, when the ratio between the Schmidt and the Prandtl numbers is less or equal to 1. Precisely, we obtain necessary and sufficient conditions of unconditional nonlinear exponential stability of the basic motionless state. 
@article{RLIN_1998_9_9_3_a6,
     author = {Mulone, Giuseppe and Rionero, Salvatore},
     title = {Unconditional nonlinear exponential stability in the {B\'enard} problem for a mixture: necessary and sufficient conditions},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {221--236},
     year = {1998},
     volume = {Ser. 9, 9},
     number = {3},
     zbl = {0922.76171},
     mrnumber = {MR1683010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_1998_9_9_3_a6/}
}
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Mulone, Giuseppe; Rionero, Salvatore. Unconditional nonlinear exponential stability in the Bénard problem for a mixture: necessary and sufficient conditions. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 9 (1998) no. 3, pp. 221-236. http://geodesic.mathdoc.fr/item/RLIN_1998_9_9_3_a6/