On instability and stability of three-dimensional gravity driven viscous flows in a bounded domain |
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Authors: | Fei Jiang Song Jiang |
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Institution: | 1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China;2. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China |
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Abstract: | We investigate the instability and stability of some steady-states of a three-dimensional nonhomogeneous incompressible viscous flow driven by gravity in a bounded domain Ω of class C2. When the steady density is heavier with increasing height (i.e., the Rayleigh–Taylor steady-state), we show that the steady-state is linear unstable (i.e., the linear solution grows in time in H2) by constructing a (standard) energy functional and exploiting the modified variational method. Then, by introducing a new energy functional and using a careful bootstrap argument, we further show that the steady-state is nonlinear unstable in the sense of Hadamard. When the steady density is lighter with increasing height, we show, with the help of a restricted condition imposed on steady density, that the steady-state is linearly globally stable and nonlinearly asymptotically stable in the sense of Hadamard. |
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Keywords: | 76D05 76E09 |
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