On the convergence conditions for the method with boundary condition splitting in Sobolev spaces of high smoothness and the compatibility conditions for the nonstationary stokes problem

2010 ◽  
Vol 82 (3) ◽  
pp. 931-935 ◽  
Author(s):  
B. V. Pal’tsev
Keyword(s):  
Boundary Condition ◽  
Sobolev Spaces ◽  
Stokes Problem ◽  
Compatibility Conditions ◽  
Convergence Conditions ◽  
High Smoothness ◽  
Nonstationary Stokes Problem

scholarly journals Instability of the abstract Rayleigh–Taylor problem and applications

2020 ◽  
Vol 30 (12) ◽  
pp. 2299-2388 ◽  
Author(s):  
Fei Jiang ◽  
Song Jiang ◽  
Weicheng Zhan
Keyword(s):  
Initial Data ◽  
Stokes Problem ◽  
Strong Solutions ◽  
Viscous Flows ◽  
Existence Theory ◽  
Lagrangian Coordinates ◽  
Compatibility Conditions ◽  
Unstable Solutions ◽  
Taylor Problem ◽  
Proper Values

Based on a bootstrap instability method, we prove the existence of unstable strong solutions in the sense of [Formula: see text]-norm to an abstract Rayleigh–Taylor (RT) problem arising from stratified viscous fluids in Lagrangian coordinates. In the proof we develop a method to modify the initial data of the linearized abstract RT problem by exploiting the existence theory of a unique solution to the stratified (steady) Stokes problem and an iterative technique, such that the obtained modified initial data satisfy the necessary compatibility conditions on boundary of the original (nonlinear) abstract RT problem. Applying an inverse transform of Lagrangian coordinates to the obtained unstable solutions and taking then proper values of the parameters, we can further obtain unstable solutions of the RT problem in viscoelastic, magnetohydrodynamics (MHD) flows with zero resistivity and pure viscous flows (with/without interface intension) in Eulerian coordinates.

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