On instability of Rayleigh–Taylor problem for incompressible liquid crystals under $L^{1}$-norm

We investigate the nonlinear Rayleigh–Taylor (RT) instability of a nonhomogeneous incompressible nematic liquid crystal in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady state solution. Thus we construct solutions of the linearized problem that grow in time in the Sobolev space $H^{4}$ H 4 , then we show that the RT equilibrium state is linearly unstable. With the help of the established unstable solutions of the linearized problem and error estimates between the linear and nonlinear solutions, we establish the nonlinear instability of the density, the horizontal and vertical velocities under $L^{1}$ L 1 -norm.

Instability of the abstract Rayleigh–Taylor problem and applications

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.

On effect of surface tension in the Rayleigh–Taylor problem of stratified viscoelastic fluids

In this article, we investigate the effect of surface tension in the Rayleigh–Taylor (RT) problem of stratified incompressible viscoelastic fluids. We prove that there exists an unstable solution to the linearized stratified RT problem with a largest growth rate Λ under the instability condition (i.e., the surface tension coefficient ϑ is less than a threshold $\vartheta _{c}$ ϑ c ). Moreover, for this instability condition, the largest growth rate $\varLambda _{\vartheta }$ Λ ϑ decreases from a positive constant to 0, when ϑ increases from 0 to $\vartheta _{c}$ ϑ c , which mathematically verifies that the internal surface tension can constrain the growth of the RT instability during the linear stage.