Instability of the abstract Rayleigh–Taylor problem and applications

Fei Jiang, Song Jiang, Weicheng Zhan

2020Mathematical Models and Methods in Applied Sciences50 citationsDOI

Abstract

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.

Topics & Concepts

Lagrangian and Eulerian specification of the flow fieldInstabilityMathematicsMagnetohydrodynamicsRayleigh–Taylor instabilityEulerian pathMathematical analysisNonlinear systemBoundary value problemApplied mathematicsLagrangianPhysicsMechanicsMagnetic fieldQuantum mechanicsNavier-Stokes equation solutionsFluid Dynamics and Turbulent FlowsComputational Fluid Dynamics and Aerodynamics