scholarly journals Fundamental solutions for second-order parabolic systems with drift terms

2018 ◽  
Vol 146 (7) ◽  
pp. 3019-3029 ◽  
Author(s):  
Hongjie Dong ◽  
Seick Kim
Keyword(s):  
Parabolic Systems ◽  
Fundamental Solutions ◽  
Second Order ◽  
Drift Terms

scholarly journals Maximum principles for parabolic systems coupled in both first-order and zero-order terms

1994 ◽  
Vol 17 (4) ◽  
pp. 813-816
Author(s):  
Chiping Zhou
Keyword(s):  
Parabolic Systems ◽  
Zero Order ◽  
Second Order ◽  
Maximum Principles ◽  
First Order

Some generalized maximum principles are established for linear second-order parabolic systems in which both first-order and zero-order terms are coupled.

scholarly journals Nonlinear stability in three-layer channel flows

2017 ◽  
Vol 829 ◽  
Author(s):  
E. S. Papaefthymiou ◽  
D. T. Papageorgiou
Keyword(s):  
Interfacial Tension ◽  
Nonlinear Stability ◽  
Evolution Equations ◽  
Nonlinear Models ◽  
Travelling Waves ◽  
Parabolic Systems ◽  
Second Order ◽  
Weakly Nonlinear ◽  
Flux Function ◽  
Inclination Angles

The nonlinear stability of viscous, immiscible multilayer flows in plane channels driven both by a pressure gradient and gravity is studied. Three fluid phases are present with two interfaces. Weakly nonlinear models of coupled evolution equations for the interfacial positions are derived and studied for inertialess, stably stratified flows in channels at small inclination angles. Interfacial tension is demoted and high-wavenumber stabilisation enters due to density stratification through second-order dissipation terms rather than the fourth-order ones found for strong interfacial tension. An asymptotic analysis is carried out to demonstrate how these models arise. The governing equations are $2\times 2$ systems of second-order semi-linear parabolic partial differential equations (PDEs) that can exhibit inertialess instabilities due to interaction between the interfaces. Mathematically this takes place due to a transition of the nonlinear flux function from hyperbolic to elliptic behaviour. The concept of hyperbolic invariant regions, found in nonlinear parabolic systems, is used to analyse this inertialess mechanism and to derive a transition criterion to predict the large-time nonlinear state of the system. The criterion is shown to predict nonlinear stability or instability of flows that are stable initially, i.e. the initial nonlinear fluxes are hyperbolic. Stability requires the hyperbolicity to persist at large times, whereas instability sets in when ellipticity is encountered as the system evolves. In the former case the solution decays asymptotically to its uniform base state, while in the latter case nonlinear travelling waves can emerge that could not be predicted by a linear stability analysis. The nonlinear analysis predicts threshold initial disturbances above which instability emerges.

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