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Reaction-diffusion-convection problems in unbounded cylinders.

Rozenn Texier-PicardVitaly A. Volpert — 2003

Revista Matemática Complutense

The work is devoted to reaction-diffusion-convection problems in unbounded cylinders. We study the Fredholm property and properness of the corresponding elliptic operators and define the topological degree. Together with analysis of the spectrum of the linearized operators it allows us to study bifurcations of solutions, to prove existence of convective waves, and to make some conclusions about their stability.

Modelling of miscible liquids with the Korteweg stress

Ilya KostinMartine MarionRozenn Texier-PicardVitaly A. Volpert — 2003

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

When two miscible fluids, such as glycerol (glycerin) and water, are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients exist during some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress. We prove the global existence...

existence of solutions for an elliptic-algebraic system describing heat explosion in a two-phase medium

Cristelle BarillonGeorgy M. MakhviladzeVitaly A. Volpert — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

The paper is devoted to analysis of an elliptic-algebraic system of equations describing heat explosion in a two phase medium filling a star-shaped domain. Three types of solutions are found: classical, critical and multivalued. Regularity of solutions is studied as well as their behavior depending on the size of the domain and on the coefficient of heat exchange between the two phases. Critical conditions of existence of solutions are found for arbitrary positive source function.

Modelling of Miscible Liquids with the Korteweg Stress

Ilya KostinMartine MarionRozenn Texier-PicardVitaly A. Volpert — 2010

ESAIM: Mathematical Modelling and Numerical Analysis

When two miscible fluids, such as glycerol (glycerin) and water, are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients exist during some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress. We prove the global...

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