Properness and topological degree for general elliptic operators.
Normal solvability of general linear elliptic problems.
Construction of the Leray-Schauder degree for elliptic operators in unbounded domains
Different types of solvability conditions for differential operators.
Reaction-diffusion-convection problems in unbounded cylinders.
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.
Marangoni Convection in a Photo-Chemically Reacting Liquid
Marangoni convection caused by a photochemical reaction of the type A B in a deep liquid layer is studied. Linear stability analysis is performed and the conditions for Marangoni convection to occur are obtained. It is shown that increasing the rate of the direct reaction, for example, by increasing the light intensity, destabilizes the steady state and causes convective motion of the fluid, whereas increasing the rate of the inverse reaction stabilizes the steady state. A weakly nonlinear analysis...
Modelling of miscible liquids with the Korteweg stress
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
Numerical Modelling of Contact Elastic-Plastic Flows
Wilkins' method has been successfully used since early 60s for numerical simulation of high velocity contact elastic-plastic flows. The present work proposes some effective modifications of this method including more sophisticated material model including the Baushinger effect; modification of the algorithm based on correction of the initial configuration of a solid; a contact algorithm based on the idea of a mild contact.
existence of solutions for an elliptic-algebraic system describing heat explosion in a two-phase medium
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
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...
Spatial Dynamics of Contact-Activated Fibrin Clot Formation and in Haemophilia B: Effects of Severity and Ahemphil B Treatment
Spatial dynamics of fibrin clot formation in non-stirred system activated by glass surface was studied as a function of FIX activity. Haemophilia B plasma was obtained from untreated patients with different levels of FIX deficiency and from severe haemophilia B patient treated with FIX concentrate (Ahemphil B) during its clearance with half-life t=12 hours. As reported previously (Ataullakhanov et al. Biochim Biophys Acta 1998; 1425: 453-468), clot growth in space showed two distinct phases: activation...
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