Non-local parabolic systems: applications in the three-phase capillary fluid filtration

2005 ◽  
Vol 16 (4) ◽  
pp. 493-517 ◽  
Author(s):  
V. V. SHELUKHIN ◽  
C. I. KONDO
Keyword(s):  
Parabolic Systems ◽  
Initial Boundary ◽  
Pressure Control ◽  
Global Solvability ◽  
Fluid Filtration ◽  
Three Phase ◽  
Non Local ◽  
Degenerate Parabolic ◽  
Initial Boundary Value ◽  
Diffusion Tensors

Non-local degenerate parabolic systems arise in three-phase capillary flows in porous media under a pressure control at the inflow- and outflow-boundaries. A mathematical study of such systems is performed for a class of capillarity pressure functions corresponding to triangular capillarity-diffusion tensors. To this end a theory of non-degenerate parabolic approximations is developed: the unique global solvability of initial boundary-value problems is proved.

scholarly journals On the existence of global solution of the system of equations of liquid movement in porous medium

2021 ◽  
Vol 234 ◽  
pp. 00095
Author(s):  
Margarita Tokareva ◽  
Alexander Papin
Keyword(s):  
Porous Medium ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Solvability ◽  
Compressible Liquid ◽  
Fluid Filtration ◽  
One Dimensional ◽  
Lagrangian Variables ◽  
Initial Boundary Value ◽  
Liquid Movement

The initial-boundary value problem for the system of one-dimensional isothermal motion of viscous liquid in deformable viscous porous medium is considered. Local theorem of existence and uniqueness of problem is proved in case of compressible liquid. In case of incompressible liquid the theorem of global solvability in time is proved in Holder classes. A feature of the model of fluid filtration in a porous medium considered in this paper is the inclusion of the mobility of the solid skeleton and its poroelastiс properties. The transition from Euler variables to Lagrangian variables is used in the proof of the theorems.

Estimates of blow-up time for a non-local problem modelling an Ohmic heating process

2002 ◽  
Vol 13 (3) ◽  
pp. 337-351 ◽  
Author(s):  
N. I. KAVALLARIS ◽  
C. V. NIKOLOPOULOS ◽  
D. E. TZANETIS
Keyword(s):  
Blow Up ◽  
Ohmic Heating ◽  
Numerical Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Solutions ◽  
Upper And Lower Bounds ◽  
Heating Process ◽  
Non Local ◽  
Initial Boundary Value

We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f (u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2 for 0 < (λ−λ*) [Lt ] 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.

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