scholarly journals Existence result for degenerate cross-diffusion system with application to seawater intrusion

2018 ◽  
Vol 24 (4) ◽  
pp. 1735-1758 ◽  
Author(s):  
Jana Alkhayal ◽  
Samar Issa ◽  
Mustapha Jazar ◽  
Régis Monneau
Keyword(s):  
Seawater Intrusion ◽  
Existence Result ◽  
Degenerate Parabolic System ◽  
Strongly Coupled ◽  
Cross Diffusion ◽  
General Existence ◽  
Uniqueness Question ◽  
Degenerate Parabolic ◽  
Entropy Estimate ◽  
Medium Type

In this paper we study a degenerate parabolic system, which is strongly coupled. We prove general existence result, but the uniqueness question remains open. Our proof of existence is based on a crucial entropy estimate which controls the gradient of the solution together with its non-negativity. Our system is of porous medium type which is applicable to models in seawater intrusion.

scholarly journals Non-negative global weak solutions for a degenerate parabolic system modelling thin films driven by capillarity

2012 ◽  
Vol 142 (5) ◽  
pp. 1071-1085 ◽  
Author(s):  
Bogdan-Vasile Matioc
Keyword(s):  
Parabolic System ◽  
Weak Solutions ◽  
Driving Mechanism ◽  
System Modelling ◽  
Global Weak Solutions ◽  
Degenerate Parabolic System ◽  
Strongly Coupled ◽  
Fluid Layers ◽  
Order Degenerate ◽  
Degenerate Parabolic

We prove the global existence of non-negative weak solutions for a strongly coupled, fourth-order degenerate parabolic system governing the motion of two thin fluid layers in a porous medium when capillarity is the sole driving mechanism.

scholarly journals Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion

2004 ◽  
Vol 10 (3) ◽  
pp. 719-730 ◽  
Author(s):  
Y. S. Choi ◽  
◽  
Roger Lui ◽  
Yoshio Yamada ◽  
◽  
...  
Keyword(s):  
Global Solutions ◽  
Strongly Coupled ◽  
Cross Diffusion

scholarly journals Higher integrability for weak solutions to a degenerate parabolic system with singular coefficients

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yan Dong ◽  
Guangwei Du ◽  
Kelei Zhang
Keyword(s):  
Parabolic System ◽  
Weak Solutions ◽  
Vector Fields ◽  
Degenerate Parabolic System ◽  
Higher Integrability ◽  
Singular Coefficients ◽  
Degenerate Parabolic ◽  
Alpha Beta ◽  
Hörmander’S Condition ◽  
Reverse Hölder

Abstract In this paper, we study the degenerate parabolic system $$ u_{t}^{i} + X_{\alpha }^{*} \bigl(a_{ij}^{\alpha \beta }(z){X_{\beta }} {u^{j}}\bigr) = {g_{i}}(z,u,Xu) + X_{\alpha }^{*} f_{i}^{\alpha }(z,u,Xu), $$ u t i + X α ∗ ( a i j α β ( z ) X β u j ) = g i ( z , u , X u ) + X α ∗ f i α ( z , u , X u ) , where $X=\{X_{1},\ldots,X_{m} \}$ X = { X 1 , … , X m } is a system of smooth real vector fields satisfying Hörmander’s condition and the coefficients $a_{ij}^{\alpha \beta }$ a i j α β are measurable functions and their skew-symmetric part can be unbounded. After proving the $L^{2}$ L 2 estimates for the weak solutions, the higher integrability is proved by establishing a reverse Hölder inequality for weak solutions.

scholarly journals A generalised quasi-variational inequality without upper semicontinuity

1996 ◽  
Vol 54 (2) ◽  
pp. 247-254 ◽  
Author(s):  
Paolo Cubiotti ◽  
Xian-Zhi Yuan
Keyword(s):  
Variational Inequality ◽  
Convex Subset ◽  
Existence Result ◽  
Upper Semicontinuity ◽  
Nonempty Closed Convex Subset ◽  
Positive Answer ◽  
Upper Semicontinuous ◽  
Quasi Variational Inequality ◽  
General Existence ◽  
Image Position

In this note we deal with the following problem: given a nonempty closed convex subset X of Rn and two multifunctions Γ : X → 2X and , to find ( such thatWe prove a very general existence result where neither Γ nor Φ are assumed to be upper semicontinuous. In particular, our result give a positive answer to an open problem raised by the first author recently.

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