scholarly journals Global weak solutions for a degenerate parabolic system modeling the spreading of insoluble surfactant

2011 ◽  
Vol 60 (6) ◽  
pp. 1975-2020 ◽  
Author(s):  
Joachim Escher ◽  
Mattieu Hillairet ◽  
Philippe Laurencot ◽  
Cristoph Walker
Keyword(s):  
Parabolic System ◽  
Weak Solutions ◽  
System Modeling ◽  
Global Weak Solutions ◽  
Degenerate Parabolic System ◽  
Insoluble Surfactant ◽  
Degenerate Parabolic

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 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.

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