degenerate parabolic system
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2020 ◽  
Vol 30 (03) ◽  
pp. 513-537 ◽  
Author(s):  
Gaspard Jankowiak ◽  
Diane Peurichard ◽  
Anne Reversat ◽  
Christian Schmeiser ◽  
Michael Sixt

A two-dimensional mathematical model for cells migrating without adhesion capabilities is presented and analyzed. Cells are represented by their cortex, which is modeled as an elastic curve, subject to an internal pressure force. Net polymerization or depolymerization in the cortex is modeled via local addition or removal of material, driving a cortical flow. The model takes the form of a fully nonlinear degenerate parabolic system. An existence analysis is carried out by adapting ideas from the theory of gradient flows. Numerical simulations show that these simple rules can account for the behavior observed in experiments, suggesting a possible mechanical mechanism for adhesion-independent motility.


2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yan Dong ◽  
Guangwei Du ◽  
Kelei Zhang

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.


2018 ◽  
Vol 24 (4) ◽  
pp. 1735-1758 ◽  
Author(s):  
Jana Alkhayal ◽  
Samar Issa ◽  
Mustapha Jazar ◽  
Régis Monneau

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.


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