Homogenization of a nonlinear elliptic system with a transport term in L 2

2021 ◽  
pp. 1-16
Author(s):  
Juan Casado-Díaz
Keyword(s):  
Elliptic System ◽  
Existence Of Solutions ◽  
Nonlinear Elliptic System ◽  
Convection Term ◽  
Renormalized Solutions ◽  
Linear Diffusion ◽  
Nonlinear Elliptic ◽  
Non Linear ◽  
Non Linear System ◽  
Transport Term

We consider the homogenization of a non-linear elliptic system of two equations related to some models in chemotaxis and flows in porous media. One of the equations contains a convection term where the transport vector is only in L 2 and a right-hand side which is only in L 1 . This makes it necessary to deal with entropy or renormalized solutions. The existence of solutions for this system has been proved in reference (Comm. Partial Differential Equations 45(7) (2020) 690–713). Here, we prove its stability by homogenization and that the correctors corresponding to the linear diffusion terms still provide a corrector for the solutions of the non-linear system.

A non-local thermistor problem

1995 ◽  
Vol 6 (1) ◽  
pp. 83-94 ◽  
Author(s):  
W. Allegretto ◽  
H. Xie
Keyword(s):  
Fixed Point ◽  
Elliptic System ◽  
Existence Of Solutions ◽  
Current Source ◽  
Input Current ◽  
Nonlinear Elliptic System ◽  
Nonlinear Elliptic ◽  
Large Current ◽  
Non Local ◽  
A Current

We consider the question of the existence/non-existence of solutions for the non-local nonlinear elliptic system which models a thermistor driven by a current source. Specifically, we show that for small input current there exists a solution, while this will not in general be the case for a sufficiently large current. A feature of our estimates is that the conditions for non-existence are determined by local criteria on the domain and the coefficients. Our basic tools for existence involve truncation, L2.μ estimates and fixed point arguments. Non-existence is obtained by averaging procedures and an application of Barta's Inequality.

On Some Generalized Compound Problem for a Nonlinear Elliptic System

1986 ◽  
Vol 126 (1) ◽  
pp. 55-62
Author(s):  
J. Wolska-Bochenek ◽  
L. Von Wolfersdorf
Keyword(s):  
Elliptic System ◽  
Nonlinear Elliptic System ◽  
Nonlinear Elliptic

On the Dirichlet problem for weakly non-linear elliptic partial differential equations

1977 ◽  
Vol 76 (4) ◽  
pp. 283-300 ◽  
Author(s):  
E. N. Dancer
Keyword(s):  
Dirichlet Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence Of Solutions ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Weakly Nonlinear ◽  
Partial Derivatives ◽  
Nonlinear Elliptic ◽  
Non Linear

SynopsisWe study the existence of solutions of the Dirichlet problem for weakly nonlinear elliptic partial differential equations. We only consider cases where the nonlinearities do not depend on any partial derivatives. For these cases, we prove the existence of solutions for a wide variety of nonlinearities.

Sign in / Sign up

Export Citation Format

Share Document