Euler–Poincaré flows and leibniz structure of nonlinear reaction–diffusion type systems

2006 ◽  
Vol 56 (9) ◽  
pp. 1736-1751 ◽  
Author(s):  
Partha Guha
Keyword(s):  
Reaction Diffusion ◽  
Type Systems ◽  
Diffusion Type ◽  
Nonlinear Reaction

The entropy dissipation method for spatially inhomogeneous reaction–diffusion-type systems

2008 ◽  
Vol 464 (2100) ◽  
pp. 3273-3300 ◽  
Author(s):  
Marco Di Francesco ◽  
Klemens Fellner ◽  
Peter A Markowich
Keyword(s):  
Reaction Diffusion ◽  
Nonlinear Problems ◽  
Type Systems ◽  
Rates Of Convergence ◽  
Energy Functional ◽  
Convergence To Equilibrium ◽  
Solid State Physics ◽  
Diffusion Type ◽  
Entropy Dissipation ◽  
Diffusion Convection

We study the long-time asymptotics of reaction–diffusion-type systems that feature a monotone decaying entropy (Lyapunov, free energy) functional. We consider both bounded domains and confining potentials on the whole space for arbitrary space dimensions. Our aim is to derive quantitative expressions for (or estimates of) the rates of convergence towards an (entropy minimizing) equilibrium state in terms of the constants of diffusion and reaction and with respect to conserved quantities. Our method, the so-called entropy approach, seeks to quantify convergence to equilibrium by using functional inequalities, which relate quantitatively the entropy and its dissipation in time. The entropy approach is well suited to nonlinear problems and known to be quite robust with respect to model variations. It has already been widely applied to scalar diffusion–convection equations, and the main goal of this paper is to study its generalization to systems of partial differential equations that contain diffusion and reaction terms and admit fewer conservation laws than the size of the system. In particular, we successfully apply the entropy approach to general linear systems and to a nonlinear example of a reaction–diffusion–convection system arising in solid-state physics as a paradigm for general nonlinear systems.

Turbulent Spatio-Temporal Dynamics in Reacting-Diffusing Systems: Results for a Soluble Model

1978 ◽  
Vol 33 (5) ◽  
pp. 588-596
Author(s):  
P. Ortoleva ◽  
M. DelleDonne
Keyword(s):  
Temporal Dynamics ◽  
Temporal Evolution ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Type Systems ◽  
Model Systems ◽  
Dimensional Manifold ◽  
Perturbation Scheme ◽  
Diffusion Type ◽  
Spatio Temporal

A class of soluble three "species" reaction-diffusion type systems is presented Exact solutions are obtained which show turbulent spatio-temporal evolution All homogeneous evolution tends asymptotically toward an attractor which is shown to be a two layered two dimensional manifold in the three dimensional species space. Sustained aperiodic spatio-temporal solutions are also found.By considering particular model systems we show that turbulent solutions may exit as finite amplitude instabilities or as bifurcations which are aperiodic arbitrarily close to the bifurcation point and hence do not arise as a transition starting out essentially periodically.A perturbation scheme is used to show that d parameter families of spatio-temporal evolution are admitted by more general systems with attracting d dimensional manifolds in the homogeneous chemical kinetics

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