Convective Turing bifurcation with conservation laws
Abstract
Generalizing results of MC,S and HSZ for certain model reaction-diffusion and reaction-convection-diffusion equations, we derive and rigorously justify weakly nonlinear amplitude equations governing general Turing bifurcation in the presence of conservation laws. In the nonconvective, reaction-diffusion case, this is seen similarly as in MC,S to be a real Ginsburg-Landau equation coupled with a diffusion equation in a large-scale mean-mode vector comprising variables associated with conservation laws. In the general, convective case, by contrast, the amplitude equations as noted in HSZ consist of a complex Ginsburg-Landau equation coupled with a singular convection-diffusion equation featuring rapidly-propagating modes with speed 1/ where measures amplitude of the wave as a disturbance from a background steady state. Different from the partially coupled case considered in HSZ in the context of B\'enard-Marangoni convection/inclined flow, the Ginzburg Landau and mean-mode equations are here fully coupled, leading to substantial new difficulties in the analysis. Applications are to biological morphogenesis, in particular vasculogenesis, as described by the Murray-Oster and other mechanochemical/hydrodynamical models
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