Department of

July 2022 August 2022 September 2022 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 1 2 3 4 5 6 1 2 3 3 4 5 6 7 8 9 7 8 9 10 11 12 13 4 5 6 7 8 9 10 10 11 12 13 14 15 16 14 15 16 17 18 19 20 11 12 13 14 15 16 17 17 18 19 20 21 22 23 21 22 23 24 25 26 27 18 19 20 21 22 23 24 24 25 26 27 28 29 30 28 29 30 31 25 26 27 28 29 30 31

Friday, April 8, 2022

**Abstract:** Generalizing results of Matthews-Cox/Sukhtayev for a model reaction-diffusion equation, 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 Matthews-Cox, Sukhtayev to be a real Ginsburg-Landau equation weakly 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 consist of a complex Ginsburg-Landau equation weakly coupled with a singular convection-diffusion equation featuring rapidly-propagating modes with speed $\sim 1/\epsilon$ where $\epsilon$ measures amplitude of the wave as a disturbance from a background steady state. Applications are to biological morphogenesis, in particular vasculogenesis, as described by the Murray-Oster and other mechanochemical/hydrodynamical models. This work is joint with Kevin Zumbrun.