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A simple réaction-diffusion model is used to demonstrate the existence of asymptotic (i e long time) knotted solutions of réaction-transport problems. The knots are attained with respect to the surface of constant concentration. These solutions cannot be mapped continuously onto the plane and as such have no two dimensional analogue — they are strictly three dimensional structures The existence of knotted solutions is first argued for intuitively using the properties of a simple reaction-diffusion System. A variational theorem for this system is then denved Extrema of the associated « energy » functional with knotted topology are obtained numencally The existence of a rich class of knotted and other strictly three dimensional solutions is also discussed When the reaction-diffusion medium is subject to mechanical stresses, flows may result These flows may interact with emerging dissipative structure when the time scales for flow and reaction are comparable Imposed shears may orient compositional patterns Ifthe rhéologie properties of the medium depend on composition, vortices may émerge under conditions far below the cntical Taylor shear rate I. A FREE BOUNDARY REACTION-DIFFUSION MODEL Consider the knotted structures of figures 1, 2. Our purpose here is to demonstrate that a surface of equal concentration for the long time solution of a reaction-transport system can take on such topologies. For concreteness consider the reaction-diffusion problem ^ i = D,Vc, +Fl(c) (1.1) (*) Research supported in part by the Mathematics, Chemistry, and Geosciences Programs of the U.S. National Science Foundation. (*) Department of Chemistry, Indiana University, Bloomington, IN 47405 MAN Modélisation mathématique et Analyse numérique 507/11/$ 3.10 Mathematical Modellmg and Numencal Analysis © AFCET Gauthier-Vülars

Nonequilibrium reaction-diffusion structures in rigid and visco-elastic media : knots and unstable noninertial flows