Ooid growth: Uniqueness of time-invariant, smooth shapes in 2D

Evolution of planar curves under a nonlocal geometric equation is investigated. It models the simultaneous contraction and growth of carbonate particles called ooids in geosciences. Using classical ODE results and a bijective mapping we demonstrate that the steady parameters associated with the physical environment determine a unique, time-invariant, compact shape among smooth, convex curves embedded in $R^2$. It is also revealed that any time-invariant solution possesses $D_2$ symmetry. The model predictions remarkably agree with ooid shapes observed in nature.