Remarks on Parabolicity in a One-Dimensional Interdiffusion Model with the Vegard Rule

Until 1948 the interdiffusion theory was based on the Onsager phenomenology, namely thermodynamics of irreversible processes, and a drift was not included. Its main limitation is practical impossibility of the experimental as well as theoretical determination of mobilities (diffusivities) in multicomponent systems (r> 2). After experimental discovery of the drift by Smigelskas and Kirkendall (Trans AIME 171:130–142, 1947), Darken (Trans AIME 175:184–201, 1948) formulated his famous model for the binary system. Consequently, the bi-velocity approach dominates interdiffusion studies (e.g. in more than 500 papers in 2020). In this paper, we consider the diffusional transport in a one-dimensional r-component solid solution. The model is expressed by the nonlinear system of strongly coupled evolution differential equations with initial and nonlinear coupled boundary conditions. We present a non-trivial proof of a theorem called the criterion of parabolicity, which implies the generalized parabolicity condition formulated without a proof in our previous works. This condition is a key in the proofs of our previous theorems on existence, uniqueness and properties of global weak solutions of the differential problem studied. The criterion of parabolicity works if diffusion coefficients are not too dispersed, and it is true in many physical systems. The numerical simulations consistent with real experiments for which our criterion works are given.