The concept of separation of diffusional and drift flows, i.e. , the postulate that the total mass flow is a sum of diffusion flux and translation only, is applied for the general case of diffusional transport in an rcomponent compound (process defined as interdiffusion in a one-dimensional mixture). The equations of local mass conservation (continuity equations), the appropriate expressions describing the cruxes (drift flux and diffusional flux), and momentum conservation equation (equation of motion) allow a complete quantitative description of diffusional transport process (in one-dimensional mixture showing constant concentration) to be formulated. The equations describing the interdiffusion process (mixing) in the general case where the components diffusivities vary with composition are derived. If certain regularity assumptions and a quantitative condition (concerning the diffusion coefficientsproviding a parabolic type of the final equation) are fulfilled, then there exists the unique solution of the interdiffusion problem. Good agreement between the numerical solution obtained with the use of Faedo-Galerkin method and experimental data is shown. An effective algebraic criterion allows us to determine the parabolic type of a particular problem. A condition for the "up-hill diffusion" in the three component mixture is given and a universal example of such effect is demonstrated. The results extend the standard Darken approach. The phenomenology allows the quantitative data on the dynamics of the processes to be obtained within an interdiffusion zone.

Abstract. We discuss some properties of an orthogonal projection onto a subset of a Euclidean space. The special stress is laid on projection's regularity and characterization of the interior of its domain.0. Introduction. Let M be a non-empty subset of a metric space Z. We define a relation P ⊂ Z × M , which we call the orthogonal projection onto M . Its domain is dom P := {z ∈ Z : there exists a unique point zwhere d denotes the metric of Z andThe orthogonal projection of z ∈ dom P is defined to be the unique point (z ′ =) P(z) ∈ M which realizes the distance of z to M . If M is a closed linear subspace of a Hilbert space Z, then P is the well-known linear orthogonal projection:The need of considering orthogonal projections onto non-linear sets has been noticed since a long time. For example, if Z = R n and M is a smooth (or analytic) submanifold, then the composition f • P| int dom P is the most natural smooth (analytic) extension of a given smooth (analytic) function f : M → R on an open neighbourhood of M (because in this case M ⊂ int dom P (see the generalization (3.8) of the classical result of Federer [5] and (4.1))). Of course, there are other methods of extending such functions, e.g. in the non-analytic case by local straightening of M or by applying Whitney's theory. However, in numerous problems the extension f • P is most useful, since it is simple and effective. The set int dom P is in some sense a star-shaped neighbourhood of M (see (1.5), (3.13)), so the retraction P| int dom P is helpful in studies on differentiable homotopy, e.g. for a given solenoidal vector field v : G → R n vanishing on the boundary of a 1991 Mathematics Subject Classification: 58B10, 51Kxx. Key words and phrases: projection's regularity and interior of its domain.

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A nonhomogeneous boundary value problem for interdiffusion in a solid multicomponent one‐dimensional mixture showing constant concentration is analyzed. Darken's concept of separation of diffusional and drift flows is applied for the general case of diffusional transport in an r‐component compound. The equations of local mass conservation, Darken's flux formula, the postulate of constant molar volume of the mixture, and the initial and boundary conditions form a self‐consistent interdiffusion problem (the quantitative dynamical model). This problem is analyzed in open as well as closed systems and when the component diffusivities vary with composition. The variational form of the interdiffusion problem and the criterion of parabolicity are presented. Results of numerical simulation of interdiffusion in binary and ternary solid solutions of finite dimensions are presented. The development of the “uphill diffusion” concentration profile in the ternary alloy is displayed.