Mathematical Models and Methods in Applied Sciences

In this paper we investigate the compressible Navier–Stokes–Cahn–Hilliard equations (the so-called NSCH model) derived by Lowengrub and Truskinovsky. This model describes the flow of a binary compressible mixture; the fluids are supposed to be macroscopically immiscible, but partial mixing is permitted leading to narrow transition layers. The internal structure and macroscopic dynamics of these layers are induced by a Cahn–Hilliard law that the mixing ratio satisfies. The PDE constitute a strongly coupled hyperbolic–parabolic system. We establish a local existence and uniqueness result for strong solutions.

Simulations of a flow over a roughness are prohibitively expensive for small-scale structures. If the interest is only on some macroscale quantity it will be sufficient to model the influence of the unresolved microscale effects. Such multiscale models rely on an appropriate upscaling strategy. Here the strategy originally developed by Achdou et al. [Effective boundary conditions for laminar flows over periodic rough boundaries, J. Comput. Phys. 147 (1998) 187–218] for incompressible flows is extended to compressible high Reynolds number flow. For proof of concept a laminar flow over a flat plate with partially embedded roughness is simulated. The results are compared with computations on a rough domain.

We present a mathematical model for stock market volatility flocking. Our proposed model consists of geometric Brownian motions with time-varying volatilities coupled with Cucker–Smale (C–S) flocking and regime switching mechanisms. For all-to-all interactions, we assume that all assets' volatilities are coupled to each other with a constant interaction weight, and we show that the common volatility emerges asymptotically and discuss its financial applications. We also provide several numerical simulations and compare them to existing analytical results.

The model of quasistatic rate-independent evolution of a delamination at small strains in the so-called mixed mode, i.e. distinguishing opening (Mode I) from shearing (Mode II), devised in [Delamination and adhesive contact models and their mathematical analysis and numerical treatment, Chap. 9, in Mathematical Methods and Models in Composites, ed. V. Mantič (Imperial College Press, 2014), pp. 349–400; and in Quasistatic mixed-mode delamination model, Discrete Contin. Dynam. Syst. Ser. S 6 (2013) 591–610], is rigorously analyzed in the context of a concept of stress-driven local solutions. The model has separately convex stored energy and is associative, namely the one-homogeneous potential of dissipative forces driving the delamination depends only on rates of internal parameters. An efficient fractional-step-type semi-implicit discretization in time is shown to converge to (specific, stress-driven like) local solutions that may approximately obey the maximum-dissipation principle. Making still a spatial discretization, this convergence as well as relevancy of such solution concept are demonstrated on a nontrivial two-dimensional example.

We study the homogenization of elliptic systems of equations in divergence form where the coefficients are compositions of periodic functions with a random diffeomorphism with stationary gradient. This is done in the spirit of scalar stochastic homogenization by Blanc, Le Bris and Lions. An application of the abstract result is given for Maxwell's equations in random dissipative bianisotropic media.

We consider crack propagation in brittle nonlinear elastic materials in the context of quasi-static evolutions of energetic type. Given a sequence of self-similar domains nΩ on which the imposed boundary conditions scale according to Bažant's law, we show, in agreement with several experimental data, that the corresponding sequence of evolutions converges (for n → ∞) to the evolution of a crack in a brittle linear-elastic material.