Mathematical Models and Methods in Applied Sciences

We present a non-planar fracture model in a poro-elastic medium. The medium in which the fracture is embedded is governed by the standard Biot equations of linear poro-elasticity and the flow of the fluid within the fracture is governed by the lubrication equation. We establish existence and uniqueness of the linearized coupled system under weak assumptions on the data. Two discretizations of the problem are formulated: one with a continuous Galerkin method and one with a mixed method for the flow in the reservoir and in the fracture. We perform the numerical analysis of the former and we provide an algorithm and numerical experiments for the latter.

In this paper we construct and analyze a two-well Hamiltonian on a 2D atomic lattice. The two wells of the Hamiltonian are prescribed by two rank-one connected martensitic twins, respectively. By constraining the deformed configurations to special 1D atomic chains with position-dependent elongation vectors for the vertical direction, we show that the structure of ground states under appropriate boundary conditions is close to the macroscopically expected twinned configurations with additional boundary layers localized near the twinning interfaces. In addition, we proceed to a continuum limit, show asymptotic piecewise rigidity of minimizing sequences and rigorously derive the corresponding limiting form of the surface energy.

We consider a model of gas–solid combustion with free interface proposed by L. Kagan and G. I. Sivashinsky. Our approach is twofold: (I) We eliminate the front and get to a fully nonlinear system with boundary conditions; (II) We use a fourth-order pseudo-differential equation for the front to achieve asymptotic regimes in rescaled variables. In both cases, we implement a numerical algorithm based on spectral method and represent numerically the evolution of the char. Fingering pattern formation occurs when the planar front is unstable. A series of simulations is presented to demonstrate the evolution of sparse fingers (I) and chaotic fingering (II).

We study non-negative solutions to the chemotaxis system

We develop and apply the theory of lower and upper compensated convex transforms introduced in [K. Zhang, Compensated convexity and its applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) 743–771] to define multiscale, parametrized, geometric singularity extraction transforms of ridges, valleys and edges of function graphs and sets in ℝⁿ. These transforms can be interpreted as "tight" opening and closing operators, respectively, with quadratic structuring functions. We show that these geometric morphological operators are invariant with respect to translation, and stable under curvature perturbations, and establish precise locality and tight approximation properties for compensated convex transforms applied to bounded functions and continuous functions. Furthermore, we establish multiscale and Hausdorff stable versions of such transforms. Specifically, the stable ridge transforms can be used to extract exterior corners of domains defined by their characteristic functions. Examples of explicitly calculated prototype mathematical models are given, as well as some numerical experiments illustrating the application of these transforms to 2d and 3d objects.