Mathematical Models and Methods in Applied Sciences

We study a model of phase segregation of the Allen–Cahn type, consisting in a system of two differential equations, one partial and the other ordinary, respectively interpreted as balances of microforces and microenergy; the two unknowns are the order parameter entering the standard A–C equation and the chemical potential. We introduce a notion of maximal solution to the o.d.e., parametrized on the order-parameter field; and, by substitution in the p.d.e. of the so-obtained chemical potential field, we give the latter equation the form of an Allen–Cahn equation for the order parameter, with a memory term. Finally, we prove the existence and uniqueness of global-in-time smooth solutions to this modified A–C equation, and we give a description of the relative ω-limit set.

We consider two models of flow and transport in porous media, the first one for consolidational flow in compressible sedimentary basins, the second one for flow in partially saturated media. Despite the differences in these physical settings, they lead to quite similar mathematical models with a strong pressure coupling. The first model is a coupled system of pde's of parabolic type. The second one involves a coupled system of pdes of degenerate parabolic–hyperbolic type. We state an existence result of weak solutions for both models.

Motivated by a mean field games stylized model for the choice of technologies (with externalities and economy of scale), we consider the associated optimization problem and prove an existence result. To complement the theoretical result, we introduce a monotonic algorithm to find the mean field equilibria. We close with some numerical results, including the multiplicity of equilibria describing the possibility of a technological transition.

In this paper, we prove the regularity and exponential stability of solutions in Hⁱ (i = 2, 4) for a pth power Newtonian fluid undergoing one-dimensional longitudinal motions. Some new ideas and more delicate estimates are introduced to prove these results.

We analyze the mathematical and physical properties of, and present numerical experiments for, the recently proposed NS-α-deconvolution model of fluid turbulence. This family of models has the well-known NS-α model as its zeroth-order model, and applies the Nth van Cittert deconvolution operator to the filtered terms in NS-α to create the Nth-order NS-α-deconvolution model. This model is proposed in Ref. 29, where it is shown to have consistency error O(α^{2N + 2}). Herein we prove that the model admits unique regular solutions, is frame invariant, and improves the consistency error of NS-α from O(α²) to O(α^{2N + 2}) while requiring (N + 1)^1/2 times more (typically 1 ≤ N ≤ 5) degrees of freedom for complete resolution. Numerical experiments show that adding approximate deconvolution to NS-α significantly improves accuracy in computations.

Double porosity models for the liquid filtration in a naturally fractured reservoir is derived from the homogenization theory. The governing equations on the microscopic level consist of the stationary Stokes system for an incompressible viscous fluid, occupying a crack-pore space (liquid domain), and stationary Lame equations for an incompressible elastic solid skeleton, coupled with the corresponding boundary conditions on the common boundary "solid skeleton-liquid domain". We assume that the liquid domain is a union of two independent systems of cracks (fissures) and pores, and that the dimensionless size δ of pores depends on the dimensionless size ε of cracks: δ = ε^r with r > 1. The rigorous justification is fulfilled for homogenization procedure as the dimensionless size of the cracks tends to zero, while the solid body is geometrically periodic. As the result we derive the well-known Biot–Terzaghi system of liquid filtration in poroelastic media, which consists of the usual Darcy law for the liquid in cracks coupled with anisotropic Lame's equation for the common displacements in the solid skeleton and in the liquid in pores and a continuity equation for the velocity of a mixture. The proofs are based on the method of reiterated homogenization, suggested by Allaire and Briane. As a consequence of the main result we derive the double porosity model for the filtration of the incompressible liquid in an absolutely rigid body.

In this paper we study a problem in radiotherapy treatment planning. This problem is formulated as an optimization problem of a functional of the radiative flux. It is constrained by the condition that the radiative flux, which depends on position, energy and direction of the particles, is governed by a Boltzmann integro-differential equation. We show the existence, uniqueness and regularity of solutions to this constrained optimization problem in an appropriate function space. The main new difficulty is the treatment of the energy loss term. Furthermore, we characterize optimal controls by deriving first-order optimality conditions.