Mathematical Models and Methods in Applied Sciences

We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite-dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite-dimensional. Nevertheless, in the infinite-dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite-dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one- and two-dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.

Motivated by simulations of carbon nanocones (see [S. P. Jordan and V. H. Crespi, Theory of carbon nanocones: Mechanical chiral inversion of a micron-scale three-dimensional object, Phys. Rev. Lett. 93 (2004) 255504]), we consider a variational plate model for an elastic cone under compression in the direction of the cone symmetry axis. Assuming radial symmetry, and modeling the compression by suitable Dirichlet boundary conditions at the center and the boundary of the sheet, we identify the energy scaling law in the von Kármán plate model. Specifically, we find that three different regimes arise with increasing indentation $\delta$: initially the energetic cost of the logarithmic singularity dominates, then there is a linear response corresponding to a moderate deformation close to the boundary of the cone, and for larger $\delta$ a localized inversion takes place in the central region. Then, we show that for large enough indentations minimizers of the elastic energy cannot be radially symmetric. We do so by an explicit construction that achieves lower elastic energy than the minimum amount possible for radially symmetric deformations.

In this paper, we consider the Dirichlet problem of a one-velocity viscous drift-flux model. One of the phases is compressible, the other one is weakly compressible. Under weak assumptions on the initial data, which can be discontinuous and large as well as involve transition to pure single-phase points or regions, we show existence of global bounded weak solutions. One main ingredient is that we employ a decomposition of the pressure term appearing in the mixture momentum equation into two components, one for each of the two phases. This paves the way for deriving a basic energy equality. In particular, upper bounds on the masses are extracted from the estimates provided by the energy equality. By relying on weak compactness tools we obtain an existence result within the class of weak solutions. An essential novel aspect of this analysis, compared to previous works on the same model, is that the solution space is large enough to allow for transition to single-phase flow without any constraints. In particular, one of the phases can vanish in a point while the other phase can persist. The key to achieve this result, which represents a major step forward compared to previous results for this model, is that we do not rely on any higher-order (i.e. derivatives in space) estimates on the masses or pressure, only low-order estimates provided by the energy equality and the uniform upper bounds on the liquid and gas mass.

An intrinsic approach to a mathematical model of a linearly or nonlinearly elastic body consists in considering the strain measures found in the energy of this model as the sole unknowns, instead of the displacement field in the classical approach. Such an approach thus provides a direct computation of the stresses by means of the constitutive equation. The main problem therefore consists in identifying specific compatibility conditions that these new unknowns, which are now matrix fields with components in $L^2$, should satisfy in order that they correspond to an actual displacement field. Such compatibility conditions are either of Saint-Venant type, in which case they take the form of partial differential equations, or of Donati type, in which case they take the form of ortho- gonality relations against matrix fields that are divergence-free. The main objective of this paper consists in showing how an intrinsic approach can be successfully applied to the well-known Koiter’s model of a nonlinearly elastic shallow shell, thus providing the first instance (at least to the authors’ best knowledge) of a mathematical justification of this approach applied to a nonlinear shell model (“shallow” means that the absolute value of the Gaussian curvature of the middle surface of the shell is “uniformly small enough”). More specifically, we first identify and justify compatibility conditions of Donati type guaranteeing that the nonlinear strain measures found in Koiter’s model correspond to an actual displacement field. Second, we show that the associated intrinsic energy attains its minimum over a set of matrix fields that satisfy these Donati compatibility conditions, thus providing an existence theorem for the intrinsic approach; the proof relies in particular on an interesting per se nonlinear Korn inequality on a surface. Incidentally, this existence result (once converted into an equivalent existence theorem for the classical displacement approach) constitutes a significant improvement over previously known existence theorems for Koiter’s model of a nonlinearly elastic shallow shell.

In this paper, we construct a model to describe the evolutionary epidemiology of spore producing asexual plant pathogens in a homogeneous host population. By considering the evolution in the space of the pathogen phenotypic values, we derive an integro-differential equation with nonlocal mutation terms. Our first main result is concerned with the existence and uniqueness of the endemic steady state of the model. Next assuming that the mutation kernel depends on a small parameter $\epsilon >0$ (the variance of the dispersion into the space of the pathogen phenotypic values), we investigate the concentration properties of the endemic steady state in the space of phenotypic values. In the context of this work, several Evolutionary Attractors (EAs) (as defined in classical adaptive dynamics) may exist. However, in rather general situations, our results show that only one EA persists when the populations are at equilibrium and when $\epsilon$ is small enough. Our analysis strongly relies on a refined description of the spectral properties of some integral operator with a highly concentrated kernel. We conclude the paper by presenting some numerical simulations of the model to illustrate this concentration phenomenon.