Mathematical Models and Methods in Applied Sciences

In this work we present a homogenization result for a class of degenerate elliptic functionals mimicking strongly anisotropic media. We study the limit as ε→0 of the functionals

We study, via Γ-convergence, the homogenization in L^∞ of supremal functionals of the form . We prove that the homogenized problem is still supremal and its energy density is given by a cell-problem formula.

Bloch equations give a quantum description of the coupling between atoms and a driving electric force. It is commonly used in optics to describe the interaction of a laser beam with a sample of atoms. In this paper, we address the asymptotics of these equations for a high frequency electric field, in a weak coupling regime. The electric forcing is taken quasiperiodic in time.

We prove the convergence towards a rate equation, i.e. a linear Boltzmann equation, recovering in this way the physically relevant asymptotic model. It describes the transitions amongst the various energy levels of the atoms, governed by the resonances between the electric forcing and the atoms' eigenfrequencies. We also give the explicit value for the transition rates.

The present task has already been addressed in Ref. 5 in the case when the energy levels are fixed, and for different classes of electric fields. Here, we extend the study in two directions. First, we consider almost degenerate energy levels, a natural situation in practice. In this case, almost resonances might occur. Technically, this implies that the small divisor estimates needed in Ref. 5 are false, due to the fact that the Diophantine condition is unstable with respect to small perturbations. We use an appropriate ultraviolet cutoff to restore the analysis and to sort out the asymptotically relevant frequencies. Second, since the asymptotic rate equation may be singular in time, we completely analyze the initial time-layer, as well as the associated convergence towards an equilibrium state.

Due to the lack of regularity of the solutions to the hydrostatic approximation of Navier–Stokes equations, an energy identity cannot be deduced. By including certain nonlinear perturbations to the hydrostatic approximation equations, the solutions to the perturbed problem are smooth enough so that they satisfy the corresponding energy identity. The perturbations considered in this paper are of the monotone class. Three kinds of problems are then studied. To do that, we introduce a functional setting and show in every case that the set of smooth functions with compact support is dense in the space where we search for solutions. When the perturbations are small enough in a certain sense, the solutions of the perturbed problem are close to those of the original one. As a result, this gives a new proof of the existence of solutions to the hydrostatic approximation of Navier–Stokes equations. Finally, this regularization technique has been applied to the analysis of a one-equation hydrostatic turbulence model.

We study the large-time behavior of a spherically symmetric motion of isentropic and compressible viscous gas in a field of potential force over an unbounded exterior domain in ℝⁿ(n≥2). First, we show the unique existence of a stationary solution satisfying an adhesion boundary condition and a positive spatial asymptotic condition. Then, it is shown that the stationary solution becomes a time asymptotic state to the initial boundary value problem with the same boundary and spatial asymptotic conditions. Here, the initial data can be chosen arbitrarily large if it belongs to the suitable Sobolev space. Moreover, if the external force is attractive to the center of a sphere, it can also be taken arbitrarily large. The proof of the stability theorem is based on computations, executed by using the Lagrangian coordinate. In the proof, it is the essential step to obtain the pointwise estimate for the density. It is derived through employing a representation formula of the density with the aid of the standard energy method. The Hölder regularity of the initial data is also required for translating the results in the Lagrangian coordinate to those in the Eulerian coordinate.

We describe all operators on scalar finite element spaces which appear as the restriction of a second-order (linear) differential operator. More precisely we provide a family of isomorphisms between this space of discrete differential operators and products of some exotic finite element spaces. This provides a unified framework for the Regge calculus of numerical relativity and the Nédélec edge elements of computational electromagnetism.

The main aim of this paper is to highlight that, when dealing with DG methods for linear hyperbolic equations or advection-dominated equations, it is much more convenient to write the upwind numerical flux as the sum of the usual (symmetric) average and a jump penalty. The equivalence of the two ways of writing is certainly well known (see e.g. Ref. 4); yet, it is very widespread not to consider upwinding, for DG methods, as a stabilization procedure, and too often in the literature the upwind form is preferred in proofs. Here, we wish to underline the fact that the combined use of the formalism of Ref. 3 and the jump formulation of upwind terms has several advantages. One of them is, in general, to provide a simpler and more elegant way of proving stability. The second advantage is that the calibration of the penalty parameter to be used in the jump term is left to the user (who can think of taking advantage of this added freedom), and the third is that, if a diffusive term is present, the two jump stabilizations (for the generalized upwinding and for the DG treatment of the diffusive term) are often of identical or very similar form, and this can also be turned to the user's advantage.