When two miscible fluids, such as glycerol (glycerin) and water,are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients existduring some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical modelconsisting of the diffusion equation with convective terms and ofthe Navier-Stokes equations with the Korteweg stress.We prove the global existence and uniqueness of the solution for the associated initial-boundary value problem in a two-dimensional bounded domain.We study the longtime behavior of the solution and show that it convergesto the uniform composition distribution with zero velocity field.We also present numerical simulations of miscible drops and show howtransient interfacial phenomena can change their shape.

In this work we introduce an accurate solver for theShallow Water Equations with source terms. This scheme does not need any kind of entropy correction to avoid instabilities near critical points. The scheme also solves the non-homogeneous case, in such a way that all equilibria are computed at least with second order accuracy. We perform several tests for relevant flows showing the performance of our scheme.

The limit behavior of the solutions of Signorini's type-likeproblems in periodically perforated domains with periodε is studied. The main feature of this limit behaviour isthe existence of a critical size of the perforations thatseparates different emerging phenomena as ε → 0. In the critical case, it is shown that Signorini's problemconverges to a problem associated to a new operator whichis the sum of a standard homogenized operator and an extra zeroorder term (“strange term”) coming from the geometry; itsappearance is due to the special size of the holes. The limitproblem captures the two sources of oscillations involved in thiskind of free boundary-value problems, namely, those arising fromthe size of the holes and those due to the periodic inhomogeneityof the medium. The main ingredient of the method used in the proofis an explicit construction of suitable test functions whichprovide a good understanding of the interactions between the abovementioned sources of oscillations.

We consider the time-harmonic eddy current problem in its electric formulationwhere the conductor is a polyhedral domain. By proving the convergencein energy, we justify in what sense this problem is the limit of a family of Maxwell transmission problems: Rather than a low frequency limit, this limit has to be understood in the sense of Bossavit [11].We describe the singularities of the solutions.They are related to edge and corner singularities of certain problems for the scalarLaplace operator, namely the interior Neumann problem, the exterior Dirichlet problem, and possibly, an interface problem. These singularities are the limit of the singularities of the related family of Maxwell problems.

A current procedure that takes into account the Dirichlet boundary conditionwith non-smooth data is to change it into aRobin type condition by introducing a penalization term; a major effect of thisprocedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem wherethe control variable is the Dirichlet data.We describe the Robin penalization,and we bound the gap between the penalized and the non-penalized boundary controlsfor the small penalization parameter. Some numerical results are reported on to highlightthe reliability of such an approach.

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented Lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.