Catálogo de publicaciones - libros

This paper deals with parabolic-elliptic systems of drift-diffusion type modelling gravitational interaction of particles. The main feature is presence of a nonlinear diffusion describing physically relevant density-pressure relations. We study the existence of solutions of the evolution problem, and recall results on the existence of steady states, and the blow up of solutions in cases when drift prevails the diffusion.

Existence and uniqueness, local in time, of the solution of a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material is obtained by using the Friedman-Rubinstein integral representation method through an equivalent system of two Volterra integral equations. Moreover, an explicit solution of a similarity type is presented for a non-classical heat source depending on time and heat flux on the fixed face = 0.

In this article, we present briefly the mathematical study of the dynamics of line defects called dislocations, in crystals. The mathematical model is an eikonal equation describing the motion of the dislocation line with a velocity which is a non-local function of the whole shape of the dislocation. We present some partial existence and uniqueness results. Finally we also show that the self-dynamics of a dislocation line at large scale is asymptotically described by an anisotropic mean curvature motion.

American options valuation leads to solve an optimal stopping problem or a variational inequality. These two approaches involve the knowledge of a free boundary, boundary of the so-called exercise region. As we are not able to get a closed formula for the American option value function, we will approximate the free boundary by this of a Bermudean option. Indeed a Bermudean option value function is the solution of an optimal stopping problem which can be viewed as a free boundary problem. Thanks to a maximum principle, we evaluate the difference between Bermudean and American boundaries.

We consider Bingham incompressible flows with temperature dependent viscosity and plasticity threshold and with mixed boundary conditions, including a friction type boundary condition. The coupled system of motion and energy steady-state equations may be formulated through a variational inequality for the velocity and variational methods provide a weak solution to the model. In the asymptotic limit case of a high thermal conductivity, the temperature becomes a constant solving an implicit total energy equation involving the viscosity function, the plasticity threshold and the friction yield coefficient. The limit model corresponds to a steady-state Bingham flow with nonlocal parameters, which has therefore at least one solution.

We present some results concerning two classes of P.D.E.s containing a continuous hysteresis operator. We introduce a weak formulation in Sobolev spaces for a Cauchy problem; under suitable assumptions on the hysteresis operator, we state some existence results. The presentation of the paper is quite general, as we avoid to describe all the details of the proof of the theorems involved.

In this paper, we shall prove the existence of solutions for the system of second order partial differential equations. This system is constructed by the phase field equations with a convection described by the Navier-Stokes equations in a liquid region. In our setting, this liquid region is also unknown, which is defined by the solution of the phase field equations. In order to determine the liquid region by the unknown parameter, which is called order parameter, we need to get the continuity. From the framework, we shall obtain the smoothness of the order parameter by the compactness theorem of Aubin’s type.

We consider an evolution model describing the vertical movement of water and salt in a domain split in two parts: a water reservoir and a saturated porous medium below it, in which a continuous extraction of fresh water takes place (by the roots of mangroves). The problem is formulated in terms of a coupled system of partial differential equations for the salt concentration and the water flow in the porous medium, with a dynamic boundary condition which connects both subdomains.

We study the existence and uniqueness of solutions, the stability of the trivial steady state solution, and the conditions for the root zone to reach, in finite time, the threshold value of salt concentration under which mangroves may live.

We present a numerical method for tracking breaking waves over sloping beaches. We use a fully non-linear potential model for incompressible, irrotational and inviscid flow, and consider the effects of beach topography on breaking waves. The algorithm uses a Boundary Element Method (BEM) to compute the velocity at the interface, coupled to a Narrow Band Level Set Method to track the evolving air/water interface, and an associated extension equation to update the velocity potential both on and off the interface. The formulation of the algorithm is applicable to two- and three-dimensional breaking waves; in this paper, we concentrate on two-dimensional results showing wave breaking and rollup, and perform numerical convergence studies and comparison with previous techniques.

We describe a finite difference scheme for simulating incompressible flows on nonuniform meshes using quadtree/octree data structure. A semi- Lagrangian method is used to update the intermediate fluid velocity in a standard projection framework. Two Poisson solvers on fully adaptive grids are also described. The first one is cell-centered and yields first-order accurate solutions, while producing symmetric linear systems (see Losasso, Gibou and Fedkiw []). The second is node-based and yields second-order accurate solutions, while producing nonsymmetric linear systems (see Min, Gibou and Ceniceros []). A distinguishing feature of the node-based algorithm is that gradients are found to second-order accuracy as well. The schemes are fully adaptive, i.e., the difference of level between two adjacent cells can be arbitrary. Numerical results are presented in two and three spatial dimensions.