Jesús Ildefonso Díaz

On a nonlinear parabolic problem arising in some models related to turbulent flows

This paper studies the Cauchy–Dirichlet problem associated with the equation \[ b(u)_t - {\operatorname{div}}\left( {| {\nabla u - K(b(u)){\bf e}} |^{p - 2} (\nabla u - K(b(u)){\bf e})} \right) + g(x,u) = f(t,x).\] This problem arises in the study of some turbulent regimes: flows of incompressible turbulent fluids through porous media and gases flowing in pipes of uniform cross sectional areas. The paper focuses on the class of bounded weak solutions, and shows (under suitable assumptions) their stabilization, as

An elliptic equation with singular nonlinearity

t \to \infty

On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium

, to the set of bounded weak solutions of the associated stationary problem. The existence and comparison properties (implying uniqueness) of such solutions are also investigated.

ON A DOUBLY NONLINEAR PARABOLIC OBSTACLE PROBLEM MODELLING ICE SHEET DYNAMICS

The purpose of this paper is to study the problem (P) −Δu+u−α=f in Ω, u=0 on ∂Ω, u−α∈L1(Ω), u>0 in Ω, where Ω is a bounded smooth open set of RN, f≥0, f∈L1(Ω) and 0<α<1. When α≤0 this problem corresponds to a monotone semilinear equation for which the existence and uniqueness of solutions are well known. On the other hand, semilinear equations with nonmonotone perturbations have also been considered in the literature by many different methods. The originality of our study comes from the singularity of the nonlinearity in (P).

Effective Chemical Processes in Porous Media

The main result of the paper is the uniqueness of nonnegative solutions of the Cauchy problem and of the first and mixed boundary value problems for a class of degenerate parabolic equations which includes the model equation ut=(um)xx+(un)x, where m≥1 and n>0. In particular, n is allowed to be smaller than one. The proof is based on a refined test function argument. The condition that u be nonnegative is crucial, but the restriction to one space variable is not.

ON THE COMPLEX GINZBURG¿LANDAU EQUATION WITH A DELAYED FEEDBACK

This paper deals with the weak formulation of a free (moving) boundary problem arising in theoretical glaciology. Considering shallow ice sheet flow, we present the mathematical analysis and the numerical solution of the second order nonlinear degenerate parabolic equation modelling, in the isothermal case, the ice sheet non-Newtonian dynamics. An obstacle problem is then deduced and analyzed. The existence of a free boundary generated by the support of the solution is proved and its location and evolution are qualitatively described by using a comparison principle and an energy method. Then the solutions are numerically computed with a method of characteristics and a duality algorithm to deal with the resulting variational inequalities. The weak framework we introduce and its analysis (both qualitative and numerical) are not restricted to the simple physics of the ice sheet model we consider nor to the model dimension; they can be successfully applied to more realistic and sophisticated models related to other geophysical settings.

Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion

In the book by Hornung, Chap. 6, the author proposes a homogenization strategy for the effective behavior of some chemical processes involving adsorption and reactions arising in porous media. Rigorous proofs of the convergence results are given in the case of linear adsorption rates and linear chemical reactions. The author leaves as an open question the case of a nonlinear adsorption rate. Our goal in this paper is to study two well-known examples of such nonlinear models, namely the so-called Freundlich and Langmuir kinetics.

Global bifurcation and continua of nonnegative solutions for a quasilinear elliptic problem

We show how to stabilize the uniform oscillations of the complex Ginzburg–Landau equation with periodic boundary conditions by means of some global delayed feedback. The proof is based on an abstractpseudo-linearization principle and a careful study of the spectrum of the linearized operator.

On a nonlocal stationary free-boundary problem arising in the confinement of a plasma in a Stellarator geometry

The Boussinesq system arises in Fluid Mechanics when motion is governed by density gradients caused by temperature or concentration differences. In the former case, and when thermodynamical coefficients are regarded as temperature dependent, the system consists of the Navier-Stokes equations and the non linear heat equation coupled through the viscosity, bouyancy and convective terms. According to the balance between specific heat and thermal conductivity the diffusion term in the heat equation may lead to a singular or degenerate parabolic equation. In this paper we prove the existence of solutions of the general problem as well as the uniqueness of solutions when the spatial dimension is two.

UNIQUENESS AND EXISTENCE OF SOLUTIONS IN THE BVt(Q) SPACE TO A DOUBLY NONLINEAR PARABOLIC PROBLEM

In this Note, we study the existence and multiplicity of solutions, strictly positive or nonnegative having a dead core (where the solution vanishes) of a one-dimensional equation of eigenvalue type associated to a quasilinear operator with strong absorption with respect to the diffusion.