Francisco Ortegón Gallego

Renormalized solutions to a nonlinear parabolic-elliptic system

The aim of this paper is to show the existence of renormalized solutions to a parabolic-elliptic system with unbounded diffusion coefficients. This system may be regarded as a modified version of the well-known thermistor problem; in this case, the unknowns are the temperature in a conductor and the electrical potential.

An Elliptic Equation with Blowing-Up Diffusion and Data in L1: Existence and Uniqueness

We establish some existence and uniqueness results for a nonlinear elliptic equation. The problem has a diffusion matrix A(x, u) such that A(x, s)ξξ ≥ β(s)|ξ|2, with β : (s0, + ∞) ↦ ℝ a continuous, strictly positive function which goes to infinity when s is near s0. On the other hand,

Numerical Simulation of Potential Maxwell’s Equations in the Harmonic Regime

\frac{A(x,s)}{\beta(s)}\in L^\infty(\Omega\times(s_0,+\infty))^{N\times N}

Weak Solutions to a Nonuniformly Elliptic PDE System in the Harmonic Regime

. Also, the right-hand side f belongs to L1(Ω). We make use of the concept of renormalized solutions adapted to our problem.

REGULARIZATION BY MONOTONE PERTURBATIONS OF THE HYDROSTATIC APPROXIMATION OF NAVIER–STOKES EQUATIONS

The aim of this work is to perform some numerical experiments for the resolution of a strongly coupled parabolic–elliptic system that describes the heating induction–conduction industrial process of a steel workpiece, whose unknowns are the electric potential, the magnetic vector potential, and the temperature. In order to make the numerical simulations lighter, and taking into account the different time scales between the potentials and the temperature, a new system of nonlinear partial differential equations (PDEs) has been constructed that describes the heating process in the harmonic regime.

Sur un problème elliptique non linéaire avec diffusion singulière et second membre dans L1

We study the existence of weak solutions to a nonlinear strongly coupled parabolic–elliptic PDEs arising in the heating induction-conduction process of steel hardening. In this setting, our major concern is to consider the case when the electric conductivity is nonuniformly elliptic which, together with a right hand side in L 1 in the energy balance equation, yields to a difficult theoretical situation. The existence result gives a weak solution to a similar PDEs system where the energy balance equation has been perturbed by a measure term.

The evolution thermistor problem with degenerate thermal conductivity

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.