M. Ughi

Positivity properties of viscosity solutions of a degenerate parabolic equation

We consider the problem: (I) {u t =uΔu−γ|⊇u| 2 in R N ×R + , u(x, 0)=u 0 (x) in R N , where γ is a nonnegative constant and u 0 is a bounded continuous and nonnegative function on R N

Discontinuous “viscosity” solutions of a degenerate parabolic equation

We study a nonlinear degenerate parabolic equation of the second order. Regularizing the equation by adding some artificial viscosity, we construct a generalized solution. We show that this solution is not necessarily continuous at all points.

A mathematical model of turbulent heat and mass transfer in stably stratified shear flow

It is commonly assumed that heat flux and temperature diffusivity coefficients obtained in steady-state measurements can be used in the derivation of the heat conduction equation for fluid flows. Meanwhile it is also known that the steady-state heat flux as a function of temperature gradient in stably stratified turbulent shear flow is not monotone: at small values of temperature gradient the flux is increasing, whereas it is decreasing after a certain critical value of the temperature gradient. Therefore the problem of heat conduction for large values of temperature gradient becomes mathematically ill-posed, so that its solution (if it exists) is unstable

A degenerate parabolic equation modelling the spread of an epidemic

SummaryWe consider the Cauchy problem for a degenerate parabolic equation, not in divergence form, representing the diffusive approximation of a model for the spread of an epidemic in a closed population without remotion. We prove existence and uniqueness of the weak solution, defined in a suitable way, and some qualitative properties.

Nonuniqueness of solutions of a degenerate parabolic equation

SummaryWe give some results about nonuniqueness of the solutions of the Cauchy problem for a class of nonlinear degenerate parabolic equations arising in several applications in biology and physics. This phenomenon is a truly nonlinear one and occurs because of the degeneracy of the equation at the points where u=0. For a given set of values of the parameter involved, we prove that there exists a one parameter family of weak solutions; moreover, restricting the parameter set, nonuniqueness appears even in the class of classical solutions.

Shock Propagation in a One-Dimensional Flow Through Deformable Porous Media

We consider a one-dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered to be functions of the flux intensity. The medium is initially dry and we neglect capillarity, so that a sharp wetting front proceeds into the medium. We consider the open problem of the continuation of the solution possibly in the case of onset of singularities, which can be interpreted as a local collapse of the medium. In particular we analyze the case in which the boundary pressure has a piecewise constant derivative.

ON A MODEL FOR THE PROPAGATION OF ISOTOPIC DISEQUILIBRIUM BY DIFFUSION

We consider a model for the distribution of radionuclides in the ground water around a deep repository for used nuclear fuel, based on the assumption that different isotopes of the same chemical element A contribute jointly to the chemical potential of A. In this hypothesis, the total flux Ji of a particular isotope Ai of an element A has two components, one due to the interaction of Ai with the solvent molecules B, the other with the kin isotopes. We study some qualitative properties of the solution in the physically relevant assumption that the first of these components is negligible. In this assumption the problem reduces to a parabolic equation for the total concentration of the element A, possibly coupled with hyperbolic equations for the concentrations of the single isotopes.

On the regularity at shrinking points for the porous media equation

Abstract. We consider the porous media equation in a noncylindrical domain which shrinks at P and our aim is to give conditions on the domain ensuring the continuity or discontinuity at P of any bounded solution. We make use of a recent general result which allows comparisons with the heat equation.

SHOCK PROPAGATION IN A FLOW THROUGH DEFORMABLE POROUS MEDIA: ASYMPTOTIC BEHAVIOUR AS THE POROSITY APPROXIMATES A CONSTANT

We consider a one-dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered as functions of the flux intensity. We prove that if one approximates the porosity with a constant then the solution of the hyperbolic problem converges to the classical continuous Green–Ampt solution, also in the presence of shocks. In general, however, the shocks remain present in any approximating solution.

Nonuniqueness and Irregularity Results for a Nonlinear Degenerate Parabolic Equation

Consider the problem