Climate and mathematics

Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

On the von Neumann problem and the approximate controllability of Stackelberg-Nash strategies for some environmental problems

Two problems arising in Environment are considered. The first one concerns a conjecture possed by von Neumann in 1955 on the possible modification of the albedo in order to control the Earth surface temperature. The second one is related to the approximate controllability of Stackelberg-Nash strategies for some optimization problems as, for instance, the pollution control in a lake. The results of thesecondpart were obtained in collaboration with Jacques-LouisLions

Logarithmically improved regularity criteria for the Navier-Stokes equations in homogeneous Besov spaces

Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

Beyond the classical strong maximum principle: forcing changing sign near the boundary and flat solutions

We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain $\Omega$ can be extended, under suitable conditions, to the case in which the forcing term $f(x)$ is changing sign. In addition, in the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solutions). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign is also given.Comment: 20 pages 2 Figure

Remarks on the Monge-Ampère equation: some free boundary problems in geometry

This paper deals with several qualitative properties of solutions of some stationary and parabolic equations associated to the Monge-Ampère operator. Mainly, we focus our attention in the occurrence of a free boundary (separating the region where the solution u is locally a hyperplane, and so were the Hessian D2u is vanishing from the rest of the domain). Among other thinfs, we take advantage of these proceedings to give a detailed version of some results already announced long time ago when dealing wiht other fully nonlinear equations (see the 1979 àèr by the authors on other parabolic equations, remark 2.25 of the 1985 monograph by the second author and the 1985 paper by the first author. In particular, our results apply to suitable formulations of the Gauss curvature flow and of the worn stones problems intensively studied in the literature

Global stability for convection when the viscosity has a maximum

Until now, an unconditional nonlinear energy stability analysis for thermal convection according to Navier–Stokes theory had not been developed for the case in which the viscosity depends on the temperature in a quadratic manner such that the viscosity has a maximum. We analyse here a model of non-Newtonian fluid behaviour that allows us to develop an unconditional analysis directly when the quadratic viscosity relation is allowed. By unconditional, we mean that the nonlinear stability so obtained holds for arbitrarily large perturbations of the initial data. The nonlinear stability boundaries derived herein are sharp when compared with the linear instability thresholds

Existence of weak solutions to some stationary Schrödinger equations with singular nonlinearity

We prove some existence (and sometimes also uniqueness) of weak solutions to some stationary equations associated to the complex Schrödinger operator under the presence of a singular nonlinear term. Among other new facts, with respect some previous results in the literature for such type of nonlinear potential terms, we include the case in which the spatial domain is possibly unbounded (something which is connected with some previous localization results by the authors),the presence of possible non-local terms at the equation, the case of boundary conditions different to the Dirichlet ones and, finally, the proof of the existence of solutions when the right-hand side term of the equation is beyond the usual L2-space

A sharper energy method for the localization of the support to some stationary Schrödinger equations with a singular nonlinearity

We prove the compactness of the support of the solution of some stationary Schrödinger equations with a singular nonlinear order term. We present here a sharper version of some energy methods previously used in the literature