arXiv subjects

Jesús Ildefonso Díaz

Publications and source records attributed to Jesús Ildefonso Díaz.

27 records · Page 2Linked to original sources

arXiv2018

On the exact number of monotone solutions of a simplifed Budyko climate model and their different stability

We consider a simplified version of the Budyko diffusive energy balance climate model. We obtain the exact number of monotone stationary solutions of the associated discontinuous nonlinear elliptic with absorption. We show that the bifurcation curve, in terms of the solar constant parameter, is S-shaped. We prove the instability of the decreasing part and the stability of the increasing part of the bifurcation curve. In terms of the Budyko climate problem the above results lead to an important qualitative information which is far to be evident and which seems to be new in the mathematical literature on climate models. We prove that if the solar constant is represented by $λ\in (λ_{1},λ_{2}),$ for suitable $λ_{1}<λ_{2},$ then there are exactly two stationary solutions giving rise to a free boundary (i.e. generating two symmetric polar ice caps: North and South ones) and a third solution corresponding to a totally ice covered Earth. Moreover, we prove that the solution with smaller polar ice caps is stable and the one with bigger ice caps is unstable.

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arXiv2018

Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis "nano-composite" membranes

One of the main goals of this paper is to extend some of the mathematical techniques of some previous papers by the authors showing that some very useful phenomenological properties which can be observed to the nano-scale can be simulated and justified mathematically by means of some homogenization processes when a certain critical scale is used in the corresponding framework. Here the motivating problem in consideration is formulated in the context of the reverse osmosis. We consider, on a part of the boundary of a domain $Ω\subset \mathbb R^n$, a set of very small periodically distributed semipermeable membranes having an ideal infinite permeability coefficient (which leads to Signorini type boundary conditions) on a part $Γ_1$ of the boundary. We also assume that a possible chemical reaction may take place on the membranes. We obtain the rigorous convergence of the problems to a homogenized problem in which there is a change in the constitutive nonlinearities. Changes of this type are the reason for the big success of the nanocomposite materials. Our proof is carried out for membranes not necessarily of radially symmetric shape. The definition of the associated critical scale depends on the dimension of the space (and it is quite peculiar for the special case of $n=2$). Roughly speaking, our result proves that the consideration of the critical case of the scale leads to an homogenized formulation which is equivalent to have a global semipermeable membrane, at the whole part of the boundary $Γ_1$, with a "finite permeability coefficient of this virtual membrane" which is the best we can get, even if the original problem involves a set of membranes of any arbitrary finite permeability coefficients.

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arXiv2018

The fractional Schrödinger equation with general nonnegative potentials. The weighted space approach

We study the Dirichlet problem for the stationary Schrödinger fractional Laplacian equation $(-Δ)^s u + V u = f$ posed in bounded domain $ Ω\subset \mathbb R^n$ with zero outside conditions. We consider general nonnegative potentials $V\in L^1_{loc}(Ω)$ and prove well-posedness of very weak solutions when the data are chosen in an optimal class of weighted integrable functions $f$. Important properties of the solutions, such as its boundary behaviour, are derived. The case of super singular potentials that blow up near the boundary is given special consideration. Related literature is commented.

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arXiv2018

On the well-posedness of a multiscale mathematical model for Lithium-ion batteries

We consider the mathematical treatment of a system of nonlinear partial differential equations based on a model, proposed in 1972 by J. Newman, in which the coupling between the Lithium concentration, the phase potentials and temperature in the electrodes and the electrolyte of a Lithium battery cell is considered. After introducing some functional spaces well-adapted to our framework we obtain some rigorous results showing the well-posedness of the system, first for some short time and then, by considering some hypothesis on the nonlinearities, globally in time. As far as we know, this is the first result in the literature proving existence in time of the full Newman model, which follows previous results by the third author in 2016 regarding a simplified case.

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arXiv2018

Existence and uniqueness of solutions of Schrödinger type stationary equations with very singular potentials without prescribing boundary conditions and some applications

Motivated mainly by the localization over an open bounded set $Ω$ of $\mathbb R^n$ of solutions of the Schrödinger equations, we consider the Schrödinger equation over $Ω$ with a very singular potential $V(x) \ge C d (x, \partial Ω)^{-r}$ with $r\ge 2$ and a convective flow $\vec U$. We prove the existence and uniqueness of a very weak solution of the equation, when the right hand side datum $f(x)$ is in $L^1 (Ω, d(\cdot, \partial Ω))$, even if no boundary condition is a priori prescribed. We prove that, in fact, the solution necessarily satisfies (in a suitable way) the Dirichlet condition $u = 0$ on $\partial Ω$. These results improve some of the results of the previous paper by the authors in collaboration with Roger Temam. In addition, we prove some new results dealing with the $m$-accretivity in $L^1 (Ω, d(\cdot, \partial Ω)^ α)$, where $α\in [0,1]$, of the associated operator, the corresponding parabolic problem and the study of the complex evolution Schrödinger equation in $\mathbb R^n$.

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arXiv2017

Linear diffusion with singular absorption potential and/or unbounded convective flow: the weighted space approach

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\mathbb R^N$. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain.

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arXiv2016

Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for $N\geq 3$

We prove that flat ground state solutions ($i.e.$ minimizing the energy and with gradient vanishing on the boundary of the domain) of the Dirichlet problem associated to some semilinear autonomous elliptic equations with a strong absorption term given by a non-Lipschitz function are unstable for dimensions $N=1,2$ and they can be stable for $N\geq 3$ for suitable values of the involved exponents.

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arXiv2015

Self-similar solutions with compactly supported profile of some nonlinear Schr{ö}dinger equations

This paper deals with the study of "\textit{sharp localized}" solutions of a nonlinear type Schr{ö}dinger equation in the whole space $\R^N,$ $N\ge1,$ with a zero order term, in modulus, like a power $m$ less than one of the modulus of the solution, and with a non zero external forcing term $\f.$ Our fundamental assumption is that such an exponent $m$ verifies $m\in (0,1).$ The self-similar structure of the solution is justified from the assumption that the external forcing term satisfies that $\f(t,x)=t^{-(\vp-2)/2}\F(t^{-1/2}x)$ for some complex exponent $\vp$ and for some profile function $\F$ which is assumed to be with compact support in $\R^N.$ We show the existence of solutions $\vu(t,x)=t^{\vp/2}\U(t^{-1/2}x),$ with a profile $\U,$ which also have compact support in $\R^N,$ reason why we call as "\textit{sharp localized}" solutions to this type of solutions. The proof of the localization of the support of the profile $\U$ uses some suitable energy method applied to the stationary problem satisfied by $\U$ after some unknown transformation.

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arXiv2012

Localizing Estimates of the Support of Solutions of some Nonlinear Schrödinger Equations - The Stationary Case

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrödinger equations mainly the compactness of the support and its spatial localization. This question is very related with pure essence of the derivation of the Schrödinger equation since it is well-known that if the linear Schrödinger equation is perturbated with "regular" potentials then the corresponding solution never vanishes on a positive measured subset of the domain, which corresponds with the impossibility of localize the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution becomes a compact set and so any estimate on its spatial localization implies a very rich information on places which can not be. Our results are obtained by the application of some energy methods which connect the compactness of the support with the local vanishing of a suitable "energy function" which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006.

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