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

Failure of the Hopf-Oleinik lemma for a linear elliptic problem with singular convection of non-negative divergence

In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem $-\mathrm{div}( M(x) \nabla u ) = -\mathrm{div} (E(x) u) + f$ in a bounded domain of $\mathbb R^N$ with $N \ge 3$. We are particularly interested in singular $E$ with $\mathrm{div} E \ge 0$. We start by recalling known existence results when $|E| \in L^N$ that do not rely on the sign of $\mathrm{div} E$. Then, under the assumption that $\mathrm{div} E \ge 0$ distributionally, we extend the existence theory to $|E| \in L^2$. For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of $E$ singular at one point as $Ax /|x|^2$, or towards the boundary as $\mathrm{div} E \sim \mathrm{dist}(x, \partial \Omega)^{-2-\alpha}$. In these cases the singularity of $E$ leads to $u$ vanishing to a certain order. In particular, this shows that the Hopf-Oleinik lemma, i.e. $\partial u / \partial n < 0$, fails in the presence of such singular drift terms $E$

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

Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence

In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem −div(M(x)∇u)=−div(E(x)u)+f in a bounded domain of RN with N≥3. We are particularly interested in singular E with divE≥0. We start by recalling known existence results when |E|∈LN that do not rely on the sign of divE. Then, under the assumption that divE≥0 distributionally, we extend the existence theory to |E|∈L2. For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of E singular at one point as Ax/|x|2, or towards the boundary as divE∼dist(x,∂Ω)−2−α. In these cases the singularity of E leads to u vanishing to a certain order. In particular, this shows that the strong maximum principle fails in the presence of such singular drift terms E

Steiner symmetrization for anisotropic quasilinear equations via partial discretization

In this paper we obtain comparison results for the quasilinear equation $-\Delta_{p,x} u - u_{yy} = f$ with homogeneous Dirichlet boundary conditions by Steiner rearrangement in variable $x$, thus solving a long open problem. In fact, we study a broader class of anisotropic problems. Our approach is based on a finite-differences discretization in $y$, and the proof of a comparison principle for the discrete version of the auxiliary problem $A U - U_{yy} \le \int_0^s f^*$, where $AU = (n\omega^{1/n}s^{1/n'} )^p (- U_{ss})^{p-1}$. We show that this operator is T-accretive in $L^\infty$. We extend our results for $-\Delta_{p,x}$ to general operators of the form $-\mathrm{div} (a(|\nabla_x u|) \nabla_x u)$ where $a$ is non-decreasing and behaves like $| \cdot |^{p-2}$ at infinity

Caracterización del índice de aridez en México usando dos alternativas de estimación de la evapotranspiración potencial y escenarios de cambio climático

Ponencia presentada en: VIII Congreso de la Asociación Española de Climatología celebrado en Salamanca entre el 25 y el 28 de septiembre de 2012.[ES]Se delimitaron las Áreas de Influencia Climática (AIC) para todo el territorio de México mediante la sobreposición de las capas de precipitación (P) y temperatura (T) media anual delimitadas a escala 1:250 000 con una herramienta de un Sistema de Información Geográfica. Para cada AIC se generaron las bases de datos mensuales de variables climáticas y se estimó la Evapotranspiración Potencial (ETP) por dos metodologías: a) Thorthwaite modificado por Dunne y Leopold y b) Penman modificado por Monteith. Se calculó el Índice de Aridez (IA) a partir de la relación P/ETP para el escenario base (promedio de 1970 a 2000) usando los valores de ETP calculados por los dos métodos indicados. Asimismo, en cada AIC a las variables climáticas mensuales se aplicaron las razones de cambio de los modelos generales de circulación atmosférica GFDLCM 2.0 y el HADGEM 1, en el escenario A2 para el horizonte de tiempo al 2050 y se estimaron los respectivos IA con los valores de P y ETP obtenidos. Se elaboraron los mapas y las bases de datos del IA para el escenario base y cada uno de los escenarios de cambio climático mencionados, agrupando este parámetro en cuatro categorías (Áridos, Semiáridos, Subhúmedos secos y Subhúmedos húmedos y húmedos).[EN]The Areas of Climatic Influence (ACI) were delimitated for the entire Mexican territory by overlapping the shape files of mean annual precipitation (P) and mean annual temperature (T) that were previously delimited at 1: 250 000 scale using a Geographic Information System. For each ACI a monthly climatic variables data base were generated and then estimated the Potential Evapotranspiration (PET) using two methodologies a) the Thornthwaite modified by Dunne y Leopold and b) the Penman modified by Monteith. The Aridity Index (AI) was calculated using the ratio of P/PET for the base scenario (climatic variables averages from 1970 to 2000), for each ACI two AI were estimated one for the Thorntwaite PET and the other for Penman PET. Also, for each ACI were obtained the change ratios of the climatic variables by the Atmospheric General Circulation Models GFDLCM 2.0 and HADGEM 1, for the scenario A2 to the year 2050. Those change ratios were applied to the climatic variables in each ACI and calculated the PET for the two methodologies described and with the new values of P and PET the AI were estimated. The AI maps for the base scenario and the ones calculated for each climate change scenario were constructed, grouping the AI into four categories (Arid, Semiarid, Sub humid dry and Sub humid moist and humids)

Retos de adaptación al cambio climático para la agricultura de México

Ponencia presentada en: XII Congreso de la Asociación Española de Climatología celebrado en Santiago de Compostela entre el 19 y el 21 de octubre de 2022.[ES]El estudio del impacto del cambio climático sobre la actividad agrícola ha sido explorado desde hace varios años a nivel nacional. Desde el Departamento de Suelos de la Universidad Autónoma Chapingo un equipo de trabajo multidisciplinario ha trabajado desde hace varios años en proyectos de adaptación al cambio climático. Por lo que el objetivo del presente es mostrar los principales impactos del cambio climático a partir de los estudios antes indicados y otros más en la actividad agrícola nacional. Se muestran indicadores que sugieren el seguimiento a las acciones de adaptación en el sector. Cabe recordar que la actividad agrícola es parte importante de la economía en México por lo que es menester evidenciar su vulnerabilidad a los cambios en el clima. Se muestran resultados sobre los impactos del cambio climático sobre las aptitudes actuales y futuras, con escenarios de cambio climático, de poco mas de quince cultivos de importancia; también sobre el cambio en el periodo de crecimiento que muestra los posibles impactos en esta variable. Sobre los cultivos estudiados se muestra el cambio en los rendimientos futuros para evaluar el cambio potencial y establecer un impacto por esta variable. Así mismo, los resultados incluyen el cambio en temperatura y humedad de los suelos de México.[EN]The study of the impact of climate change on agricultural activity has been explored for several years at the national level. From the Department of Soils of the Universidad Autónoma Chapingo, a multidisciplinary work team has been working for several years on climate change adaptation projects. Therefore, the objective of the present is to show the main impacts of climate change based on the aforementioned studies and others in the national agricultural activity. Indicators are shown that suggest the monitoring of adaptation actions in the sector. It should be remembered that agricultural activity is an important part of the economy in Mexico, so it is necessary to demonstrate its vulnerability to changes in the climate. Results are shown on the impacts of climate change on current and future skills, with climate change scenarios, of just over fifteen important crops; also on the change in the growth period that shows the possible impacts on this variable. On the crops studied, the change in future yields is shown to evaluate the potential change and establish an impact for this variable. Likewise, the results include temperature and humidity of the soils of Mexico

Propuesta metodológica de inclusión de evaluación de impactos del cambio climático en la ordenación del territorio

Ponencia presentada en: VI Congreso Internacional de la Asociación Española de Climatología celebrado en Tarragona del 8 al 11 de octubre de 2008.[ES]Se presenta una propuesta de Ordenamiento Territorial que considera escenarios de cambio climático. Se elaboró el Modelo de Ordenamiento bajo el enfoque tradicional en México además de incluir diversos escenarios de cambio climático en la fase de pronóstico. Los modelos de cambio climático utilizados fueron el norteamericano (GFDL), inglés (HADLEY) y alemán (ECHAM), para los escenarios de emisiones A2 y B2 y los horizontes de tiempo 2020 y 2050. Los principales resultados muestran que el cambio y la variabilidad climáticos son elementos prioritarios a considerarse en las metodologías de ordenamiento, ya que a partir de este enfoque es viable evaluar las posibles condiciones futuras de las variables climáticas y su consecuente impacto en las actividades que se pueden proponer en el presente.[EN]A methodology of “Ordenamiento Territorial” it is presented considering climate change scenarios. The “Ordenamiento” model was elaborated under the traditional approach and also including diverse climate change scenarios in the forecast phase. The climate change models applied were the North American (GFDL), English (HADLEY) and German (ECHAM), for the emission scenarios A2 and B2 for the horizon time of 2020 and 2050. The main results show that climate variability and change are high priority elements to be considered in the methodology, since this approach is a viable way to evaluate the possible future conditions of the climate variables and their consequent impact in the activities that can be proposed in the present