On the regularity of maximal operators

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb{R}) \times W^{1,q}(\mathbb{R}) \to W^{1,r}(\mathbb{R})$ with $1 <p,q < \infty$ and $r\geq 1$, boundedly and continuously. The same result holds on $\mathbb{R}^n$ when $r>1$. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.Comment: 10 page

Existence and optimal regularity theory for weak solutions of free transmission problems of quasilinear type via Leray-Lions method

We study existence and regularity of weak solutions for the following PDE -\dive(A(x,u)|\nabla u|^{p-2}\nabla u) = f(x,u),\;\;\mbox{in $B_1$}. where $A(x,s) = A_+(x)\chi_{\{s>0\}}+A_-(x)\chi_{\{s\le 0\}}$ and $f(x,s) = f_+(x)\chi_{\{s>0\}}+f_-(x)\chi_{\{s\le 0\}}$. Under the ellipticity assumption that $\frac{1}{\mu}\le A_{\pm} \le \mu$, A_{\pm}\in C(\O) and f_{\pm}\in L^N(\O), we prove that under appropriate conditions the PDE above admits a weak solution in $W^{1,p}(B_1)$ which is also $C^{0,\alpha}_{loc}$ for every $\alpha\in (0,1)$ with precise estimates. Our methods relies on similar techniques as those developed by Caffarelli to treat viscosity solutions for fully non-linear PDEs (c.f. \cite{C89}). Other key ingredients in our proofs are the \TT_{a,b} operator (which was introduced in \cite{MS22}) and Leray-Lions method (c.f. \cite{BM92}, \cite{MT03})

The Río de La Plata estuary hydrology and circulation

Este trabajo compila y resume los conocimientos actuales sobre la hidrología y circulación del Estuario del Río de la Plata, revisando la literatura existente sobre el tema. En primer lugar, se describen los principales forzantes de la circulación en el estuario -su geometría y batimetría, descarga, olas, mareas y vientos- enfatizando, cuando corresponde, en sus escalas temporales de variabilidad de intra a inter-anual. Luego, se discuten la estructura de la densidad y los principales patrones de circulación, sintetizando los resultados obtenidos a partir de estudios basados en modelos y en observaciones. La descripción de la circulación se realiza en dos partes: (i) en primer lugar, se presenta la circulación barotrópica (media vertical) con gran impacto en el transporte neto de masa y, por lo tanto, en la variabilidad del nivel del agua en el estuario, (ii) en segundo lugar, se discute lo que se conoce acerca de las corrientes baroclínicas (que varían verticalmente) en la zona frontal y su influencia en la estructura de densidad. Finalmente, se describen los caminos de las masas de agua dulce de los principales afluentes a lo largo del estuario y se discuten los mecanismos que favorecen la retención en la zona frontal.This work compiles and summarizes the current knowledge of the Río de la Plata Estuary hydrology and circulation, reviewing the existing literature on the issue. We firstly describe the main forcings of the circulation in the estuary -its geometry and bathymetry, runoff, wind-waves, tides and winds- emphasizing, when it applies, in their temporal scales of variability from intra to inter-annual. Then, we discuss the density structure and the main circulation patterns, synthesizing results derived from modelling and observational studies. The description of the circulation is done in two parts: (i) firstly reporting the barotropic (vertically averaged) motion, which strongly affects the net mass transport and sea surface height variability in the estuary and, therefore, is linked to the surges; and (ii) secondly, discussing what it is known about the baroclinic (vertically varying) currents in the frontal zone and their influence on the density structure. Finally, the paths of the freshwater masses of the main tributaries along the estuary are described and the mechanisms that favour retention in the frontal zone are discussed.Fil: Moreira, Diego. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Centro de Investigaciones del Mar y la Atmósfera. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Centro de Investigaciones del Mar y la Atmósfera; Argentina. Instituto Franco-argentino sobre Estudios del Clima y sus Impactos; ArgentinaFil: Simionato, Claudia Gloria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Centro de Investigaciones del Mar y la Atmósfera. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Centro de Investigaciones del Mar y la Atmósfera; Argentina. Instituto Franco-argentino sobre Estudios del Clima y sus Impactos; Argentin

Flipping regularity via the Harnack approach and applications to nonlinear elliptic problems

We prove an abstract result ensuring that one-sided geometric control yields two-sided estimates for functions satisfying general conditions. Our findings resonate in the context of nonlinear elliptic problems, including supersolutions to fully nonlinear elliptic equations and functions in the De Giorgi class. Among the consequences of our abstract results are regularity estimates, and conditions for a continuous function to be in the class of viscosity solutions. We also prove that one-sided geometric control yields $L^pL^\infty$-estimates. It provides a converse to the implication in the De Giorgi-Nash-Moser theory

Nonlinear Interaction Between the Tide and the Storm Surge with the Current due to the River Flow in the Río de la Plata

The Río de la Plata estuary (RdP) is characterized by the large flow of its tributary rivers (Q), with an average of 22,000 m3 s− 1 and an interannual variability range from 8000 to 90,000 m3 s− 1. In this work, the hypothesis that the current due to that flow (CDR) interacts nonlinearly with both the tides and storm surges is evaluated utilizing water level observations and numerical simulations. Two tide gauge time series gathered at the freshwater tidal zone (FTZ) of the RdP are analyzed with the novel surrogate analysis. The analysis is applied for periods of high, medium and low Q. Results show that both interactions occur at the upper half of the FTZ and increase with Q. Harmonic analyses support the surrogate analysis’ conclusions and show that tide-CDR interaction redistributes the energy among tidal harmonics, increasing asymmetry. Numerical simulations confirm that (i) both interactions maximize at the upper half of the FTZ and decrease downstream; and (ii) they are modulated by Q; a rise of about 14,000 m3 s− 1 (interquartile range) can produce an intensification of 50% and 100% of the amplitudes of the tide-CDR and surge-CDR interactions, respectively; and (iii) both interactions introduce asymmetries in the water level, with faster rises and slower falls; (iv) the quadratic bottom friction is the main source of both interactions; (v) tide-CDR interaction represents 12% of the water level associated with the tide, whereas surge-CDR interaction accounts for 5% of the surge peak; and (vi) the interactions are significant in the upper FTZ because there, the magnitude of the currents associated with the tide and the surge are comparable to CDR; downstream, the channel widens and CDR decreases.Fil: Dinapoli, Matias. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Centro de Investigaciones del Mar y la Atmósfera. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Centro de Investigaciones del Mar y la Atmósfera; Argentina. Instituto Franco-argentino sobre Estudios del Clima y sus Impactos; ArgentinaFil: Simionato, Claudia Gloria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Centro de Investigaciones del Mar y la Atmósfera. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Centro de Investigaciones del Mar y la Atmósfera; Argentina. Instituto Franco-argentino sobre Estudios del Clima y sus Impactos; ArgentinaFil: Moreira, Diego. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Centro de Investigaciones del Mar y la Atmósfera. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Centro de Investigaciones del Mar y la Atmósfera; Argentina. Instituto Franco-argentino sobre Estudios del Clima y sus Impactos; Argentin

Development and validation of a storm surge forecasting/hindcasting modelling system for the extensive Río de la Plata Estuary and its adjacent Continental Shelf

The Río de la Plata (RdP) Estuary is affected by significant surges several times per year. This phenomenon has historically caused catastrophic water-level enlargements of up to 4.44 m, threatening and claiming human lives and producing major economic and material damages. The negative surges are less frequent, but when they do occur, inhibit the access to the principal harbors and waterways and disable the drinking water intakes of the Metropolitan Area of Buenos Aires (the Capital City of Argentina) with a population of more and 16 million people. Recent works suggest that the number and strength of the surge events have been increasing with time. Nevertheless, a state-of-the-art system for the forecast of those events is not available yet. In this work, the implementation of a numerical modelling system for the forecast/hindcast of storm surges and the associated currents in the RdP and the adjacent continental shelf are presented and validated. This pre-operational system is based on an adaptation of the CROCO community ocean model to solve the dynamics associated with the surge. The model was implemented using a set of routines written in open-source programming language (Linux and Python) to be cheap and efficient and to ensure an easy future transfer to the services responsible for the alerts. For a better representation of the regional atmospheric dynamics, wind speed and sea-level pressure used to force the simulations were corrected using direct observations collected at an oceanographic buoy anchored at the estuary. The model system performance in hindcast mode was quantified by comparison with observations from tidal gauges and current meters at several locations of the estuary and the adjacent shelf. Percent errors for water level over the whole estuary and currents in the intermediate and exterior estuary drove to average results of 8 and 13%, respectively. The skill scores resulted, on average, of 0.90 and 0.80, respectively. The model performance in both hindcast and forecast modes was evaluated during historical extreme storm surges. Results support the good performance of the model to simulate even extreme events with average skill scores of 0.97 and 0.92, respectively. Results are encouraging, particularly taking into account the limitations in the atmospheric forcing for the region, where only a relatively small number of direct observations are assimilated by the reanalysis and forecast models.Fil: Dinapoli, Matias. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Centro de Investigaciones del Mar y la Atmósfera. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Centro de Investigaciones del Mar y la Atmósfera; Argentina. Instituto Franco-argentino sobre Estudios del Clima y sus Impactos; ArgentinaFil: Simionato, Claudia Gloria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Centro de Investigaciones del Mar y la Atmósfera. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Centro de Investigaciones del Mar y la Atmósfera; Argentina. Instituto Franco-argentino sobre Estudios del Clima y sus Impactos; ArgentinaFil: Moreira, Diego. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Centro de Investigaciones del Mar y la Atmósfera. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Centro de Investigaciones del Mar y la Atmósfera; Argentina. Instituto Franco-argentino sobre Estudios del Clima y sus Impactos; Argentin