Evolution Equations with a Complex Spatial Variable

It is known that, if the time variable in the heat equation is complex and belongs to a sector in ℂ, then the theory of analytic semigroups becomes a powerful tool of study. The same is true for the Laplace equation on an infinite strip in the plane, regarded as an initial value boundary value problem. Also, it is known that if both variables, time and spatial, are complex, then, e.g. the Cauchy problem for the heat equation admits as solution, only a formal power series which, in general, converges nowhere. In a recent paper (C.G. Gal, S.G. Gal, and J.A. Goldstein, Evolution equations with real time variable and complex spatial variables, Complex Var. Elliptic Eqns. 53 (2008), pp. 753–774), a complementary approach was made: the study of the complex versions of the classical heat and Laplace equations, obtained by ‘complexifying’ the spatial variable only (and keeping the time variable real). The goal of this article is to extend that study to the higher-order heat and Laplace-type equations. This ‘complexification’ is based on integral representations of the solutions in the case of a real spatial variable, by complexifying the spatial variable in the corresponding semigroups of operators. It is of interest to note that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness.

Long-term trends and wave climate variability in the South Atlantic Ocean: The influence of climate indices

This paper describes the use of a unique spatio-temporally resolved precipitation and temperature dataset to assess the spatio-temporal dynamics of water resources over a period of almost seven decades across the Sierra Nevada mountain range, which is the most southern Alpine environment in Europe. The altitude and geographical location of this isolated alpine environment makes it a good detector of climate change. The data were generated by applying geostatistical co-kriging to significant instrumental precipitation and temperature (minimum, maximum and mean) datasets. The correlation between precipitation and altitude was not particularly high and the statistical analysis yielded some surprising results in the form of mean annual precipitation maps and yearly precipitation time series. These results confirm the importance of orographic precipitation in the Sierra Nevada mountain range and show a decrease in mean annual precipitation of 33 mm per decade. Seasonality, however, has remained constant throughout the period of the study. The results show that previous studies have overestimated the altitudinal precipitation gradient in the Sierra Nevada and reveal its complex spatial variability. In addition, the results show a clear correspondence between the mean annual precipitation and the NAO index and, to a much lesser extent, the WeMO index. With respect to temperature, there is a high correlation between minimum temperature and altitude (coefficient of correlation = −0.84) and between maximum temperature and altitude (coefficient of correlation = −0.9). Thus, our spatial temperature maps were very similar to topographic maps, but the temporal trend was complex, with negative (decreasing) and positive (increasing) trends. A dynamic model of snowfall can be obtained by using the degree-day methodology. These results should be considered when checking the local performance of climatological models.

ABSTRACTUrban trees provide essential ecosystem services, notably air cooling through transpiration, which helps mitigate the urban heat island effect and enhances cities' climate resilience. However, the complex spatial variability within urban areas and extreme weather events like droughts can disrupt trees' ecohydrological dynamics. In a study conducted in Freiburg, Germany, we investigated transpiration processes in Norway maple (Acer platanoides) and small‐leaved lime (Tilia cordata) across diverse urban locations, including parks, parking lots, grass verges and tree pits. We assessed the effects of four distinct drought periods on transpiration and compared differences between tree species and growing sites. Small‐leaved lime exhibited a 5% greater reduction in transpiration during drought periods compared to Norway maple, which experienced a 34% decline in transpiration during peak sap flow compared to nondrought periods. Tree pits with 90% surface sealing induced the most significant drought‐induced transpiration reduction for small‐leaved lime (58%), with both species displaying the lowest transpiration to potential evapotranspiration ratio in these locations. Significant differences were observed in the diurnal sap velocity patterns for both species. We highlighted the site‐specific impact of surface sealing on transpiration during droughts, as well as the significant relationship between soil water deficit and relative transpiration rates. This study provides crucial insights into common urban tree species' responses to drought‐induced transpiration across varied urban settings, emphasising the role of surface sealing. Continuous monitoring of diverse urban tree species is essential for building extensive databases and enhancing our understanding of tree water relations in diverse urban landscapes.

Lithology identification from well logs remains challenging due to overlapping signals and complex spatial variability in heterogeneous formations. To address this issue, a novel method, termed LIsr (Lithology Identification with a sliding window and recurrent neural network (RNN)), is proposed. The sliding window enhances distinguishing features between lithologies and reduces prediction uncertainty by incorporating label sequences predicted by an improved RNN. This network captures spatial correlations in depth between neighbor samples, a factor often overlooked in conventional machine learning (ML) approaches. A bidirectional scanning mechanism is introduced to mitigate the impact of sedimentary sequences, while residual connections and a deep architecture enhance feature extraction. The method is validated using data from the Hangjinqi Gas Field in Ordos Basin, China. Results show that incorporating unidirectional scanning improves accuracy by 5.5%, and the residual deep structure contributes an additional 22% gain. The proposed LIsr achieves 90.8% validation accuracy and 89.9% accuracy on blind wells, demonstrating its effectiveness in complex lithology prediction.

Earth system models have deficiencies and biases that raise large uncertainties in the simulated ocean temperature and salinity. Methods of assessing vertical thermohaline structures traditionally rely on computing the mean profile in somewhat ad hoc regions. However, the global ocean has complex spatial variability, complicating regional classification. Here, to assess the global thermohaline profiles simulated by CMIP6 models, we apply an unsupervised clustering method to classify the Argo-observed profiles and compare them with the historical simulations of eight CMIP6 models. The results show that in each season, the Argo-observed thermohaline profiles in the global ocean can be divided into around 60 classes, which have distinct geographical distributions. CESM2 (MRI-ESM2-0) performs best in simulating the main climatological spatial patterns and vertical structures of temperature (salinity) in 0–1000 m, with an overall bias of 1.19°C (0.31). Temperature biases in 8-CMIP6 multimodel mean highlight several hotspots: 1) the tropical Atlantic Ocean and the Gulf of Mexico; 2) the Kuroshio Extension; 3) south of the equator in the western Pacific Ocean; and 4) the Agulhas retroflection region. Vertical structures of temperature and salinity biases vary in these four regions. Composite analysis of vertical bias patterns and dynamic characteristics suggests that region 1 may reflect remote advective influences, regions 2 and 4 may be associated with the inaccuracy of mode water simulations, and region 3 appears attributable to wind stress and mixed layer depth biases. The proposed evaluation framework will provide important insights for optimizing global ocean simulations in future modeling efforts.

In this paper the question of asymptotic stability for retarded functional reaction diffusion equations is faced. Due to the infinite dimension of the problem a numerical approach is necessary. Here we propose a technique based on a pseudospectral discretization in time and on a spectral discretization in space of the infinitesimal generator associated to the semigroup of solution operators. Some numerical experiments on a Hutchinson-type equation modeling heat conduction in a rod with spatially variable gain delayed feedback are performed to show the efficiency of the scheme. This work represents a nontrivial extension of previous work of the authors on the computation of asymptotic stability for delay differential equations.

We systematically analyze differential and analytical properties of various kinds of semigroups of linear operators, including (local) convoluted C-semigroups and ultradistribution semigroups. The study of differentiable integrated semigroups leans heavily on the unification of the approaches of Barbu (Ann Scuola Norm Sup Pisa 23:413–429, 1969) and Pazy (Semigroups of linear operators and applications to partial differential equations. Springer, Berlin, 1983). We furnish illustrative examples of operators which generate differentiable integrated semigroups, further analyze the analytic properties of solutions of the backwards heat equation, and prove that several introduced classes of differentiable semigroups persist under bounded ‘commuting’ perturbations.

The Korteweg-de Vries (KdV) equation has been generalized by Rosenau and Hyman [3] to a class of partial differential equations (PDEs) which has solitary wave solution with compact support. These solitary wave solutions are called compactons. Compactons are solitary waves with the remarkable soliton property, that after colliding with other compactons, they reemerge with the same coherent shape. These particle like waves exhibit elastic collision that are similar to the soliton interaction associated with completely integrable systems. The point where two compactons collide are marked by a creation of low amplitude compacton-anticompacton pair. These equations have only a finite number of local consevation laws. In this paper, an implicit numerical method has been developed to solve the K(2,3) equation. Accuracy and stability of the method have been studied. The analytical solution and the conserved quantities are used to assess the accuracy of the suggested method. The numerical results have shown that this compacton exhibits true soliton behavior.

Alpine lakes on the Tibetan Plateau have significantly changed under a changing climate over past decades. However, the changing patterns of the inflow sources of the lakes, i.e., rainfall and the melt water of snow and glaciers, and their response to climate change remain uncertain because obtaining accurate precipitation and melt water discharge is difficult due to the complex topography, spatial variability, and scarce stations of the alpine area. A distributed hydrological model, J2000, was employed in this study to simulate runoff component variations of the Yamzho Yumco Lake glaciated basin during 1974–2019. Except for observed daily runoff from two tributaries, a High Asia Refined (HAR) high-resolution reanalysis of precipitation data was combined with field precipitation gradient observation and snow cover area validation, all performed simultaneously to reduce the uncertainty of inflow components in the model. Results showed that the average runoff into the lake during 1974–2019 was 5.5 ± 1.4 × 108 m3/10a, whereas rainfall runoff, glacier melt runoff, snowmelt runoff, and baseflow contributed to 54.6%, 10.8%, 1.8%, and 32.7% of total runoff in mean, respectively. Seasonal runoff in spring, summer, autumn, and winter accounted for 6.7%, 60.6%, 23.9% and 8.8% of annual total runoff, respectively. In glacial areas, the reduction in total runoff after removing the precipitation trend was 1.4 times than that of temperature, and in non-glacial areas, the reduction in total runoff after removing the precipitation trend was 1.6 times than the increase in total runoff after removing the temperature trend. The proportion of rainfall runoff increased at a rate of 1.0%/10a, whereas the proportion of melt runoff decreased at a rate of 0.07%/10a during the study period.

initial value problems (Cauchy problems) are usually studied via operator semigroups. In many cases, the well-developed theory of C0-semigroups, i.e., one-parameter operator semigroups which are strongly continuous for the norm on a Banach space X , suffices and provides a powerful machinery to study such problems. The applications range from partial differential equations, Volterra integro-differential equations and dynamic boundary problems to delay equations. It seems that a linear (autonomous) evolution equation like { u′(t) = Au(t) u(0) = x, (EE) that is an equation that describes a system evolving from an initial state ”in time”, can be handled via the theory of C0-semigroups. However, as the most trivial example shows this is not the case. Consider the left shift semigroup S on the space of bounded, continuous functions Cb(R) S(t)f(s) := f(t+ s). It is clear that the orbit t 7→ S(t)f is continuous for the supremum norm, if and only if f is uniformly continuous. This shows that the shift semigroup does not fit into the framework of C0-semigroups, nevertheless S clearly describes an evolution system on R. The situation is not so bad either. If we replace the norm topology by the topology τc of uniform convergence on compact sets, then the orbits become τc-continuous. This leads us to weakening the notion of C0-semigroups, thus allowing to consider such ”pathological” cases, these therefore turn out to be less unpleasant than thought previously. There are numerous generalisations of the theory of C0-semigroups. Among these we find the approach of introducing new continuity notions, this method is we want to follow. Investigations on semigroups on locally convex spaces were started fairly long ago, see e.g., [14], [51], [52] and [64]. A nice exposition on the historical aspects can be found in [56]. To exploit the Banach space structure and the same time to introduce coarser topologies, the notion of bi-continuous semigroups was introduced recently by Kuhnemund in [56]. The theory developed therein covers a large part of previously known results, and puts these concrete examples in an abstract framework. The general theory is then applicable in concrete cases.

A stochastic numerical model for wave-induced dynamic response in a spatially heterogeneous seabed around structures

We suggest a method for constructing the Green's functions for boundary-value problems in “half-space” type domains for some classes of partial differential equations whose third derivative with respect to the first spatial variable is distinguished, give formulas for exact solutions, and prove uniqueness theorems in the Schwartz spaces.

During three low-altitude (99, 66, 66 km) flybys through the Enceladus plume in 2010 and 2011, Cassini's ion neutral mass spectrometer (INMS) made its first high spatial resolution measurements of the plume's gas density and distribution, detecting in situ the individual gas jets within the broad plume. Since those flybys, more detailed Imaging Science Subsystem (ISS) imaging observations of the plume's icy component have been reported, which constrain the locations and orientations of the numerous gas/grain jets. In the present study, we used these ISS imaging results, together with ultraviolet imaging spectrograph stellar and solar occultation measurements and modeling of the three-dimensional structure of the vapor cloud, to constrain the magnitudes, velocities, and time variability of the plume gas sources from the INMS data. Our results confirm a mixture of both low and high Mach gas emission from Enceladus' surface tiger stripes, with gas accelerated as fast as Mach 10 before escaping the surface. The vapor source fluxes and jet intensities/densities vary dramatically and stochastically, up to a factor 10, both spatially along the tiger stripes and over time between flyby observations. This complex spatial variability and dynamics may result from time-variable tidal stress fields interacting with subsurface fissure geometry and tortuosity beyond detectability, including changing gas pathways to the surface, and fluid flow and boiling in response evolving lithostatic stress conditions. The total plume gas source has 30% uncertainty depending on the contributions assumed for adiabatic and nonadiabatic gas expansion/acceleration to the high Mach emission. The overall vapor plume source rate exhibits stochastic time variability up to a factor ∼5 between observations, reflecting that found in the individual gas sources/jets. Key Words: Cassini at Saturn—Geysers—Enceladus—Gas dynamics—Icy satellites. Astrobiology 17, 926–940.

Semigroups of linear operators are closely related to the solution of the following linear differential equation: $$ \dot u(t) = Au(t).u(0) = x \in X, $$ (2.1)