Energy Space Research Articles

Global energy conservation solution for the [formula omitted] family of Camassa–Holm type equation

In this paper, we investigate the N−abc family of Camassa–Holm type equation with (N+1)-order nonlinearities. This quasi-linear equation is nonlocal with higher order nonlinearities, compared to the Camassa–Holm equation (N=1) and Novikov equation (N=2). Using both the lower order and the higher order energy conservation laws, as well as the characteristic method, we establish the global existence and uniqueness of the Hölder continuous energy weak solution to the N−abc family of Camassa–Holm type equation in the energy space H1(R)×W1,2N(R). Moreover, we show that a very natural and interesting problem is to study how the regularity of solution changes with respect to N. More precisely, we establish Hölder continuous energy weak solutions with the exponent 1−12N. This result precisely shows how the regularity of solution changes with respect to the power of nonlinear wave speed N, and it reveals an intrinsic relation between Camassa–Holm equation (linear wave speed u1), Novikov equation (quadratic wave speed u2) and the N−abc family of Camassa–Holm type equation (wave speed uN).

The scattering map on collapsing charged spherically symmetric spacetimes

In this paper we generalise our previous results (Alford 2020 Ann. Inst. Henri Poincare 21 2031–92) concerning scattering on the exterior of collapsing dust clouds to the charged case, including in particular the extremal case. We analyse the energy boundedness of solutions φ to the wave equation on the exterior of collapsing spherically symmetric charged matter clouds. We then proceed to define the scattering map on this spacetime, and look at the implications of our boundedness results on this map. More specifically, we first construct a class of spherically symmetric charged collapsing matter cloud exteriors, and then consider solutions to the wave equation with Dirichlet (reflective) boundary conditions on the surface of these clouds. We then show that the energy of φ remains uniformly bounded going forwards or backwards in time, and that the scattering map is bounded going forwards but not backwards. Therefore, the scattering map is not surjective onto the space of finite energy on I+∪H+ . Thus there does not exist a backwards scattering map from finite energy radiation fields on I+∪H+ to finite energy radiation fields on I− for these models. These results will be used to give a treatment of Hawking radiation in a companion paper (Alford 2023 arXiv:2309.03022).

Existence and uniqueness of the global conservative weak solutions to the cubic Camassa–Holm-Type equation

This paper is devoted to the cubic Camassa–Holm-Type equation with a nonlocal cubic nonlinearity suggested as an asymptotic model for the 2D full water wave dynamics which is nonlocal with higher-order nonlinearities compared to the classical Camassa–Holm equation. We establish the global existence and uniqueness of the energy conservative weak solution in the energy space.

Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation

Abstract Consideration herein is the stability issue of peaked solitary wave solution for the modified Camassa-Holm-Novikov equation, which is derived from the shallow water theory. This wave configuration accommodates the ordered trains of the modified Camassa-Holm-Novikov-peaked solitary solution. With the application of conservation laws and the monotonicity property of the localized energy functionals, we prove the orbital stability of this wave profile in the H 1 ( R ) {H}^{1}\left({\mathbb{R}}) energy space according to the modulation argument.

The Loewner–Kufarev energy and foliations by Weil–Petersson quasicircles

AbstractWe study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure has finite Loewner–Kufarev energy, defined by whenever is of the form , and set to otherwise. Moreover, if either of these two energies is finite, they are equal up to a constant factor, and in this case, the foliation leaves are Weil–Petersson quasicircles. This duality between energies has several consequences. The first is that the Loewner–Kufarev energy is reversible, that is, invariant under inversion and time reversal of the foliation. Furthermore, the Loewner energy of a Jordan curve can be expressed using the minimal Loewner–Kufarev energy of those measures that generate the curve as a leaf. This provides a new and quantitative characterization of Weil–Petersson quasicircles. Finally, we consider conformal distortion of the foliation and show that the Loewner–Kufarev energy satisfies an exact transformation law involving the Schwarzian derivative. The proof of our main theorem uses an isometry between the Dirichlet energy space on the unit disc and that we construct using Hadamard's variational formula expressed by means of the Loewner–Kufarev equation. Our results are related to ‐parameter duality and large deviations of Schramm–Loewner evolutions coupled with Gaussian random fields.

Monte Carlo Stellar Dynamics near Massive Black Holes: Two-dimensional Fokker–Planck Solutions of Multiple Mass Components

In this study we present a novel Monte Carlo code, referred to as GNC, which enables the investigation of dynamical relaxation in clusters comprising multiple mass components in the vicinity of supermassive black holes at the centers of galaxies. Our method is based on two-dimensional Fokker–Planck equations in the energy and angular momentum space, and allows the evolution of multiple mass components, including stars and compact objects. The code demonstrates remarkable flexibility in incorporating additional complex dynamics. By employing a weighting method, we effectively enhance the statistical accuracy of rare particle results. In this initial publication, we present the fundamental version of our method, focusing on two-body relaxations and loss cone effects. Through comparisons with previous studies, we establish consistent outcomes in terms of relaxation processes, energy and angular momentum distributions, density profiles, and loss cone consumption rates. We consistently observe the development of tangential anisotropy within the cluster, while the outer regions tend to retain near-isotropic characteristics. GNC holds great promise for exploring a wide range of intriguing phenomena within galactic nuclei, including relativistic stellar dynamics, providing detailed and insightful outcomes.

In Situ Performance Analysis of Hybrid Fuel Heating System in a Net-Zero Ready House

The global population’s growth and increased energy consumption have driven greenhouse gas (GHG) emissions. In Canada, the residential sector accounts for 17% of secondary energy use and 13% of GHG emissions. To mitigate GHG emissions, promoting renewable energy and efficient heating systems is crucial, especially in cold climates like Canada, where there is a heavy dependency on fossil fuels for space heating applications. A viable solution is hybrid fuel heating systems that combine electric-driven air-source heat pumps (ASHPs) with natural gas tankless water heaters (TWHs). This system can alternate its operation between the ASHP and TWH based on efficiency and real-time energy costs, reducing grid peak demand and enhancing resilience during power outages. Although lab experiments have shown its benefits, in situ performance lacks evaluation. This study analyzes the in situ energy performance of a net-zero ready house and its hybrid fuel heating system, assessing energy consumption, hourly space heating output, and system heating performance. HOT2000 is a robust simulation software designed for assessing energy consumption, space heating, cooling, and domestic hot water systems in residential buildings. An artificial neural network model was developed to predict the energy performance of the hybrid fuel system, which was used as a substitute for monitored data for evaluating the HOT2000’s simulation results under the same weather conditions. Therefore, this study proposes a comprehensive framework for the in situ performance analysis of hybrid fuel heating systems. This study then, using HOT2000 energy consumption results, evaluates the life cycle costs of the hybrid fuel system against conventional heating systems. Furthermore, this study proposes an economical control strategy using in situ data or manufacturer specifications.

On the Kadomtsev–Petviashvili equation with double-power nonlinearities

In this paper, we study the generalized KP equation with double-power nonlinearities. Our investigation covers various aspects, including the existence of solitary waves, their nonlinear stability, and instability. Notably, we address a broader class of nonlinearities represented by μ1|u|p1−1u+μ2|u|p2−1u, with p1>p2, encompassing cases where μ1>0 and μ1<0<μ2. One of the distinct features of our work is the absence of scaling, which introduces several challenges in establishing the existence of ground states. To overcome these challenges, we employ two different minimization problems, offering novel approaches to address this issue. Furthermore, our study includes a nuanced analysis to ascertain the stability of these ground states. Intriguingly, we extend our stability analysis to encompass cases where the convexity of the Lyapunov function is not guaranteed. This expansion of stability criteria represents a significant contribution to the field. Moving beyond the analysis of solitary waves, we shift our focus to the associated Cauchy problem. Here, we derive criteria that determine whether solutions exhibit finite-time blow-up or remain uniformly bounded within the energy space. Remarkably, our study unveils a notable gap in the existing literature, characterized by the absence of both theoretical evidence of blow-up and uniform boundedness. To explore this intriguing scenario, we employ the integrating factor method, providing a numerical investigation of solution behavior. This method distinguishes itself by offering spectral-order accuracy in space and fourth-order accuracy in time. Lastly, we rigorously establish the strong instability of the ground states, adding another layer of understanding to the complex dynamics inherent in the generalized KP equation.

Coupled inhomogeneous nonlinear Schrödinger system with potential in three space diemensions

This work studies the inhomogeneous Schrödinger coupled system with potential The wave function components read for . The inhomogeneous singular term exponent is . The source term is energy sub‐critical: . Moreover, in order to avoid a singularity of the term , one assumes that . The linear Schrödinger operator reads , where is a potential satisfying some assumptions which imply that the dispersive and Strichartz estimates hold. The purpose is two‐fold. First, we develop a local well‐posedness theory in the energy space . Second, we present a dichotomy of global existence and scattering versus blow‐up of energy solutions under the ground‐state threshold in the inter‐critical focusing regime. The scattering is obtained by using the new approach of Dodson–Murphy which is based on Tao's scattering criteria and Morawetz estimates. The novelty here is the presence of the potential . The challenge is to deal with three technical problems: a coupled nonlinearity, an inhomogeneous singular term , and the presence of the potential . This note naturally extends previous works by the authors about the above problem for .

Lump solutions of the fractional Kadomtsev–Petviashvili equation

Of concern is the fractional Kadomtsev–Petviashvili (fKP) equation and its lump solution. As in the classical Kadomtsev–Petviashvili equation, the fKP equation comes in two versions: fKP-I (strong surface tension case) and fKP-II (weak surface tension case). We prove the existence of nontrivial lump solutions for the fKP-I equation in the energy subcritical case α>45\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >\\frac{4}{5}$$\\end{document} by means of variational methods. It is already known that there exist neither nontrivial lump solutions belonging to the energy space for the fKP-II equation [9] nor for the fKP-I when α≤45\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\le \\frac{4}{5}$$\\end{document} [26]. Furthermore, we show that for any α>45\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >\\frac{4}{5}$$\\end{document} lump solutions for the fKP-I equation are smooth and decay quadratically at infinity. Numerical experiments are performed for the existence of lump solutions and their decay. Moreover, numerically, we observe cross-sectional symmetry of lump solutions for the fKP-I equation.

On the Cauchy problem for the generalized double dispersion equation with logarithmic nonlinearity

Abstract In this paper, we investigate the global existence and finite time blow-up of solution for the Cauchy problem of one-dimensional fifth-order Boussinesq equation with logarithmic nonlinearity. Fist we prove the existence and uniqueness of local mild solutions in the energy space by means of the contraction mapping principle. Further under some restriction on the initial data, we establish the results on existence and uniqueness of global solutions and finite time blow-up of solutions by using the potential well method. Moreover, the sufficient and necessary conditions of global existence and finite time blow-up of solutions are given.

An approximation method for exact controls of vibrating systems with numerical viscosity

We analyze a method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russell’s “stabilizability implies controllability” principle and a finite elements method of order θ with vanishing numerical viscosity. We show that the algorithm is convergent for any initial data in the energy space and that the error is of order θ for sufficiently smooth initial data. Both results are consequences of the uniform exponential decay of the discrete solutions guaranteed by the added viscosity and improve previous estimates obtained in the literature. Several numerical examples for the wave and the beam equations are presented to illustrate the method analyzed in this article.

Sapiential battery systems: beyond traditional electrochemical energy.

As indispensable energy-storage technology in modern society, batteries play a crucial role in diverse fields of 3C products, electric vehicles, and electrochemical energy storage. However, with the growing demand for future electrochemical energy devices, lithium-ion batteries as an existing advanced battery system face a series of significant challenges, such as time-consuming manual material screening, safety concerns, performance degradation, non-access in the off-grid state, poor environmental adaptability, and pollution from waste batteries. Accordingly, incorporating the characteristics of sapiential life into batteries to construct sapiential systems is one of the most engaging tactics to tackle the above issues. In this review, we introduce the concept of sapiential battery systems and provide a comprehensive overview of their core sapiential features, including materials genomics, non-destructive testing, self-healing, self-sustaining capabilities, temperature adaptation, and degradability, which endow batteries with higher performance and more functions. Moreover, the possible future research directions on sapiential battery systems are deeply discussed. This review aims to offer insights for designing beyond traditional electrochemical energy, meeting broader application scenarios such as ultra-long-endurance electric vehicles, wide-temperature energy storage, space exploration, and wearable electronic devices.

Imperatives to improve the efficiency of the energy security system of Ukraine

The increasing volatility of external and internal factors, emergence of new risks in the global energy market (implementation of sustainable development principles and spread of the green economy), aggravation of political challenges, and intensification of integration processes point to the need to review and supplement existing methods for assessing the levels of energy security of states. Forming a single European energy space increases the requirements for ensuring an uninterrupted sustainable energy supply and efficient energy management that protects the interests of all market participants. In the process of substantiating approaches and creating new models of energy security, national and supranational issues of developing a joint energy base that will form the economic basis for the strategic competitiveness of European national economies are becoming essential. A necessary condition for organizing this process is to ensure adequacy and transparency of energy governance, which should consider the current state and trends of joint and national energy security indicators.

On coupled non-linear Schrödinger systems with singular source term

&lt;p&gt;This work studies a coupled non-linear Schrödinger system with a singular source term. First, we investigate the question of the local existence of solutions. Second, one proves the existence of global solutions which scatter in some Sobolev spaces. Finally, one establishes the existence of non-global solutions. The main difficulty here is to overcome the regularity problem in the non-linearity. Indeed, because of the singularity of the source term, the classical contraction method in the energy space fails in such a regime. So, this paper is to fill such a gap in the literature. The argument follows ideas in T. Cazenave and I. Naumkin (&lt;italic&gt;Comm. Contemp. Math.&lt;/italic&gt;, &lt;bold&gt;19&lt;/bold&gt; (2017), 1650038). This consists to remark that the singularity problem is only near the origin. So, one needs to impose that the solution stays away from zero. This is not trivial, since there is no maximum principle for the Schrödinger equation. The existence of global solutions which scatter follows with the pseudo-conformal transformation via the existence of local solutions. Finally, the existence of non-global solutions follows with the classical variance method.&lt;/p&gt;

Vectors of development of energy cooperation between Russia and foreign countries under economic sanctions

The subject of this study is the modern aspects of Russia’s positioning in the global energy space. They are associated with the shift from the previously established liberal world order and the trend towards new polarisation in the world economic system, as well as with the increasing risks of turbulence in world energy markets. In this regard, it is extremely important to assess the real impact of economic sanctions on the nature of relationships and determine promising vectors for the development of energy cooperation between Russia and foreign countries. The research methodology includes analysis and synthesis methods and a historical approach, which are productively used to assess the tactical and strategic actions of our country in the system of international energy cooperation. The scientific novelty of the study lies in the substantiation of theoretical approaches and assessment of the practical implementation of energy cooperation with foreign countries. Those priority interests are considered that determine the economic sovereignty and identity of Russia in the conditions of a gradual but categorical shift of the world community from the previously established imperatives of liberalism and economic globalisation. The article is of an overview nature and describes the key areas of cooperation in the oil and gas and coal industries. Its main provisions may be used for long-term researches related to modernising the strategy of Russia’s foreign economic relations and increasing the level of its foreign economic security in the context of new global challenges.

Energy calibration of silicon drift detector - based spectrometer at ADITYA-U

The Tokamak plasma start-up is the first process to have a successful Tokamak plasma discharge. Considering the start-up condition, high electric field and neutral density with the absence of flux surfaces an enhanced electron transport to the Plasma-Facing Components (PFC) or the inner vessel wall is expected. Such electron excursion under an applied electric field induces electromagnetic emission, especially in the x-ray region. Such emission is important as it gives vital information about the initial condition of the plasma start-up and the potential for energy loss to the PFC/vessel walls. A Silicon Drift Detector (SDD) based soft x-ray (SX) spectrometer is installed at the ADITYA-U tokamak, operating within the range of 1–30 keV generating a 4 K channel spectrum. The energy calibration, spectra transformation from channel space to the energy space, is a pre-requisite for any spectroscopic measurement. The calibration is performed by natural radioactive sources 241Am, 109Cd, 133Ba, and 55Fe having micro-curie strength. Two-point and multipoint calibrations are applied to the SDD spectra. The results from the two processes establish that the multipoint calibration works well for the identification of the photon energy due to the realization of the detector linearity for a wider band.

Logarithmic Gross-Pitaevskii equation

We consider the Schrödinger equation with a logarithmic nonlinearty and non-trivial boundary conditions at infinity. We prove that the Cauchy problem is globally well posed in the energy space, which turns out to correspond to the energy space for the standard Gross-Pitaevskii equation with a cubic nonlinearity, in small dimensions. We then characterize the solitary and traveling waves in the one dimensional case.

Decolonizing the political economy of energy transitions: new energy spaces and pluriversal politics in Mexico

This paper draws on Critical Political Economy (CPE) to explore energy transitions in Mexico. It analyzes struggles over competing energy visions from a decolonial, spatial and post-development perspective. The article frames energy transformations as more than a move away from fossil fuels: They encompass a radical overhaul of universalized Eurocentric capitalist modernity that call the state’s role as both facilitator and driver of the energy transition into question. I examine two low-carbon infrastructure projects in the state of Yucatan, Mexico to build on Bridge and Gailing’s notion of new energy spaces as the sites, scales and spatialities through which broader questions of political economy and contemporary struggles are being worked out. CPE of the energy transition must ensure that low-carbon infrastructures are not deployed as new forms of extraction, but rather as the material basis of pluriversal transitions informed by affected communities and their territorial struggles. Both studies on the political economy of energy transitions and territorial struggles can benefit from an open dialogue about how socioecological transformations are imagined, designed and implemented.

Exploring the Latent Chemical Space of Oxygen Vacancy Formation Energy by a Machine Learning Ensemble

Exploring the Latent Chemical Space of Oxygen Vacancy Formation Energy by a Machine Learning Ensemble