Effects of COVID-19 on Indian Energy Consumption

Just after the Indian government issued the first lockdown rule to cope with the increasing number of COVID-19 cases in March 2020, the energy consumption in India plummeted dramatically. However, as the lockdown relaxed, energy consumption started to recover. In this study, we investigated how COVID-19 cases affected Indian energy consumption during the COVID-19 crisis by testing if the lockdown release had a positive impact on energy consumption and if richer regions were quicker to recover their energy consumption to the level before the lockdown. Using the autoregressive distributed lag (ARDL) model, the study reveals that a long-run relationship holds between the COVID-19 cases and energy consumption and that the COVID-19 cases have a positive effect on Indian energy consumption. This result indicates that as lockdown relaxed, energy consumption started to recover. However, such a positive impact was not apparent in the Eastern and North-Eastern regions, which are the poorest regions among the five regions investigated in the study. This implies that poorer regions need special aid and policy to recover their economy from the damage suffered from the COVID-19 crisis.

Energy Definition and Dark Energy: A Thermodynamic Analysis

Accepting the Komar mass definition of a source with energy-momentum tensor Tμν and using the thermodynamic pressure definition, we find a relaxed energy-momentum conservation law. Thereinafter, we study some cosmological consequences of the obtained energy-momentum conservation law. It has been found out that the dark sectors of cosmos are unifiable into one cosmic fluid in our setup. While this cosmic fluid impels the universe to enter an accelerated expansion phase, it may even show a baryonic behavior by itself during the cosmos evolution. Indeed, in this manner, while Tμν behaves baryonically, a part of it, namely, Tμν(e) which is satisfying the ordinary energy-momentum conservation law, is responsible for the current accelerated expansion.

The L1L^{1} gradient flow of a generalized scale invariant Willmore energy for radially non-increasing functions

We use the minimizing movement theory to study the gradient flow associated to a non-regular relaxation of a geometric functional derived from the Willmore energy. Thanks to the coarea formula, we can define a Willmore energy on regular functions of L^{1}(\mathbb{R}^{d}). This functional is extended to every {L^{1}} function by taking its lower semicontinuous envelope. We study the flow generated by this relaxed energy for radially non-increasing functions (functions with balls as superlevel sets). In the first part of the paper, we prove a coarea formula for the relaxed energy of such functions. Then, we show that the flow consists of an erosion of the initial data. The erosion speed is given by a first order ordinary equation.

Curvature-dependent energies: a geometric and analytical approach

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

On the Allen—Cahn/Cahn—Hilliard system with a geometrically linear elastic energy

We present an extension of the Allen-Cahn/Cahn-Hilliard system that incorporates a geometrically linear ansatz for the elastic energy of the precipitates. The model contains both the elastic Allen-Cahn system and the elastic Cahn-Hilliard system as special cases, and accounts for the microstructures on the microscopic scale. We prove the existence of weak solutions to the new model for a general class of energy functionals. We then give several examples of functionals that belong to this class. This includes the energy of geometrically linear elastic materials for dimensions D < 3. Moreover, we show this for D = 3 in the setting of scalar-valued deformations, which corresponds to the case of anti-plane shear. All this is based on explicit formulae for relaxed energy functionals newly derived in this article for D = 1 and D = 3. In these cases we can also prove the uniqueness of the weak solutions.