On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law

<p style='text-indent:20px;'>In this paper, we define a renormalized dissipative measure-valued (rDMV) solution of the compressible magnetohydrodynamics (MHD) equations with non-monotone pressure law. We prove the existence of the rDMV solutions and establish a suitable relative energy inequality. And we obtain the weak (measure-valued)-strong uniqueness property of this rDMV solution with the help of the relative energy inequality.</p>

National and Subnational Drivers for Cooking Energy Consumption in Rural India

BackgroundThe energy transition pattern in India highlights that cooking continues to be the weakest link in the energy transition process for rural households. The government is trying to subsidize LPG for rural families to increase usage of LPG in India for a clean cooking energy transition. The paper endeavors to fill the void by revisiting the nature and degree of the rural energy transition for cooking in India. MethodologyUsing a multinomial logit model, the potential drivers at the individual household and group level have been identified. The group effect analysis has been conducted purposely to understand if social and cultural norms or -level factors within a village society affect the cooking energy transition of households in rural India and if that prevails over the income effect. At the sub-national level, an estimate of energy inequality has been derived by applying social choice-based Atkinson Inequality measure to examine the connection between higher energy inequality in primary fuel used for cooking and higher income inequality. In addition, the Brock-Dechert-Scheinkman (BDS) test on firewood consumption has been conducted for poor energy households from 38 districts of Bihar, one of the highest energy-poor states, to statistically examine the perceived nonlinearity in the dynamics of energy transition/fuel choice in cooking for rural households. ResultsThe analysis at the national level indicates the importance of local and cultural factors that leads to the nonlinearity in the probability of the switchover from firewood to other clean fuel options. Conclusions SummaryThe paper highlights if subsidies on modern fuel and/or other cooking alternatives alone drive the transition process and examines the validity of the energy ladder hypothesis in the case of rural cooking energy transition and the drivers of the energy transition at the national and sub-national level. Potential ImplicationsThe state-level analysis done for Bihar and across various districts corroborates this finding and provides an important direction towards local context-specific policy-making in the long term.

On the supercritical defocusing NLW outside a ball

We study a defocusing semilinear wave equation, with a power nonlinearity $$|u|^{p-1}u$$ | u | p - 1 u , defined outside the unit ball of $$\mathbb {R}^{n}$$ R n , $$n\ge 3$$ n ≥ 3 , with Dirichlet boundary conditions. We prove that if $$p>n+3$$ p > n + 3 and the initial data are nonradial perturbations of large radial data, there exists a global smooth solution. The solution is unique among energy class solutions satisfying an energy inequality. The main tools used are the Penrose transform and a Strichartz estimate for the exterior linear wave equation perturbed with a large, time dependent potential.

Dissipative Solutions to the Stochastic Euler Equations

We study the three-dimensional incompressible Euler equations subject to stochastic forcing. We develop a concept of dissipative martingale solutions, where the nonlinear terms are described by generalised Young measures. We construct these solutions as the vanishing viscosity limit of solutions to the corresponding stochastic Navier–Stokes equations. This requires a refined stochastic compactness method incorporating the generalised Young measures. As a main novelty, our solutions satisfy a form of the energy inequality which gives rise to a weak–strong uniqueness result (pathwise and in law). A dissipative martingale solution coincides (pathwise or in law) with the strong solution as soon as the latter exists.

MODELING OF FLOWS THROUGH A CHANNEL BY THE NAVIER–STOKES VARIATIONAL INEQUALITIES

We deal with a mathematical model of a flow of an incompressible Newtonian fluid through a channel with an artificial boundary condition on the outflow. We explain how several artificial boundary conditions formally follow from appropriate variational formulations and the wayone expresses the dynamic stress tensor. As the boundary condition of the “do nothing”–type, that is predominantly considered to be the most appropriate from the physical point of view, does not enable one to derive an energy inequality, we explain how this problem can be overcome by using variational inequalities. We derive a priori estimates, which are the core of the proofs, and present theorems on the existence of solutions in the unsteady and steady cases.

On a two-dimensional fractional thermoelastic system with nonlocal constraints describing a fractional Kirchhoff plate

We show herein the existence and uniqueness of solutions for coupled fractional order partial differential equations modeling a thermoelastic fractional Kirchhoff plate model associated with initial, Dirichlet, and nonlocal boundary conditions involving fractional Caputo derivative. Some efficient results of existence and uniqueness are obtained by employing the energy inequality method.