On the energy equality for very weak solutions to 3D MHD equations

In this paper, we consider the energy equality of the 3D Cauchy problem for the magneto-hydrodynamics (MHD) equations. We show that if a very weak solution of MHD equations belongs to $L^{4}(0,\,T;L^{4}(\mathbb {R}^{3}))$ , then it is actually in the Leray–Hopf class and therefore must satisfy the energy equality in the time interval $[0,\,T]$ .

Inequality in energy transitions: Subnational air pollution disparities resulting from national decarbonization strategies

Energy transitions and decarbonization require rapid changes to a nation’s generation mix. There are a host of possible decarbonization pathways, yet there is vast uncertainty about how different decarbonization pathways will advance or derail the nation’s energy equality goals. We present a framework for investigating how decarbonization pathways, driven by a least cost paradigm, will lead to air pollution inequality across different vulnerabilities (e.g., low-income, energy poverty). If an equitable energy transition is the goal (i.e., one that reaches total equality), using least cost optimization capacity expansion models without strict renewable energy technology mandates will not accomplish this. Thus, it is imperative that decisions regarding national regarding national decarbonization pathways have strict mandates for equality outcomes or be driven by an equality focused paradigm.

Regularity criterion on the energy conservation for the compressible Navier–Stokes equations

This paper concerns the energy conservation for the weak solutions of the compressible Navier–Stokes equations. Assume that the density is positively bounded, we work on the regularity assumption on the gradient of the velocity, and establish a L p –L s type condition for the energy equality to hold in the distributional sense in time. We mention that no regularity assumption on the density derivative is needed any more.