Energy Equality in Compressible Fluids with Physical Boundaries

Abstract

We study the energy balance for the weak solutions of the three-dimensional compressible Navier-Stokes equations in a bounded domain. We establish an L p -L q regularity condition on the velocity field for the energy equality to hold, provided that the density is bounded and satisfies √ ρ ∈ L ∞ t H 1 x . The main idea is to construct a global mollification combined with an independent boundary cut-off, and then take a double limit to prove the convergence of the resolved energy.

Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain

“…Remark 5.1. Note that our result is consistent with the result [5] for the compressible Navier-Stokes equations in a bounded domain. In addition, in the absence of vacuum, as proved in [24], the condition ∇ √ ρ ∈ L ∞ (0, T ; L 2 (Ω)) in (5. where…”

“…There are also some results for the inhomogeneous incompressible case and the compressible case for different types of domains. See [5,20,24,33] and the references therein.…”

“…Now, as in [5], for t 0 > 0, we take some positive τ and α satisfying τ + α < t 0 , and define which is exactly (1.4).…”

Energy conservation for the weak solutions to the equations of compressible magnetohydrodynamic flows in three dimensions

“…In [27], the case of density-dependence viscosity is also considered. Recently, Chen, Liang, Wang, Xu [5] nicely extended Yu's results to the Dirichlet problem. The purpose of this paper is to provide a sufficient condition for the energy conservation of the weak solution of (1.1)-(1.2), which is motivated by Yu's work [27] (see also [5]).…”

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

“…Remark 1.6. Our results work for bounded domains with C 1 smooth boundaries, by slightly modifying the argument of [8].…”