Relative entropy and a weak–strong uniqueness principle for the compressible Navier–Stokes equations on moving domains

Abstract

We establish a relative entropy inequality for the compressible Navier-Stokes equations, posed on domains with time-dependent moving boundaries. Using the relative entropy, a weak-strong uniqueness result is shown for the class of finite energy weak solutions.

Measure-valued solutions and weak-strong uniqueness for the incompressible inviscid fluid-rigid body interaction

“…For an analogous result for a cavity filled by compressible fluid see [14]. We also refer to [9], [26] for problems on moving domains. But in these previous mentioned references regarding weak-strong uniqueness for fluid-structure interaction problem, the authors have always taken the motion of rigid body in a viscous fluid.…”

Compressible Navier-Stokes system on a moving domain in the $L_p-L_q$ framework

“…These results have been recently generalized to the complete system with heat conductivity in [13] and [14]. The first weak-strong uniqueness result on a moving domain has been shown in [2] in case of no-slip boundary condition. A generalization of this to slip conditions as well as a local existence result for strong solution for both types boundary conditions can be found in [15].…”

Local existence of strong solutions and weak–strong uniqueness for the compressible Navier–Stokes system on moving domains

“…For clarity of the proof we have so far restricted our presentation to the case of slip boundary conditions. In case of homogeneous Dirichlet boundary condition (u − V)| Γτ = 0 for any τ ≥ 0 (7.1) i6a the weak-strong uniqueness principle has been shown recently in [5]. However, the existence of regular solutions has remained so far open question.…”

“…In [10] the authors used relative entropy inequality to prove the weak-strong uniqueness property. In the case of moving domain with no-slip boundary condition Doboszczak [5] proved both the relative entropy inequality as well as the weak-strong uniqueness property under assumption of an existence of local strong solution.…”