Navier--Stokes--Fourier System on Unbounded Domains: Weak Solutions, Relative Entropies, Weak-Strong Uniqueness

Abstract

We investigate the Navier-Stokes-Fourier system describing the motion of a compressible, viscous and heat conducting fluid on large class of unbounded domains with no slip and slip boundary conditions. We propose a definition of weak solutions, that is particularly convenient for the treatment of the Navier-Stokes-Fourier system on unbounded domains. We prove existence of weak solutions for arbitrary large initial data for potential forces with an arbitrary growth at large distances. We show, that any weak solution satisfies the so called relative entropy inequality. Finally we prove the weak-strong uniqueness principle, meaning that the weak solutions coincide with strong solutions emanating from the same initial data (as long as the latter exist), at least when the potential force vanishes at large distances.

Vanishing dissipation limit for the Navier–Stokes–Fourier system

“…As shown in [10], any weak solution [̺, ϑ, u] to the Navier-Stokes-Fourier system with the boundary conditions (1.16), (1.17) satisfies the relative energy inequality [17] proposed for the incompressible Euler system, we may define dissipative solutions to the Navier-Stokes-Fourier system requiring solely certain regularity of [̺, ϑ, u] and the relative energy inequality (4.1) to be satisfied for any trio [r, Θ, U] as in (4.2). Such an approach was used by Jesslé, Jin, Novotný [14] to attack problems on unbounded spatial domains.…”

Asymptotic solutions of weakly compressible Newtonian Poiseuille flows with pressure-dependent viscosity

“…The reader is referred, for example, to the work of Lions [23] and to the papers by Lanzendörfer [24] and Jesslé et al [25] for relevant discussions. However, it should be pointed out that the asymptotic solutions derived in Section 3 are unique, due to the fact that the governing equations that result from the perturbation procedure are linear.…”

Inviscid Incompressible Limits Under Mild Stratification: A Rigorous Derivation of the Euler–Boussinesq System

“…Remark 3.1 Under the hypotheses (2.7 -2.15), the existence of dissipative weak solutions to the Navier-Stokes-Fourier system in (0, T ) × Ω was shown in [11].…”

“…Remark 2.1 Note that the above definition of dissipative weak solutions on unbounded domains, proposed in [11], is different from that on bounded domains introduced in [6]. In [6], the relative entropy inequality (2.6) is replaced by the total energy balance, whereas (2.6) is automatically satisfied for any weak solution to the Navier-Stokes-Fourier system.…”