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We consider the Navier-Stokes-Fourier system describing the motion of a compressible, viscous, and heat conducting fluid in a bounded domain Ω ⊂ R d , d = 2, 3, with general nonhomogeneous Dirichlet boundary conditions for the velocity and the absolute temperature, with the associated boundary conditions for the density on the inflow part. We introduce a new concept of weak solution based on the satisfaction of the entropy inequality together with a balance law for the ballistic energy. We show the weak-strong uniqueness principle as well as the existence of global-in-time solutions.

The goal of the present paper is to study the weak-strong uniqueness problem for the compressible Navier-Stokes system with a general barotropic pressure law. Our results include the case of a hard sphere pressure of Van der Waals type with a non-monotone perturbation and a Lipschitz perturbation of a monotone pressure. Although the main tool is the relative energy inequality, the results are conditioned by the presence of viscosity and do not seem extendable to the Euler system. 2010 Mathematics Subject Classification. Primary: 35 Q 30 . Secondary: 35 B 30.

In this paper our goal is to define a renormalised dissipative measure-valued (rDMV) solution of compressible Navier-Stokes system for fluids with nonmonotone pressure-density relation. We prove existence of rDMV solutions and establish a suitable relative energy inequality. Moreover we obtain the Weak (Measure-valued)-Strong uniqueness property of this rDMV solution with the help of relative energy inequality.• Pressure Law: In an isentropic setting, the pressure p and the density ̺ of the fluid are interrelated by :(1.4) Remark 1.1. The Consideration of h in (1.4) has been motivated from isentropic equation of state given by h(̺) = a̺ γ with γ ≥ 1 and a > 0.• Here we consider no slip boundary condition for velocity i.e. u| {∂Ω×(0,T )}=0 .( 1.5) The compressible Navier-Stokes equations admit global-in-time weak solution(s) for general finite energy initial data and a large class of pressure-density constitutive relations. Considering q ≡ 0 in (1.4) and following the literatures of Antontsev et al.[1], Lions[15], Feireisl[8], Plotnikov et al.[18] and many others, we observe globalin-time weak solution for adiabatic exponent γ ≥ 1 for d = 1, 2 and γ > 3 2 for d = 3. Even for non-monotone pressure, Feireisl in [7] has proved a similar result and recent work by Bresch and Jabin [2] indicates that for p ∈ C 1 [0, ∞) ≥ 0, p(0) = 0, lim ̺→∞ p ′ (̺) ̺ γ−1 = a > 0 and γ ≥ 2 the system admits a weak solution. So it may seem unnecessary to develop the theory of measure valued solution that extends the class of generalised solutions but in the following discussion we will try to justify why we still choose to consider it.

The goal of the present paper is to prove that if a weak limit of a consistent approximation scheme of the compressible complete Euler system in full space $$ \mathbb {R}^d,\; d=2,3 $$ R d , d = 2 , 3 is a weak solution of the system, then the approximate solutions eventually converge strongly in suitable norms locally under a minimal assumption on the initial data of the approximate solutions. The class of consistent approximate solutions is quite general and includes the vanishing viscosity and heat conductivity limit. In particular, they may not satisfy the minimal principle for entropy.