Abstract. We consider the compressible Navier-Stokes system with variable entropy. The pressure is a nonlinear function of the density and the entropy/potential temperature which, unlike in the NavierStokes-Fourier system, satisfies only the transport equation. We provide existence results within three alternative weak formulations of the corresponding classical problem. Our constructions hold for the optimal range of the adiabatic coefficients from the point of view of the nowadays existence theory. -IntroductionThe purpose of this paper is to analyze the model of flow of compressible viscous fluid with variable entropy. Such flow can be described by the compressible Navier-Stokes equations coupled with an additional equation describing the evolution of the entropy. In case when the conductivity is neglected, the changes of the entropy are solely due to the transport and the whole system can be written as:(1.1a) 1c) where the unknowns are the density ̺ : (0, T ) × Ω → R + ∪ {0}, the entropy s : (0, T ) × Ω → R + and the velocity of fluid u : (0, T ) × Ω → R 3 , and where Ω is a three dimensional domain with a smooth boundary ∂Ω. The momentum, the continuity and the entropy equations are additionally coupled by the form of the pressure p, we assume thatwhere T (·) is a given smooth and strictly monotone function from R + to R + , in particular T (s) > 0 for s > 0. We assume that the fluid is Newtonian and that the viscous part of the stress tensor is of the following formwith D(u) = 1 2 (∇u + ∇u T ). Viscosity coefficients µ and η are assumed to be constant, hence we can write div S(∇u) = µ∆u + (µ + λ)∇ div u with λ = η − 2 3 µ. To keep the ellipticity of the Lamé operator we require that µ > 0, 3λ + 2µ > 0.(1.3)

A common feature of systems of conservation laws of continuum physics is that they are endowed with natural companion laws which are in such cases most often related to the second law of thermodynamics. This observation easily generalizes to any symmetrizable system of conservation laws; they are endowed with nontrivial companion conservation laws, which are immediately satisfied by classical solutions. Not surprisingly, weak solutions may fail to satisfy companion laws, which are then often relaxed from equality to inequality and overtake the role of physical admissibility conditions for weak solutions. We want to answer the question: what is a critical regularity of weak solutions to a general system of conservation laws to satisfy an associated companion law as an equality? An archetypal example of such a result was derived for the incompressible Euler system in the context of Onsager's conjecture in the early nineties. This general result can serve as a simple criterion to numerous systems of mathematical physics to prescribe the regularity of solutions needed for an appropriate companion law to be satisfied.

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We consider the initial value problem for the inviscid Primitive and Boussinesq equations in three spatial dimensions. We recast both systems as an abstract Eulertype system and apply the methods of convex integration of De Lellis and Székelyhidi to show the existence of infinitely many global weak solutions of the studied equations for general initial data. We also introduce an appropriate notion of dissipative solutions and show the existence of suitable initial data which generate infinitely many dissipative solutions.