Semiflow selection for the compressible Navier–Stokes system

Abstract

Although the existence of dissipative weak solutions for the compressible Navier–Stokes system has already been established for any finite energy initial data, uniqueness is still an open problem. The idea is then to select a solution satisfying the semigroup property, an important feature of systems with uniqueness. More precisely, we are going to prove the existence of a semiflow selection in terms of the three state variables: the density, the momentum, and the energy. Finally, we will show that it is possible to introduce a new selection defined only in terms of the initial density and momentum; however, the price to pay is that the semigroup property will hold almost everywhere in time.

Statistical Solutions to the Barotropic Navier–Stokes System

“…In the present setting, axiom [A1] is satisfied in view of the existence result stated in Proposition 4.1. Axioms [A4], [A5] can be verified in the same way as in Basarič [3], [7]. Finally, as observed in [7], axioms [A2], [A3] follow from the property of weak sequential stability stated below.…”

“…We adopt a variant of the Skorokhod topology D([0, ∞); R) as a trajectory space for the energy, see Jakubowski [29]. Finally, we remark that a similar procedure in the context of the barotropic Navier-Stokes system with homogeneous boundary conditions has been performed by Basarič [3], where the phase space for the energy is taken L 1 loc [0, ∞). The Skorokhod topology seems more convenient as the pointwise in time values of the total energy are well defined while they correspond merely to the Lebesgue points in the L 1 loc setting.…”

“…Similarly to Basarič [3], we apply the method of Krylov [30] adapted to the deterministic problems by Cardona and Kapitanski [12].…”

Semiflow Selection to Models of General Compressible Viscous Fluids

“…where As already done for the Euler and Navier-Stokes systems, cf. [4,9], among all the dissipative solutions emanating from the same initial data it is possible to select only the admissible ones, i.e., satisfying the physical principal of minimizing the total energy or equivalently, that are minimal with respect to relation ≺ defined as…”

“…The idea of the proof is to reduce iteratively the set U(x) for a fixed x ∈ D, selecting the minimum points of particular functionals in order to obtain finally a single point in T , which will define u(x). The procedure has been proposed by Cardona and Kapitanski [11] in the context of continuous trajectories and later adapted to more general setting in [9] and [4]. We introduce the functionals I λ,k : T → R defined for every Φ ∈ T as…”