Two-velocity hydrodynamics and thermodynamics

Two-velocity hydrodynamics in fluid mechanics: Part II Existence of global κ-entropy solutions to the compressible Navier–Stokes systems with degenerate viscosities

“…the density κ is equal to 0). This property was also explained in the works [38] and [20] where the authors discuss the capillary-temperature.…”

“…(1 − κ)ρu e κ + |u| 2 2 + κρu e κ + |u + 2∇ϕ( )| 2 2 similarly to energy from [38]. This quantity may be treated as a generalization of the κ-entropy, found for the barotropic case, to the heat-conducting case.…”

“…That means that the κ-entropy demonstrates two-velocity structure in the usual compressible Navier-Stokes system. For an introduction to the two-velocity hydrodynamics and thermodynamics we refer to [38] and [20] . Indeed, using the identity…”

“…Remark 7 To see the link between the above system and the one from [38], one should take the main velocity equal to w = u + 2κ∇ϕ( ) and the drift equal to v = 2∇ϕ( ).…”

“…Using estimates (37), (38), (34) and (35), one can check that T is a continuous mapping of the ball B M,τ into itself and for sufficiently small τ = T (n) it is a contraction. Therefore, it possesses a unique fixed point which is a solution to (32) and (33) for T = T (n).…”