Mathematical Theory of Compressible Viscous Fluids

Abstract

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On a class of compressible viscoelastic rate-type fluids with stress-diffusion

“…To construct the weak solutions to problem (2.1-2.3) we adapt the multilevel approximation scheme from [6,Chapter 7]:…”

“…For fixed m > 0, ε > 0, the approximate problem (6.1-6.3) can be solved by a fixed point argument, similarly to [6,Chapter 7]. More specifically, given b ∈ C([0, T ]; W 1,2 (Ω)), the system (6.1), (6.2) admits a unique solution (…”

“…We recall that all the a priori bounds derived formally in Section 3 remain valid at this level of approximation as they are based the energy estimates that are compatible with the Galerkin approximation. Exactly as in [6,Chapter 7], we can pass to the limit m → ∞ obtaining the second level approximate solutions satisfying…”

Continuous, Semi-discrete, and Fully Discretised Navier-Stokes Equations

“…If the volume of a fluid parcel changes over time, the flow is called compressible. In this case, the mass density ρ is modelled as an unknown function and related to the pressure p through constitutive or so-called state equations, like the ideal gas low ; see [FKP16].…”

On the Classification of Incompressible Fluids and a Mathematical Analysis of the Equations That Govern Their Motion

“…In the following lemma, several assertions characterizing weak compactness in L 1 , namely the Dunford-Pettis criterion (ii), uniform integrability (iii), and the de la Vallé-Poussin criterion (iv), are provided. The exact statement is taken from [29,p. 21,Theorem 10].…”