Large data existence theory for unsteady flows of fluids with pressure- and shear-dependent viscosities

Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion

“…[3] for the existence analysis, [2,4,30] for regularity theory for the Stokes system and [7] for a conditional regularity result for Navier-Stokes system. The key di erence and also the main mathematical advantage of the Navier slip boundary conditions is, that for smooth domains, namely if Ω ∈ C , , we can introduce the pressure p as an integrable function, e.g., by using an additional layer of approximation as in [11], see also [15,16] or [8] which discuss the treatment of the pressure in evolutionary models subject to the Navier boundary condition. Nevertheless, since we shall always deal with formulation without the pressure (see the De nition), we can also treat the Dirichlet boundary condition, as well as very general implicitly speci ed boundary conditions see e.g.…”

On a Navier–Stokes–Fourier-like system capturing transitions between viscous and inviscid fluid regimes and between no-slip and perfect-slip boundary conditions

“…Consequently, we will be enabled to use the standard Carathéodory theory to construct some stepping-stone solutions, since instead of the relatively complicated S and s in the original problem, we will deal with explicitly given S k (Dv, e) and s k (v τ , e) := −(S k (Dv, e)n) τ , respectively. Thirdly, due to the approximation's ability to take the velocity as a test function in the balance of linear momentum, there is no need for inequality (5) as the balance of total energy (3) is equivalent to the balance of internal energy (4).…”

“…with p 1 + p 2 = p. We will only sketch the procedure as the detailed version is to be found in [4] and the very origin lies in [7]. According to (42), for any ϕ ∈ W 2,2 (Ω) such that ∇ϕ · n = 0 at ∂Ω and a.e.…”

“…Attainment of the initial condition (22) for the velocity is shown in a standard way; see e.g. [4,Sec. 5.4].…”

“…Inasmuch as we deal merely with weak solutions here, this equivalence cannot in general be verified. However, we still require (4) to be satisfied at least as an inequality (in the weak sense; see (21))…”