We consider the unsteady compressible Navier-Stokes equations in a perforated three-dimensional domain, and show that the limit system for the diameter of the holes going to zero is the same as in ...the perforated domain provided the perforations are small enough. The novelty of this result is the lower adiabatic exponent γ>3 instead of the known value γ>6. The proof is based on the use of two different restriction operators leading to two different types of pressure estimates. We also discuss the extension of this result for the unsteady Navier-Stokes-Fourier system as well as the optimality of the known results in arbitrary space dimension for both steady and unsteady problems.
The principle purpose of this work is to investigate a “viscous” version of a “simple” but still realistic bi-fluid model described in
Bresch
et al. (in:
Giga
,
Novotný
(eds) Handbook of mathematical ...analysis in mechanics of viscous fluids,
2018
) whose “non-viscous” version is derived from physical considerations in
Ishii
and
Hibiki
(Thermo-fluid dynamics of two-phase flow, Springer, Berlin,
2006
) as a particular sample of a multifluid model with algebraic closure. The goal is to show the existence of weak solutions for large initial data on an arbitrarily large time interval. We achieve this goal by transforming the model to a transformed two-densities system which resembles the compressible Navier–Stokes equations, with, however, two continuity equations and a momentum equation endowed with the pressure of a complicated structure dependent on two variable densities. The new “transformed two-densities system” is then solved by an adaptation of the Lions–Feireisl approach for solving compressible Navier–Stokes equation, completed with several observations related to the DiPerna–Lions transport theory inspired by
Maltese
et al. (J Differ Equ 261:4448–4485,
2016
) and
Vasseur
et al. (J Math Pures Appl 125:247–282,
2019
). We also explain how these techniques can be generalized to a model of mixtures with more than two species. This is the first result on the existence of weak solutions for any realistic multifluid system.
We study the homogenization of stationary compressible Navier–Stokes–Fourier system in a bounded three dimensional domain perforated with a large number of very tiny holes. Under suitable assumptions ...imposed on the smallness and distribution of the holes, we show that the homogenized limit system remains the same in the domain without holes.
Based on the recent result from Chaudhuri and Feireisl (Navier–Stokes–Fourier system with Dirichlet boundary conditions, 2021.
arXiv:2106.05315
) for the evolutionary compressible ...Navier–Stokes–Fourier equations we present the proof of existence of a weak solution for the steady system with Dirichlet boundary condition for the temperature without any restriction on the size of the data. The weak formulation of the equations for the temperature is based on the total energy balance and entropy inequality with compactly supported test functions and a steady version of the ballistic energy inequality which allows to obtain estimates of the temperature.
We investigate the existence of weak solutions to a certain system of partial differential equations, modeling the behavior of a compressible non‐Newtonian fluid for small Reynolds number. We ...construct the weak solutions despite the lack of the
L∞$$ {L}^{\infty } $$ estimate on the divergence of the velocity field. The result was obtained by combining the regularity theory for singular operators with a certain logarithmic integral inequality for
BMO$$ BMO $$ functions, which allowed us to adjust the method from Feireisl et al. (2015) to more relaxed conditions on the velocity.
We consider a system of partial differential equations which describes steady flow of a compressible heat conducting chemically reacting gaseous mixture. We extend the result from Giovangigli et al. ...(2015) in the sense that we introduce the variational entropy solution for this model and prove existence of a weak solution for γ>43 and existence of a variational entropy solution for any γ>1. The proof is based on improved density estimates.
We consider the regularity criteria for the incompressible Navier--Stokes equations connected with one velocity component. Based on the method from Cao and Titi (2008 Indiana Univ. Math. J. 57 ...2643--61) we prove that the weak solution is regular, provided , , \frac {10}{3} ' src='http://ej.iop.org/images/0951-7715/23/5/004/non327722in003.gi f ' align=middle> or provided , if or if s (3, {infinity}. As a corollary, we also improve the regularity criteria expressed by the regularity of or .
We investigate the creation and properties of eventual vacuum regions in the weak solutions of the continuity equation, in general, and in the weak solutions of compressible Navier–Stokes equations, ...in particular. The main results are based on the analysis of renormalized solutions to the continuity and pure transport equations and their inter-relations which are of independent interest.
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