On the flow of chemically reacting gaseous mixture

We consider the Cauchy problem for the system of equations governing flow of isothermal reactive mixture of compressible gases. Our main contribution is to prove sequential stability of weak solutions when the state equation essentially depends on the species concentration and the viscosity coefficients vanish on vacuum. Moreover, under additional assumption on the "cold" component of the pressure in the regions of small density, we prove the existence of weak solutions for arbitrary large initial data

Analysis of semidiscretization of the compressible Navier-Stokes equations

The objective of this work is to investigate the time discretization of two-dimensional Navier-Stokes system with the slip boundary conditions. First, the existence of weak solutions for fixed time step δt > 0 is presented and then the limit passage as δt→0+ is carried out. The proof is based on a new technique established for the steady Navier-Stokes equations by P.B. Mucha and M. Pokornỳ [P.B. Mucha, M. Pokornỳ, On a new approach to the issue of existence and regularity for the steady compressible Navier-Stokes equations, Nonlinearity 19 (8) (2006) 1747-1768] which enables to estimate the growth of L∞ norm of the density when δt goes to 0. © 2011 Elsevier Inc

Analysis of nonlocal model of compressible fluid in 1-D

We study a nonlocal modification of the compressible Navier–Stokes equations in mono-dimensional case with a boundary condition characteristic for the free boundaries problem. From the formal point of view, our system is an intermediate between the Euler and Navier–Stokes equations. Under certain assumptions, imposed on initial data and viscosity coefficient, we obtain the local and global existence of solutions. Particularly, we show the uniform in time bound on the density of fluid

On the steady flow of a multicomponent, compressible, chemically reacting gas

We consider the system of equations governing the steady flow of a polyatomic isothermal reactive gas mixture. The model covers situations when the pressure depends on species concentration and when the diffusion coefficients of each of the species are density-dependent. It is shown that this problem admits a weak solution provided the adiabatic exponent for the mixture γ is greater than 7/3

Heat-Conducting, Compressible Mixtures with Multicomponent Diffusion: Construction of a Weak Solution

We investigate a coupling between the compressible Navier'stokes-Fourier system and the full Maxwell'stefan equations. This model describes the motion of a chemically reacting heat-conducting gaseous mixture. The viscosity coefficients are density-dependent functions vanishing in a vacuum and the internal pressure depends on species concentrations. By several levels of approximation we prove the global-in-time existence of weak solutions on the three-dimensional torus

Transport of congestion in two-phase compressible/incompressible flows

We study the existence of weak solutions to the two-phase fluid model with congestion constraint. The model encompasses the flow in the uncongested regime (compressible) and the congested one (incompressible) with the free boundary separating the two phases. The congested regime appears when the density in the uncongested regime achieves a threshold value that describes the comfort zone of individuals. This quantity is prescribed initially and transported along with the flow. We prove that this system can be approximated by the fully compressible Navier–Stokes system with a singular pressure, supplemented with transport equation for the congestion density. We also present the application of this approximation for the purposes of numerical simulations in the one-dimensional domain

Approximate solutions to a model of two-component reactive flow

We consider a model of motion of binary mixture, based on the compressible Navier-Stokes system. The mass balances of chemically reacting species are described by the reaction-diffusion equations with generalized form of multicomponent difiusion ux. Under a special relation between the two density dependent viscosity coefficients and for singular cold pressure we construct the weak solutions passing through several levels of approximation

On the maximal Lp-Lq regularity of solutions to a general linear parabolic system

We show the existence of solution in the maximal regularity framework to a class of symmetric parabolic problems on a uniformly domain in . Our approach consist in showing - boundedness of families of solution operators to corresponding resolvent problems first in the whole space, then in half-space, perturbed half-space and finally, using localization arguments, on the domain. Assuming additionally boundedness of the domain we also show exponential decay of the solution. In particular, our approach does not require assuming a priori the uniform Lopatinskii - Shapiro condition

Multicomponent mixture model: The issue of existence via time discretization

We prove the existence of global-in-time weak solutions to a model of chemically reacting mixture. We consider a coupling between the compressible Navier-Stokes system and the reaction diffusion equations for chemical species when the thermal effects are neglected. We first prove the existence of weak solutions to the semi-discretization in time. Based on this, the existence of solutions to the evolutionary system is proven

Chemically reacting mixtures in terms of degenerated parabolic setting

The paper analyzes basic mathematical questions for a model of chemically reacting mixtures. We derive a model of several (finite) component compressible gas taking rigorously into account the thermodynamical regime. Mathematical description of the model leads to a degenerate parabolic equation with hyperbolic deviation. The thermodynamics implies that the diffusion terms are non-symmetric, not positively defined, and cross-diffusion effects must be strongly marked. The mathematical goal is to establish the existence of weak solutions globally in time for arbitrary number of reacting species. A key point is an entropy-like estimate showing possible renormalization of the system. © 2013 AIP Publishing LLC