On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain $\Omega = \Omega_0 \times (0,L) \in \mathbb{R}^3$. We show existence of a solution $(v,\rho) \in W^2_p(\Omega) \times W^1_p(\Omega)$, where $v$ is the velocity of the fluid and $\rho$ is the density, that is a small perturbation of a constant flow $(\bar v \equiv [1,0,0], \bar \rho \equiv 1)$. We also show that this solution is unique in a class of small perturbations of $(\bar v,\bar \rho)$. The term $u \cdot \nabla w$ in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence $(v^n,\rho^n)$ that is bounded in $W^2_p(\Omega) \times W^1_p(\Omega)$ and satisfies the Cauchy condition in a larger space $L_{\infty}(0,L;L_2(\Omega_0))$ what enables us to deduce that the weak limit of a subsequence of $(v^n,\rho^n)$ is in fact a strong solution to our problem.Comment: 27 pages, 1 figure. Proof of Theorem 1 corrected, some misprints remove

On steady solutions to a model of chemically reacting heat conducting compressible mixture with slip boundary conditions

We consider a model of chemically reacting heat conducting compressible mixture. We investigate the corresponding system of partial differential equations in the steady regime with slip boundary conditions for the velocity and, in dependence on the model parameters, we establish existence of either weak or variational entropy solutions. The results extend the range of parameters for which the existence of weak solutions is known in the case of homogeneous Dirichlet boundary conditions for the velocity.Comment: arXiv admin note: text overlap with arXiv:1612.0544

Steady compressible Navier–Stokes flow in a square

AbstractWe investigate a steady flow of compressible fluid with inflow boundary condition on the density and slip boundary conditions on the velocity in a square domain Q∈R2. We show existence if a solution (v,ρ)∈Wp2(Q)×Wp1(Q) that is a small perturbation of a constant flow (v¯≡[1,0], ρ¯≡1). We also show that this solution is unique in a class of small perturbations of the constant flow (v¯,ρ¯). In order to show the existence of the solution we adapt the techniques known from the theory of weak solutions. We apply the method of elliptic regularization and a fixed point argument

On strong dynamics of compressible two-component mixture flow

We investigate a system describing the flow of a compressible two-component mixture. The system is composed of the compressible Navier-Stokes equations coupled with non-symmetric reaction-diffusion equations describing the evolution of fractional masses. We show the local existence and, under certain smallness assumptions, also the global existence of unique strong solutions in $L_p-L_q$ framework. Our approach is based on so called entropic variables which enable to rewrite the system in a symmetric form. Then, applying Lagrangian coordinates, we show the local existence of solutions applying the $L_p$-$L_q$ maximal regularity estimate. Next, applying exponential decay estimate we show that the solution exists globally in time provided the initial data is sufficiently close to some constants. The nonlinear estimates impose restrictions $2<p<\infty, \ 3<q<\infty$. However, for the purpose of generality we show the linear estimates for wider range of $p$ and $q$