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On the incompressible limit for the Navier-Stokes-Fourier system in domains with wavy bottoms. (English) Zbl 1158.35072

The motion of a compressible viscous and heat conducting fluid occupying a domain ${\Omega }\subset {ℝ}^{3}$ is described by a triple of functions – the density $\rho \left(x,t\right)$, the velocity $u\left(x,t\right)$, the temperature $\vartheta \left(x,t\right)$. These functions satisfy the Navier-Stokes-Fourier system of equations

$\begin{array}{cc}\hfill \frac{\partial \rho }{\partial t}& +\text{div}\phantom{\rule{0.166667em}{0ex}}\left(\rho u\right)=0,\hfill \\ \hfill \frac{\partial }{\partial t}\left(\rho u\right)+\text{div}\phantom{\rule{0.166667em}{0ex}}\left(\rho u\otimes u\right)& +\frac{1}{{\text{Ma}}^{2}}\nabla p\left(\rho ,\vartheta \right)=\text{div}\phantom{\rule{0.166667em}{0ex}}S+\frac{1}{{\text{Fr}}^{2}}\rho \nabla F,\hfill \\ \hfill \frac{\partial }{\partial t}\left(\rho s\left(\rho ,\vartheta \right)\right)& +\text{div}\phantom{\rule{0.166667em}{0ex}}\left(\rho s\left(\rho ,\vartheta \right)u\right)+\text{div}\phantom{\rule{0.166667em}{0ex}}\left(\frac{q}{\vartheta }\right)=\sigma ,\hfill \\ \hfill \frac{d}{dt}{\int }_{{\Omega }}& \left(\frac{{\text{Ma}}^{2}}{2}\rho {|u|}^{2}+\rho e\left(\rho ,\vartheta \right)-\frac{{\text{Ma}}^{2}}{{\text{Fr}}^{2}}\rho F\right)dx=0,\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $S$ is a viscous stress tensor, the heat flux $q$ obeys Fourier’s law

$q=-k\left(\vartheta \right)\nabla \vartheta ,$

$\sigma$ is the entropy production, $p$ is the pressure, $s$ is the specific entropy, $e$ is the specific internal energy, Ma and Fr denote the Mach and Froude numbers.

Let $\overline{\vartheta }$ and $\overline{\rho }$ are average quantities

$\overline{\vartheta }=\frac{1}{|{\Omega }|}{\int }_{{\Omega }}\vartheta \phantom{\rule{0.166667em}{0ex}}dx,\phantom{\rule{1.em}{0ex}}\overline{\rho }=\frac{1}{|{\Omega }|}{\int }_{{\Omega }}\rho \phantom{\rule{0.166667em}{0ex}}dx·$

Setting Ma=$\epsilon$, Fr=$\sqrt{\epsilon }$, where $\epsilon$ is a small parameter, the triple of unknown functions is represented by

$\rho ={\rho }_{\epsilon }=\overline{\rho }+\epsilon {r}_{\epsilon },\phantom{\rule{1.em}{0ex}}u={u}_{\epsilon },\phantom{\rule{1.em}{0ex}}\vartheta ={\vartheta }_{\epsilon }=\overline{\vartheta }+\epsilon {\theta }_{\epsilon }·$

It is proved that the limits

${r}_{\epsilon }\to r,\phantom{\rule{1.em}{0ex}}{u}_{\epsilon }\to u,\phantom{\rule{1.em}{0ex}}{\theta }_{\epsilon }\to \theta \phantom{\rule{1.em}{0ex}}\text{as}\phantom{\rule{4pt}{0ex}}\epsilon \to 0$

satisfy to the Oberbeck-Boussinesq system

$\begin{array}{cc}\hfill \text{div}\phantom{\rule{0.166667em}{0ex}}u=0& ,\hfill \\ \hfill \overline{\rho }\left(\frac{\partial u}{\partial t}+\text{div}\phantom{\rule{0.166667em}{0ex}}\left(u\otimes u\right)\right)+\nabla P& =\overline{\mu }{\Delta }u-r\nabla F,\hfill \\ \hfill \overline{\rho }{\overline{c}}_{p}\left(\frac{\partial \theta }{\partial t}+\text{div}\phantom{\rule{0.166667em}{0ex}}\left(\theta u\right)\right)-& \text{div}\phantom{\rule{0.166667em}{0ex}}\left(\overline{k}\nabla \theta \right)=0,\hfill \\ \hfill r+\overline{\rho }\overline{\alpha }\left(\theta -\overline{\theta }\right)& =0,\hfill \end{array}$

where the viscosity coefficient $\overline{\mu }$, the specific heat at constant pressure ${\overline{c}}_{p}$, the heat conductivity coefficient $\overline{k}$ and the coefficient of thermal expansion $\overline{\alpha }$ are evaluated at $\overline{\rho },\overline{\vartheta }$.

It is shown that the oscillations of the sound waves are effectively damped by the presence of a “wavy bottom” of physical domain.

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 35B35 Stability of solutions of PDE 76N15 Gas dynamics, general