## 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 $$\Omega\subset \mathbb{R}^3$$ is described by a triple of functions – the density $$\rho(x,t)$$, the velocity $$u(x,t)$$, the temperature $$\vartheta(x,t)$$. These functions satisfy the Navier-Stokes-Fourier system of equations \begin{aligned} \frac{\partial \rho}{\partial t}&+\text{div}\,(\rho u)=0,\\ \frac{\partial }{\partial t}(\rho u)+\text{div}\,(\rho u\otimes u)&+ \frac{1}{\text{Ma}^2}\nabla p(\rho,\vartheta)=\text{div}\,S+\frac{1}{\text{Fr}^2}\rho\nabla F,\\ \frac{\partial }{\partial t}(\rho s(\rho,\vartheta))&+\text{div}\,(\rho s(\rho,\vartheta)u)+\text{div}\,\left(\frac{q}{\vartheta}\right)=\sigma,\\ \frac{d}{dt}\int_\Omega&\left(\frac{\text{Ma}^2}{2}\rho | u| ^2 +\rho e(\rho,\vartheta)-\frac{\text{Ma}^2}{\text{Fr}^2}\rho F \right)dx=0, \end{aligned}\tag{1} where $$S$$ is a viscous stress tensor, the heat flux $$q$$ obeys Fourier’s law $q=-k(\vartheta)\nabla\vartheta,$ $$\sigma$$ 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 $$\bar{\vartheta}$$ and $$\bar{\rho}$$ are average quantities $\bar{\vartheta}=\frac{1}{| \Omega| }\int_\Omega\vartheta\,dx,\quad \bar{\rho}=\frac{1}{| \Omega| }\int_\Omega\rho\,dx.$ Setting Ma=$$\varepsilon$$, Fr=$$\sqrt{\varepsilon}$$, where $$\varepsilon$$ is a small parameter, the triple of unknown functions is represented by $\rho=\rho_\varepsilon=\bar{\rho}+\varepsilon r_\varepsilon,\quad u=u_\varepsilon,\quad \vartheta=\vartheta_\varepsilon=\bar{\vartheta}+\varepsilon \theta_\varepsilon.$ It is proved that the limits $r_\varepsilon\rightarrow r,\quad u_\varepsilon\rightarrow u,\quad \theta_\varepsilon\rightarrow \theta \quad \text{as}\;\varepsilon\rightarrow 0$ satisfy to the Oberbeck-Boussinesq system \begin{aligned} \text{div}\,u=0&,\\ \bar{\rho}\big(\frac{\partial u}{\partial t}+\text{div}\,(u\otimes u)\big)+\nabla P&=\bar{\mu}\Delta u-r\nabla F,\\ \bar{\rho}\bar{c}_p\big(\frac{\partial \theta}{\partial t}+\text{div}\,(\theta u)\big)-& \text{div}\,(\bar{k}\nabla \theta)=0,\\ r+\bar{\rho}\bar{\alpha}(\theta-\bar{\theta})&=0, \end{aligned} where the viscosity coefficient $$\bar{\mu}$$, the specific heat at constant pressure $$\bar{c}_p$$, the heat conductivity coefficient $$\bar{k}$$ and the coefficient of thermal expansion $$\bar{\alpha}$$ are evaluated at $$\bar{\rho},\bar{\vartheta}$$.
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 Navier-Stokes equations 35B35 Stability in context of PDEs 76N15 Gas dynamics (general theory)
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### References:

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