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Navier-Stokes equations (NSE) for compressible flow

The NSE for compressible flow in a bounded domain $G\subset \R^3$ express the law of conservation of mass:
\begin{eqnarray*}\partial _t\rho + div(\rho u)=0\end{eqnarray*}
and balance of momenta
\begin{eqnarray*}\partial _t(\rho u) +div(\rho u\otimes u)-\mu\Delta -(\lambda +\mu)\nabla div u+ a\nabla \rho^\gamma=\rho f\end{eqnarray*}
Here denotes $\rho=\rho(t,x)$ the density, $u=u(t,x)$ the velocity. $t\in\R^+$ and $x\in G$. Assume the no-slip condition
\begin{eqnarray*}u\vert _\partial G=0\end{eqnarray*}
The viscosity coefficients $\lambda ,\mu$ satisfy the positivity conditions $\lambda +\mu\ge 0$ and $\mu >0$. The adiabatic constant $\gamma >1$ and $a>0$. Given is the force density $f$ which should be bounded and measurable i.g.
\begin{eqnarray*}\begin{array}{l}{ess\; sup}\\ { t\in \R^+,x\in G}\end{array}\vert f(t,x)\vert\le K\end{eqnarray*}
Existence theory is done for so-called finite energy solutions$(\rho,u)$ i.e.
\begin{eqnarray*}&&\rho\ge 0,\quad \rho\in L_{loc}^\infty(\R^+;L^\gamma(G)), u^......nt\limits _G\vert div u\vert^2dx\le\int\limits _G\rho f\cdot udx\end{eqnarray*}
Last inequality is satisfied in ${\cal D}'(\R^+)$. The energy $E$ is given by
\begin{eqnarray*}E(t)=E[\rho,u](t)=\frac12\int\limits _G\rho(t)\vert u(t)\vert^2dx+\frac{a}{\gamma-1}\int\limits _G\rho^\gamma(t)dx.\end{eqnarray*}
A detailed description the reader can find in the book by P.-L. Lions (1998) Mathematical Topics in Fluid Dynamics, Vol 2, Compressible Models, Oxford Science Publication, Oxford.
 
 

A late reference: E. Feiereisl, (2000) Global attractors for the Navier-Stokes equations of three-dimensional compressible flow, Équations aux dérivées partielles, C.R. Acad. Sci Paris, t.331, Série I, p. 35-39.
 


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Nächste Seite:Fluid flow in porousAufwärts:Models of Mathematical PhysicsVorherige Seite:Frictional viscoelastic contact problem
collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg