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A semigroup generated by the linearized Navier-Stokes equations for compressible fluid and its uniform growth bound in Hölder spaces. (English) Zbl 0954.35130
Salvi, Rodolfo (ed.), Navier-Stokes equations. Theory and numerical methods. Proceedings of the international conference, Varenna, Lecco, Italy, June 2-6, 1997. Harlow: Longman. Pitman Res. Notes Math. Ser. 388, 86-100 (1998).
A linearized version of the Navier-Stokes equations in a domain \(\Omega\) \[ {\partial u\over\partial t}+ u\cdot\nabla u= f-{\nabla p(\rho)\over \rho}+ {\mu_1\over \rho}\Delta u+{\mu_2\over \rho}\nabla\text{ div }u, \]
\[ {\partial\rho\over\partial t}+ \text{div}(\rho u)= 0, \] leads the autor to a semigroup of linear operators \(e^{Lt}\), on the space \[ {\mathcal X}= \left\{\begin{pmatrix} v\\ \sigma\end{pmatrix}; v\in C^{0+(\alpha)}(\overline\Omega)^3, \sigma\in C^{1+(\alpha)}(\overline\Omega), \int_\Omega\sigma dx= 0\right\}. \] He (laboriously) obtains an upper estimate for the uniform growth bound of the \(C_0\)-semigroup \(e^{Lt}\) \[ \omega_0= \inf\{\omega\in \mathbb{R}; \exists M_\omega> 0:\|e^{Lt}\|_{{\mathcal X}\to{\mathcal X}}\leq M_\omega e^{\omega t}\}. \]
For the entire collection see [Zbl 0927.00032].
MSC:
35Q30 Navier-Stokes equations
47D03 Groups and semigroups of linear operators
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