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2018 | OriginalPaper | Chapter

36. Local and Global Existence of Strong Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Maximal Regularity

Author : Matthias Kotschote

Published in: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Publisher: Springer International Publishing

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Abstract

The purpose of this contribution is to show how the maximal regularity method can serve to prove existence of strong solutions to the Navier-Stokes equations. In order to illustrate the method, existence and uniqueness of global solutions to the Navier-Stokes equations for compressible fluids with or without heat conductivity in bounded domains shall be proved. The initial data have to be near equilibria that may be nonconstant due to considering large external potential forces. The exponential stability of equilibria in the phase space is shown and, above all, the problem is studied in Eulerian coordinates. The latter seems to be a novelty, since in works by other authors, global strong L_p-solutions have been investigated only in Lagrangian coordinates; Eulerian coordinates are even declared as impossible to deal with, cf. on page 418 in Mucha, Zaja̧czkowski (ZAMM 84(6):417–424, 2004). The proof is based on a careful derivation and study of the associated linear problem.

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previous chapter Fourier Analysis Methods for the Compressible Navier-Stokes Equations

next chapter Local and Global Solvability of Free Boundary Problems for the Compressible Navier-Stokes Equations Near Equilibria

Let A be the generator of the strongly continuous semigroup T(t). The growth bound ω₀(A) of A is defined by \(\omega _{0}(A) =\inf \{\omega \in \mathbb{R}:\exists M_{\omega } \geq 1,\text{ so that }\vert T(t)\vert \leq M_{\omega }e^{\omega t},\:\forall t \geq 0\}\).

Let x ∈ X an element of the Banach space X and X^∗ denotes its dual space. Then it is set \(\mathcal{J} (x) =\{ x^{{\ast}}\in X^{{\ast}}:\langle x\,,\,x^{{\ast}}\rangle _{X,X^{{\ast}}} = \vert x\vert _{X}^{2} = \vert x^{{\ast}}\vert _{X^{{\ast}}}^{2}\}\).

Although the duality spaces of \(\mathcal{X}_{p}^{2}\) and \(\mathcal{X}_{p}^{3}\) are not determined, one easily sees \(j(\theta ) \in \mathcal{J} (\theta )\) and \(j(\varrho ) \in \mathcal{J} (\varrho )\), e.g., there holds \(\langle \varrho \,,\,j(\varrho )\rangle _{\mathcal{X}_{p}^{3},(\mathcal{X}_{p}^{3})^{{\ast}}} =\|\varrho \|_{ \mathcal{X}_{p}^{3}}^{2}\) and \(\|\varrho \|_{\mathcal{X}_{p}^{3}} \leq \| j(\varrho )\|_{(\mathcal{X}_{p}^{3})} =\sup \{\langle \phi \,,\,j(\varrho )\rangle _{\mathcal{X}_{p}^{3},(\mathcal{X}_{p}^{3})^{{\ast}}}:\phi \in \mathcal{X}_{p}^{3},\vert \phi \vert _{\mathcal{X}_{p}^{3}} \leq 1\} \leq \|\varrho \|_{\mathcal{X}_{p}^{3}}\).

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Title: Local and Global Existence of Strong Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Maximal Regularity
Author: Matthias Kotschote
Publisher: Springer International Publishing
Book: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids
Print ISBN: 978-3-319-13343-0

Electronic ISBN: 978-3-319-13344-7

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-13344-7_50

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