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Local and Global Existence of Strong Solutions Near General Equilibria

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Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

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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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Notes

  1. 1.

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

  2. 2.

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

  3. 3.

    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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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Konstanz, D-78457, Konstanz, Germany

    Matthias Kotschote

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  1. Tokyo, Tokyo, Japan

    Yoshikazu Giga

  2. Toulon, Var, France

    Antonin Novotny

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Kotschote, M. (2017). Local and Global Existence of Strong Solutions Near General Equilibria. In: Giga, Y., Novotny, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-10151-4_50-1

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  • DOI: https://doi.org/10.1007/978-3-319-10151-4_50-1

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