Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in ℝ n

Reinhard Farwig; Hermann Sohr

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 1, page 61-79
  • ISSN: 0011-4642

Abstract

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For a bounded domain Ω ⊂ ℝ n , n ≥ 3 , we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system - Δ u + u · ∇ u + ∇ p = f , div u = k , u | ∂ Ω = g with u ∈ L q , q ≥ n , and very general data classes for f , k , g such that u may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669–717, where the existence of a weak solution which is locally regular is proved.

How to cite

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Farwig, Reinhard, and Sohr, Hermann. "Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in $\mathbb {R}^n$." Czechoslovak Mathematical Journal 59.1 (2009): 61-79. <http://eudml.org/doc/37908>.

@article{Farwig2009,
abstract = {For a bounded domain $\Omega \subset \mathbb \{R\} ^n$, $n\ge 3,$ we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system $-\Delta u + u \cdot \nabla u + \nabla p=f$, $\operatorname\{div\}u = k$, $u_\{|_\{\partial \Omega \}\}=g$ with $u \in L^q$, $q \ge n$, and very general data classes for $f$, $k$, $g$ such that $u$ may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669–717, where the existence of a weak solution which is locally regular is proved.},
author = {Farwig, Reinhard, Sohr, Hermann},
journal = {Czechoslovak Mathematical Journal},
keywords = {stationary Stokes and Navier-Stokes system; very weak solutions; existence and uniqueness in higher dimensions; regularity classes in higher dimensions; stationary Stokes system; Navier-Stokes system, very weak solution; regularity class; higher dimension},
language = {eng},
number = {1},
pages = {61-79},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in $\mathbb \{R\}^n$},
url = {http://eudml.org/doc/37908},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Farwig, Reinhard
AU - Sohr, Hermann
TI - Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in $\mathbb {R}^n$
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 1
SP - 61
EP - 79
AB - For a bounded domain $\Omega \subset \mathbb {R} ^n$, $n\ge 3,$ we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system $-\Delta u + u \cdot \nabla u + \nabla p=f$, $\operatorname{div}u = k$, $u_{|_{\partial \Omega }}=g$ with $u \in L^q$, $q \ge n$, and very general data classes for $f$, $k$, $g$ such that $u$ may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669–717, where the existence of a weak solution which is locally regular is proved.
LA - eng
KW - stationary Stokes and Navier-Stokes system; very weak solutions; existence and uniqueness in higher dimensions; regularity classes in higher dimensions; stationary Stokes system; Navier-Stokes system, very weak solution; regularity class; higher dimension
UR - http://eudml.org/doc/37908
ER -

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