Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spatially-Periodic Solutions for Evolution Anisotropic Variable-Coefficient Navier-Stokes Equations: I. Existence

Sergey E. Mikhailov

Abstract

We consider evolution (non-stationary) space-periodic solutions to the $n$-dimensional non-linear Navier-Stokes equations of anisotropic fluids with the viscosity coefficient tensor variable in space and time and satisfying the relaxed ellipticity condition. Employing the Galerkin algorithm with the basis constituted by the eigenfunctions of the periodic Bessel-potential operator, we prove the existence of a global weak solution.

Spatially-Periodic Solutions for Evolution Anisotropic Variable-Coefficient Navier-Stokes Equations: I. Existence

Abstract

We consider evolution (non-stationary) space-periodic solutions to the -dimensional non-linear Navier-Stokes equations of anisotropic fluids with the viscosity coefficient tensor variable in space and time and satisfying the relaxed ellipticity condition. Employing the Galerkin algorithm with the basis constituted by the eigenfunctions of the periodic Bessel-potential operator, we prove the existence of a global weak solution.
Paper Structure (15 sections, 10 theorems, 163 equations)

This paper contains 15 sections, 10 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Periodic function spaces
  3. Weak formulation of the evolution spatially-periodic anisotropic Navier-Stokes problem
  4. Existence for evolution spatially-periodic anisotropic Navier-Stokes problem
  5. (a)
  6. (b)
  7. (c)
  8. (d)
  9. Auxiliary results
  10. Advection term properties
  11. Some point-wise multiplication results
  12. Spectrum of the periodic Bessel potential operator
  13. Isomorphism of divergence and gradient operators in periodic spaces
  14. Some functional analysis results
  15. Data availability statement

Key Result

THEOREM 2.1

Let $s\in{\mathbb R}$ and $n\ge 2$. (a) The space $\dot{\mathbf H}_{\#}^{s}$ has the Helmholtz-Weyl decomposition, $\dot{\mathbf H}_{\#}^{s}=\dot{\mathbf H}_{\# g}^{s}\oplus\dot{\mathbf H}_{\# \sigma}^{s},$ that is, any $\mathbf F\in \dot{\mathbf H}_{\#}^{s}$ can be uniquely represented as $\mathbf

Theorems & Definitions (19)

  • THEOREM 2.1
  • DEFINITION 3.1
  • LEMMA 3.2
  • proof
  • REMARK 3.3
  • THEOREM 4.1
  • proof
  • THEOREM 5.1
  • proof
  • THEOREM 5.2
  • ...and 9 more