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An Analysis of the Evolution Equations for a Generalized Bioconvective Flow

J. L. Boldrini, B. Climent-Ezquerra, M. A. Rojas-Medar, M. D. Rojas-Medar

TL;DR

It is proved results on existence and uniqueness of solutions of a system of equations modeling the evolution of a generalized bioconvective flow that describes the convective motion generated by the upward swimming of a culture of microorganisms under the influence of vertical gravitational forces.

Abstract

We prove results on existence and uniqueness of solutions of a system of equations modeling the evolution of a generalized bioconvective flow. The mathematical model considered in the present work describes the convective motion generated by the upward swimming of a culture of microorganisms under the influence of vertical gravitational forces, in an incompressible viscous fluid whose viscosity may depend on the concentration of microorganisms.

An Analysis of the Evolution Equations for a Generalized Bioconvective Flow

TL;DR

It is proved results on existence and uniqueness of solutions of a system of equations modeling the evolution of a generalized bioconvective flow that describes the convective motion generated by the upward swimming of a culture of microorganisms under the influence of vertical gravitational forces.

Abstract

We prove results on existence and uniqueness of solutions of a system of equations modeling the evolution of a generalized bioconvective flow. The mathematical model considered in the present work describes the convective motion generated by the upward swimming of a culture of microorganisms under the influence of vertical gravitational forces, in an incompressible viscous fluid whose viscosity may depend on the concentration of microorganisms.
Paper Structure (18 sections, 26 theorems, 243 equations)

This paper contains 18 sections, 26 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Notation, functional spaces and some technical lemmas
  3. Notation and some interpolation results
  4. Specific functional spaces and some known technical results
  5. Existence of global weak evolution solutions
  6. Proof Theorem Theorem \ref{['WeakSolutionExistence']}
  7. Existence of local strong evolution solutions
  8. Proof of Theorem \ref{['localsolMainResults']}
  9. Preparation for the proof of Proposition \ref{['localsol']}
  10. Higher order estimates
  11. Proof of Proposition \ref{['localsol']}
  12. Proof of the uniqueness of strong evolution solutions
  13. Existence of global strong evolution solutions
  14. Proof of existence of global strong evolution solutions
  15. Exponential $L^2$-stability
  16. ...and 3 more sections

Key Result

Lemma 1

(Some interpolation inequalities Adams adams)

Theorems & Definitions (32)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • ...and 22 more