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The Large Flux Problem to the Navier-Stokes Equations

Global Strong Solutions in Cylindrical Domains

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  • © 2019

Overview

Authors:
  1. Joanna Rencławowicz

    1. Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

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  2. Wojciech M. Zajączkowski

    1. Institute of Mathematics, Polish Academy of Sciences, Institute of Mathematics and Cryptology, Military University of Technology, Warsaw, Poland

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Considers the motion of incompressible fluids described by the Navier-Stokes equations with large inflow and outflow
  • Proves global existence of regular solutions without any restrictions on the magnitude of the initial velocity, the external force, or the flux
  • Utilizes a sophisticated method of increasing regularity of weak solutions through an application of weighted Sobolev spaces

Part of the book series: Advances in Mathematical Fluid Mechanics (AMFM)

Part of the book sub series: Lecture Notes in Mathematical Fluid Mechanics (LNMFM)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-vi
    Download chapter PDF

  2. Introduction

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 1-9

  3. Notation and Auxiliary Results

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 11-30

  4. Energy Estimate: Global Weak Solutions

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 31-57

  5. Local Estimates for Regular Solutions

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 59-82

  6. Global Estimates for Solutions to Problem on (v, p)

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 83-93

  7. Global Estimates for Solutions to Problem on (h, q)

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 95-97

  8. Estimates for h t

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 99-105

  9. Auxiliary Results: Estimates for (v, p)

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 107-115

  10. Auxiliary Results: Estimates for (h, q)

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 117-128

  11. The Neumann Problem (3.6) in L 2-Weighted Spaces

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 129-141

  12. The Neumann Problem (3.6) in L p-Weighted Spaces

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 143-155

  13. Existence of Solutions (v, p) and (h, q)

    • Joanna Rencławowicz, Wojciech M. Zajączkowski
    Pages 157-168

  14. Back Matter

    Pages 169-179
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Keywords

About this book

This monograph considers the motion of incompressible fluids described by the Navier-Stokes equations with large inflow and outflow, and proves the existence of global regular solutions without any restrictions on the magnitude of the initial velocity, the external force, or the flux. To accomplish this, some assumptions are necessary: The flux is close to homogeneous, and the initial velocity and the external force do not change too much along the axis of the cylinder. This is achieved by utilizing a sophisticated method of deriving energy type estimates for weak solutions and global estimates for regular solutions—an approach that is wholly unique within the existing literature on the Navier-Stokes equations. To demonstrate these results, three main steps are followed: first, the existence of weak solutions is shown; next, the conditions guaranteeing the regularity of weak solutions are presented; and, lastly, global regular solutions are proven. This volume is ideal for mathematicians whose work involves the Navier-Stokes equations, and, more broadly, researchers studying fluid mechanics.

Authors and Affiliations

  • Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

    Joanna Rencławowicz

  • Institute of Mathematics, Polish Academy of Sciences, Institute of Mathematics and Cryptology, Military University of Technology, Warsaw, Poland

    Wojciech M. Zajączkowski

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