Time global existence and finite time blow-up criterion for solutions to the Keller-Segel system coupled with the Navier-Stokes fluid

Hideo Kozono, Masanari Miura, Yoshie Sugiyama

Research output: Contribution to journalArticle

Abstract

We will deal with the chemotaxis model under the effect of the Navier-Stokes fluid, i.e., the incompressible viscous fluid. We shall show the existence of a local mild solution for large initial data and a global mild solution for small initial data in the scale invariant class demonstrating that n0∈L1(R2) and u0∈Lσ 2(R2). Our method is based on the perturbation of linearization together with the Lp−Lq-estimates of the heat semigroup. As a by-product of our method, we shall prove the smoothing effect and uniqueness of our mild solution. In addition, we shall show a blow-up criterion which almost covers the well-known threshold number 8π of the size ‖n0L1(R2) under the rest state of the fluid motion. Furthermore, the blow-up rate will be also discussed.

Original languageEnglish
JournalJournal of Differential Equations
DOIs
Publication statusPublished - 2019 Jan 1

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Blow-up Criterion
Finite Time Blow-up
Mild Solution
Navier-Stokes
Coupled System
Global Existence
Fluid
Fluids
Heat Semigroup
Smoothing Effect
Blow-up Rate
Chemotaxis
Scale Invariant
Linearization
Viscous Fluid
Incompressible Fluid
Byproducts
Uniqueness
Cover
Perturbation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "We will deal with the chemotaxis model under the effect of the Navier-Stokes fluid, i.e., the incompressible viscous fluid. We shall show the existence of a local mild solution for large initial data and a global mild solution for small initial data in the scale invariant class demonstrating that n0∈L1(R2) and u0∈Lσ 2(R2). Our method is based on the perturbation of linearization together with the Lp−Lq-estimates of the heat semigroup. As a by-product of our method, we shall prove the smoothing effect and uniqueness of our mild solution. In addition, we shall show a blow-up criterion which almost covers the well-known threshold number 8π of the size ‖n0‖L1(R2) under the rest state of the fluid motion. Furthermore, the blow-up rate will be also discussed.",
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AU - Miura, Masanari

AU - Sugiyama, Yoshie

PY - 2019/1/1

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