On partial regularity problem for 3D Boussinesq equations

Daoyuan Fang, Chun Liu, Chenyin Qian

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In this paper, we study the partial regularity of the solutions for the three-dimensional Boussinesq equations. We first prove a criterion of local Hölder continuous of the suitable weak solutions of the Boussinesq equations, and show that one-dimensional Hausdorff measure of the singular point set is zero. Secondly, we present a local uniform gradient estimate on the suitable weak solutions and assert that the local behavior of the solution can be dominated by some scaled quantities, such as the scaled local L3-norm of the velocity. Besides, when the initial data v0 and θ0 decay sufficiently rapidly at ∞, the distribution of the regular point set of the suitable weak solutions is also considered. Based on it, one can find that MHD equations are more similar to Navier–Stokes equations than Boussinesq equations. Finally, we give a local regularity criterion of the suitable weak solutions near the boundary.

Original languageEnglish (US)
Pages (from-to)4156-4221
Number of pages66
JournalJournal of Differential Equations
Volume263
Issue number7
DOIs
StatePublished - Oct 5 2017

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Partial Regularity
Suitable Weak Solutions
Boussinesq Equations
Point Sets
MHD Equations
Magnetohydrodynamics
Regularity Criterion
Regular Sets
Gradient Estimate
Singular Set
Hausdorff Measure
Uniform Estimates
Singular Point
Navier-Stokes Equations
Decay
Norm
Three-dimensional
Zero

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

Fang, Daoyuan ; Liu, Chun ; Qian, Chenyin. / On partial regularity problem for 3D Boussinesq equations. In: Journal of Differential Equations. 2017 ; Vol. 263, No. 7. pp. 4156-4221.
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On partial regularity problem for 3D Boussinesq equations. / Fang, Daoyuan; Liu, Chun; Qian, Chenyin.

In: Journal of Differential Equations, Vol. 263, No. 7, 05.10.2017, p. 4156-4221.

Research output: Contribution to journalArticle

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