A logarithmically improved regularity criterion for the Boussinesq equations in a bounded domain

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ORIGINAL PAPER

A logarithmically improved regularity criterion for the Boussinesq equations in a bounded domain Ahmad M. Alghamdi1 • Sadek Gala2,3



Maria Alessandra Ragusa3,4

Received: 24 January 2020 / Accepted: 5 October 2020 Ó Springer Nature Switzerland AG 2020

Abstract The paper is concerned with the regularity of solutions of the Boussinesq equations for incompressible fluids without heat conductivity. The main goal is to prove a regularity criterion in terms of the vorticity for the initial boundary value problem in a bounded domain X of R3 with Navier-type boundary conditions and we prove that if Z T kxð; tÞkBMOðXÞ   dt\1; 0 log e þ kxð; tÞk BMOðXÞ where x :¼curl u is the vorticity, then the unique local in time smooth solution of the 3D Boussinesq equations can be prolonged up to any finite but arbitrary time. Keywords Boussinesq equations  Bounded domain  Smooth solutions  BMO spaces Mathematics Subject Classification 35Q35  76D03

This article is part of the section ‘‘Applications of PDEs’’ edited by Hyeonbae Kang. & Sadek Gala [email protected] Ahmad M. Alghamdi [email protected] Maria Alessandra Ragusa [email protected] 1

Department of Mathematical Science, Faculty of Applied Science, Umm Alqura University, P. O. Box 14035, Mecca 21955, Saudi Arabia

2

Department of Mathematics, ENS of Mostaganem, Box 227, 27000 Mostaganem, Algeria

3

Dipartimento di Matematica e Informatica, Universita` di Catania, Viale Andrea Doria, 6, 95125 Catania, Italy

4

RUDN University, 6 Miklukho, Maklay St, Moscow 117198, Russia SN Partial Differential Equations and Applications

41 Page 2 of 11

SN Partial Differ. Equ. Appl. (2020)1:41

1 Introduction and main result. Let X be a bounded, simply connected domain in R3 with oX 2 C1 and n ¼ ðn1 ; n2 ; n3 Þ be the outward unit normal vector field along boundary oX. In this note, we consider the classical problem of regularity conditions for fluid mechanics equations. Precisely, we consider the initial boundary value problem for the 3D Boussinesq equations without heat conductivity modeling the flow of an incompressible fluid with Navier-type boundary conditions : 8 ot u þ ðu  rÞu  Du þ rp ¼ he3 ; in X  ð0; 1Þ; > > > > > in X  ð0; 1Þ; > < ot h þ ðu  rÞh ¼ 0; ð1:1Þ r  u ¼ 0; in X  ð0; 1Þ; > > > > u  n ¼ 0; ðr  uÞ  n ¼ 0; on oX; > > : ðu; hÞðx; 0Þ ¼ ðu0 ; h0 Þ in X; where u ¼ uðx; tÞ and h ¼ hðx; tÞ denote the unknown velocity vector field and the scalar temperature. Initial data u0 is assumed to satisfy a compatibility condition : r  u0 ðxÞ ¼ 0 in X. e3 ¼ ð0; 0; 1Þt . p ¼ pðx; tÞ is the pressure of fluid at the point ðx; tÞ 2 X  ð0; 1Þ. The Boussinesq system has important roles in atmospheric sciences (see, e.g., [18]). When h ¼ 0, (1.1)1 and (1.1)3 are the well-known Navier-Stokes system. Giga [13], Kim [15] and Kang and Kim [14] have proved some Serrin type regularity criteria. These type of regularity results are very well-known in literature and they all started with the improvement of

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