# Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations In this paper, we study the Cauchy problem for the 3D generalized Navier-Stokes-Boussinesq equations with fractional diffusion: $$\left\{ {\begin{array}{*{20}{c}}{{u_t} + \left( {u \cdot \nabla } \right)u + v{\Lambda ^{2a}}u = -\nabla p + \theta {e_3},\;{e_3} = {{\left( {0,0,1} \right)}^T},} \\ {{\theta _t} + \left( {u \cdot \nabla } \right)t = 0,} \\ {Divu = 0.} \end{array}} \right.$$ With the help of the smoothing effect of the fractional diffusion operator and a logarithmic estimate, we prove the global well-posedness for this system with α ≥ 5/4. Moreover, the uniqueness and continuity of the solution with weaker initial data is based on Fourier localization technique. Our results extend ones on the 3D Navier-Stokes equations with fractional diffusion. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

, Volume 32 (1) – Apr 5, 2016
16 pages      /lp/springer-journals/global-well-posedness-for-3d-generalized-navier-stokes-boussinesq-akhcahO4Vj
Publisher
Springer Journals
Copyright © 2016 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-016-0539-z
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we study the Cauchy problem for the 3D generalized Navier-Stokes-Boussinesq equations with fractional diffusion: $$\left\{ {\begin{array}{*{20}{c}}{{u_t} + \left( {u \cdot \nabla } \right)u + v{\Lambda ^{2a}}u = -\nabla p + \theta {e_3},\;{e_3} = {{\left( {0,0,1} \right)}^T},} \\ {{\theta _t} + \left( {u \cdot \nabla } \right)t = 0,} \\ {Divu = 0.} \end{array}} \right.$$ With the help of the smoothing effect of the fractional diffusion operator and a logarithmic estimate, we prove the global well-posedness for this system with α ≥ 5/4. Moreover, the uniqueness and continuity of the solution with weaker initial data is based on Fourier localization technique. Our results extend ones on the 3D Navier-Stokes equations with fractional diffusion.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Apr 5, 2016

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