On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations

Lucas C. F. Ferreira; Jhean E. Pérez-López; Élder J. Villamizar-Roa

doi:10.3934/cpaa.2018115

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2018, Volume 17, Issue 6: 2423-2439. Doi: 10.3934/cpaa.2018115

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On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations

1.
Universidade Estadual de Campinas, IMECC-Departamento de Matemática, CEP 13083-859, Campinas-SP, Brazil
2.
Universidad Industrial de Santander, Escuela de Matemáticas, A.A. 678, Bucaramanga, Colombia

* Corresponding author: Élder J. Villamizar-Roa
* Corresponding author: Élder J. Villamizar-Roa

Received: August 2017

Revised: February 2018

Published: June 2018

The first author has been partially supported by CNPq and FAPESP, Brazil. The second author has been supported by CNPq, Brazil, and by Universidad Industrial de Santander. The third autor has been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander, and Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas, contrato Colciencias FP 44842-157-2016.

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Abstract

This paper is devoted to the Boussinesq equations that models natural convection in a viscous fluid by coupling Navier-Stokes and heat equations via a zero order approximation. We consider the problem in $ \mathbb{R}^{n}$ and prove the existence of stationary solutions in critical Besov-Lorentz-Morrey spaces. For that, we prove some estimates for the product of distributions in these spaces, as well as Bernstein inequalities and Mihlin multiplier type results in our setting. Considering in particular the decoupled case, our existence result provides a new class of stationary solutions for the Navier-Stokes equations in critical spaces.

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Mathematics Subject Classification: Primary: 35Q35, 76Rxx; Secondary: 76D03, 42B35.

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References

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