The Well Posedness for Nonhomogeneous Boussinesq Equations

The Well Posedness for Nonhomogeneous Boussinesq Equations

This paper is devoted to studying the Cauchy problem for non-homogeneous Boussinesq equations. We built the results on the critical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow>...

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Autores principales: Yan Liu, Baiping Ouyang
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
non homogenous boussinesq equations
global well-posedness
littlewood-paley decomposition
Mathematics
QA1-939
Acceso en línea:https://doaj.org/article/4fef51d036ba43dda3c7a13d1c5e8ac2
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spelling oai:doaj.org-article:4fef51d036ba43dda3c7a13d1c5e8ac22021-11-25T19:06:49ZThe Well Posedness for Nonhomogeneous Boussinesq Equations10.3390/sym131121102073-8994https://doaj.org/article/4fef51d036ba43dda3c7a13d1c5e8ac22021-11-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2110https://doaj.org/toc/2073-8994This paper is devoted to studying the Cauchy problem for non-homogeneous Boussinesq equations. We built the results on the critical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>∈</mo><msubsup><mi>L</mi><mi>T</mi><mo>∞</mo></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi></mrow></msubsup><mo>)</mo></mrow><mo>×</mo><msubsup><mi>L</mi><mi>T</mi><mo>∞</mo></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow><mo>⋂</mo><msubsup><mi>L</mi><mi>T</mi><mn>1</mn></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn><mi>N</mi></mrow></semantics></math></inline-formula>. We proved the global existence of the solution when the initial velocity is small with respect to the viscosity, as well as the initial temperature approaches a positive constant. Furthermore, we proved the uniqueness for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo>≤</mo><mi>N</mi></mrow></semantics></math></inline-formula>. Our results can been seen as a version of symmetry in Besov space for the Boussinesq equations.Yan LiuBaiping OuyangMDPI AGarticlenon homogenous boussinesq equationsglobal well-posednesslittlewood-paley decompositionMathematicsQA1-939ENSymmetry, Vol 13, Iss 2110, p 2110 (2021)
institution DOAJ
collection DOAJ
language EN
topic non homogenous boussinesq equations
global well-posedness
littlewood-paley decomposition
Mathematics
QA1-939
spellingShingle non homogenous boussinesq equations
global well-posedness
littlewood-paley decomposition
Mathematics
QA1-939
Yan Liu
Baiping Ouyang
The Well Posedness for Nonhomogeneous Boussinesq Equations
description This paper is devoted to studying the Cauchy problem for non-homogeneous Boussinesq equations. We built the results on the critical Besov spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>θ</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>∈</mo><msubsup><mi>L</mi><mi>T</mi><mo>∞</mo></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi></mrow></msubsup><mo>)</mo></mrow><mo>×</mo><msubsup><mi>L</mi><mi>T</mi><mo>∞</mo></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow><mo>⋂</mo><msubsup><mi>L</mi><mi>T</mi><mn>1</mn></msubsup><mrow><mo>(</mo><msubsup><mover accent="true"><mi>B</mi><mo>˙</mo></mover><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn><mi>N</mi></mrow></semantics></math></inline-formula>. We proved the global existence of the solution when the initial velocity is small with respect to the viscosity, as well as the initial temperature approaches a positive constant. Furthermore, we proved the uniqueness for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo>≤</mo><mi>N</mi></mrow></semantics></math></inline-formula>. Our results can been seen as a version of symmetry in Besov space for the Boussinesq equations.
format article
author Yan Liu
Baiping Ouyang
author_facet Yan Liu
Baiping Ouyang
author_sort Yan Liu
title The Well Posedness for Nonhomogeneous Boussinesq Equations
title_short The Well Posedness for Nonhomogeneous Boussinesq Equations
title_full The Well Posedness for Nonhomogeneous Boussinesq Equations
title_fullStr The Well Posedness for Nonhomogeneous Boussinesq Equations
title_full_unstemmed The Well Posedness for Nonhomogeneous Boussinesq Equations
title_sort well posedness for nonhomogeneous boussinesq equations
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/4fef51d036ba43dda3c7a13d1c5e8ac2
work_keys_str_mv AT yanliu thewellposednessfornonhomogeneousboussinesqequations
AT baipingouyang thewellposednessfornonhomogeneousboussinesqequations
AT yanliu wellposednessfornonhomogeneousboussinesqequations
AT baipingouyang wellposednessfornonhomogeneousboussinesqequations
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