| Summary: | 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.
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