Regularity of weak solutions to the 3D stationary tropical climate model

Abstract This article studies the regularity of weak solutions to the 3D stationary tropical climate model. We prove that when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>U</m:mi> <m:mo>,</m:mo> <m:mi>V</m:mi> <m:mo>,</m:mo> <m:mi>θ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(U,V,\theta ) belongs to the homogeneous Morrey space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mo>˙</m:mo> </m:mrow> </m:mover> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\dot{M}}^{2,p}\left({{\mathbb{R}}}^{3}) with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>&gt;</m:mo> <m:mn>3</m:mn> </m:math> p\gt 3 , then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>U</m:mi> <m:mo>,</m:mo> <m:mi>V</m:mi> <m:mo>,</m:mo> <m:mi>θ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(U,V,\theta )\in {C}^{\infty }\left({{\mathbb{R}}}^{3}) .