Stability and exponential decay for two-dimensional Boussinesq equations with partial viscosity and thermal diffusion

In this work, we study the two-dimensional anisotropic Boussinesq equations, incorporating only horizontal dissipation in the tangential velocity and horizontal thermal diffusion. When the spatial domain is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">T</m:mi> <m:mo>×</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:math> T×R , this paper addresses the stability problem and characterizes the exact long-time dynamics of the perturbations without needing any structural assumptions on the initial data. Furthermore, we show that the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mfenced close="‖" open="‖"> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>∂</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>∂</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mi>θ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msub> </m:math> (∂ ₁u,∂ ₁θ )_H¹ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mfenced close="‖" open="‖"> <m:mrow> <m:msubsup> <m:mrow> <m:mi>∂</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msubsup> <m:mi>θ</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msub> </m:math> ∂ ₁³θ _L² decay exponentially in time. As a result, the H 1 -norm of the oscillatory part <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>̃</m:mo> </m:mover> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mo>̃</m:mo> </m:mover> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (ũ,θ ̃) also decays exponentially to zero, and thus ( u , θ ) converges to its horizontal average <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>̄</m:mo> </m:mover> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mover accent="true"> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mo>̄</m:mo> </m:mover> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (ū,θ ̄) . In particular, we improve the result of Dong–Wu–Xu–Zhu (Calc. Var. Partial Differ. Equations 60 (2021), no. 3, 116), where the horizontal dissipation for normal velocity is needed. This result illustrates the stratification phenomenon that is typical of buoyancy-driven fluids.