| Summary: | This paper is concerned with the nonlinear stability and instability of the two-dimensional (2D) Boussinesq-MHD equations around the equilibrium state <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mover> <mi>u</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"></mspace> <mover> <mi>B</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"></mspace> <mover> <mi>θ</mi> <mo>¯</mo> </mover> <mo>=</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> with the temperature-dependent fluid viscosity, thermal diffusivity and electrical conductivity in a channel. We prove that if <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mo>+</mo> </msub> <mo>≥</mo> <msub> <mi>a</mi> <mo>−</mo> </msub> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <mfrac> <msup> <mi>d</mi> <mn>2</mn> </msup> <mrow> <mi>d</mi> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>κ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> or <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo><</mo> <mfrac> <msup> <mi>d</mi> <mn>2</mn> </msup> <mrow> <mi>d</mi> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>κ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>β</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula>, with <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>β</mi> <mn>0</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> small enough constant, and then this equilibrium state is nonlinearly asymptotically stable, and if <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mo>+</mo> </msub> <mo><</mo> <msub> <mi>a</mi> <mo>−</mo> </msub> </mrow> </semantics> </math> </inline-formula>, this equilibrium state is nonlinearly unstable. Here, <inline-formula> <math display="inline"> <semantics> <msub> <mi>a</mi> <mo>+</mo> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>a</mi> <mo>−</mo> </msub> </semantics> </math> </inline-formula> are the values of the equilibrium temperature <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> on the upper and lower boundary.
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