The hydrostatic approximation for the primitive equations by the scaled Navier–Stokes equations under the no-slip boundary condition
The hydrostatic approximation for the primitive equations by the scaled Navier–Stokes equations under the no-slip boundary condition
Abstract In this paper, we justify the hydrostatic approximation of the primitive equations in maximal $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:math> - $$L^q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mi>q</mml:mi></mml:msup></mml:math> -settings in the three-dimensional layer domain $$\varOmega = \mathbb {T} ^2 \times (-1, 1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Ω</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> under the no-slip (Dirichlet) boundary condition in any time interval (0, T …