Well-Posedness and Exponential Stability for Boussinesq Systems on Real Hyperbolic Manifolds and Application

We investigate the global existence and exponential decay of mild solutions for the Boussinesq systems in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mspace width=".1em"/> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:math> -phase spaces on the framework of a real hyperbolic manifold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>d</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>2</mml:mn> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>⩽</mml:mo> <mml:mi>d</mml:mi> </mml:math> . We consider a couple of Ebin–Marsden’s Laplace and Laplace–Beltrami operators associated with the corresponding linear system which provides a vectorial matrix semigroup. First, we show the existence and the uniqueness of the bounded mild solution for the linear system by using dispersive and smoothing estimates of the vectorial matrix semigroup. Next, using the fixed point arguments, we can pass from the linear system to the semilinear system to establish the existence of the bounded mild solutions. By using Gronwall’s inequality, we establish the exponential stability of such solutions. Finally, we give an application of stability to the existence of periodic mild solutions for the Boussinesq systems.