Well-Posedness of a Generalized Stokes Operator on Smooth Bounded Domains via Layer-Potentials

Abstract We prove the invertibility of the relevant single and double layer potentials associated to some generalizations of the Stokes operator on bounded domains. In order to do that, we first develop an “algebra tool kit” to deal with limit and jump relations of layer operators. We do that first on $$\mathbb {R}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> for operators acting on a distribution supported on $$\{x_{n} = 0\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> and then in general on (possibly non-compact) manifolds. We use these results to study the limit and jump relations of the layer potential operators associated to our generalized Stokes operators. As an application, we obtain well-posedness results for the Stokes system.