The inviscid inflow-outflow problem via analyticity

Abstract We consider the incompressible Euler equations on an analytic domain $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> with a nonhomogeneous boundary condition $$u\cdot {\textsf{n}} = {\overline{u}}\cdot {\textsf{n}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>·</mml:mo> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mover> <mml:mi>u</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>·</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> on $$\partial \Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> , where $${\overline{u}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>u</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> is a given divergence-free analytic vector field. We establish the local well-posedness for u in analytic spaces without any compatibility conditions in all space dimensions. We also prove the global well-posedness in the 2D case if $${\overline{u}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>u</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> decays in time sufficiently fast.