$L_p$-estimates of the resolvent of the Stokes operator in infinite tubes

The well-known $L_p$-estimates of the resolvent of the Stokes operator in bounded domains are extended to the case of an infinite tube. It is proved, in particular, that the Stokes operator generates an analytic subgroup in $L_p(\mathfrak D)$, $1<p<\infty$. To solve the equations defining the resolvent of the Stokes operator, a representation of the pressure as the sum of a terms fading at infinity and a term with non-zero pressure differential is obtained. An explicit formula for the pressure differential is presented.