Regularity and integrability of 3D Euler and Navier–Stokes equations for rotating fluids

Abstract

We consider 3D Euler and Navier–Stokes equations describing dynamics of uniformly rotating fluids. Periodic (as well as zero vertical flux) boundary conditions are imposed, the ratios of domain periods are assumed to be generic (non-resonant). We show that solutions of 3D Euler/Navier–Stokes equations can be decomposed as U(t,x 1 ,x 2 ,x 3 )=Ũ(t,x 1 ,x 2 ) +V(t,x 1 ,x 2 ,x 3 )+r, where Ũ is a solution of the 2D Euler/Navier–Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical e 3 ). The vector field V(t,x 1 ,x 2 ,x 3 ) is exactly solved in terms of the phases Ωt, τ 1 (t) and τ 2 (t). The phases τ 1 (t) and τ 2 (t) explicitly expressed in terms of vertically averaged vertical vorticity curl Ū(t)·e 3 and velocity Ū 3 (t). The remainder r is uniformly estimated from above by a majorant of order a 3 /Ω, a 3 is the vertical aspect ratio (shallowness) and Ω is non-dimensional rotation parameter based on horizontal scales. The resolution of resonances and a non-standard small divisor problem for 3D rotating Euler are the basis for error estimates. Contribution of 3-wave resonances is estimated in terms of the measure of almost resonant aspect ratios. Global solvability of the limit equations and estimates of the error r are used to prove existence on a long time interval T * of regular solutions to 3D Euler equations (T * →+∞, as 1/Ω→0); and existence on infinite time interval of regular solutions to 3D Navier–Stokes equations with smooth arbitrary initial data in the case of small 1/Ω.