首页|Nonlinear asymptotic profiles for the Navier--Stokes--Coriolis\newline equations with horizontal viscosity on $\mathbb T^2\times\mathbb R$
Nonlinear asymptotic profiles for the Navier--Stokes--Coriolis\newline equations with horizontal viscosity on $\mathbb T^2\times\mathbb R$
Zefu Feng
Nonlinear asymptotic profiles for the Navier--Stokes--Coriolis\newline equations with horizontal viscosity on $\mathbb T^2\times\mathbb R$
Nonlinear asymptotic profiles for the Navier--Stokes--Coriolis\newline equations with horizontal viscosity on $\mathbb T^2\times\mathbb R$
摘要
We determine the nonlinear large-time profile of the incompressibleNavier--Stokes--Coriolis equations with horizontal viscosity on$\mathbb T^2\times\mathbb R$ for an arbitrary constant rotation rate. Writing$u=\bar u+\widetilde u$ and denoting by $R_\Omega(t)$ the planar inertialrotation, we prove that\[ R_\Omega(-t)\bar u_h(t)\longrightarrow A_\infty =\bar u_h(0)-\int_0^\infty R_\Omega(-s)\partial_3 \overline{\widetilde u_3\widetilde u_h}(s)\,\mathrm ds,\]and that the full solution converges exponentially to the inertial orbit$(R_\Omega(t)A_\infty,0)$. Thus the limiting profile is selected by theaccumulated rotated Reynolds stress and is not, in general, determined by theinitial horizontal mean. For zero-horizontal-mean data $u_0=af$, the profilemap has the quadratic expansion$A_\infty=a^2\mathcal B_\Omega(f)+O_{H^{m-2}}(a^3)$ for every integer$m\ge2$. For each viscosity and rotation rate, we also construct arbitrarilysmall smooth data for which $A_\infty\ne0$, showing that interactions amongdamped horizontal modes can generate a persistent inertial oscillation fromzero initial mean. These asymptotic results are built on a global stabilitytheory at the $H^2$ level: sufficiently small divergence-free $H^m$ datagenerate a unique global solution, with a smallness threshold independent ofthe rotation rate, and the nonzero horizontal modes decay exponentially in$H^{m-1}$. The analysis uses the horizontal spectral gap, anisotropic productestimates, and mean--oscillation cancellations, without dispersive estimatesor a fast-rotation assumption. Fourier pseudo-spectral computations within anexact $x_2$-independent invariant subspace quantitatively support theReynolds-stress representation and the quadratic scaling of the selectedamplitude, and illustrate the dependence of its leading-order profile on therotation rate and horizontal viscosity.