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In this paper, we study stationary structures near the planar Couette flow in Sobolev spaces on a channel T × [ − 1 , 1 ] , and asymptotic behavior of Couette flow in Gevrey spaces on T × R for the β -plane equation. Let T > 0 be the horizontal period of the channel and α = 2 π T be the wave number. We obtain a sharp region O in the whole (α , β) half-plane such that non-shear traveling waves do not exist for (α , β) ∈ O and such traveling waves indeed exist for (α , β) in the remaining regions, near Couette flow for H ≥ 5 velocity perturbation. The borderlines between the region O and its remaining are determined by two curves of the principal eigenvalues of singular Rayleigh-Kuo operators. Our results reveal that there exists β ⁎ > 0 such that if | β | ≤ β ⁎ , then non-shear traveling waves do not exist for any T > 0 , while if | β | > β ⁎ , then there exists a critical period T β > 0 so that such traveling waves exist for T ∈ [ T β , ∞) and do not exist for T ∈ (0 , T β) , near Couette flow for H ≥ 5 velocity perturbation. This contrasting dynamics plays an important role in studying the long time dynamics near Couette flow with Coriolis effects. Moreover, for any β ≠ 0 and T > 0 , we prove that there exist no non-shear traveling waves with traveling speeds converging in (− 1 , 1) near Couette flow for H ≥ 5 velocity perturbation, in contrast to this, we construct non-shear stationary solutions near Couette flow for H < 5 2 velocity perturbation, which is a generalization of Theorem 1 in [22] but the construction is more difficult due to the Coriolis effects. Finally, we prove nonlinear inviscid damping for Couette flow in some Gevrey spaces by extending the method of [4] to the β -plane equation on T × R. [ABSTRACT FROM AUTHOR]

In this paper, we establish the continuous dependence for the non-resistive MHD equations in Sobolev spaces. Our obtained result fills considerably the recent result [5] for the continuous dependence on initial data. We also show that this result is optimal. More precisely, we proved that the data-to-solution map is not uniform continuous dependence in Sobolev spaces. [ABSTRACT FROM AUTHOR]