We study the inviscid damping of Couette flow with an exponentially stratified density. The optimal decay rates of the velocity field and the density are obtained for general perturbations with minimal regularity. For Boussinesq approximation model, the decay rates we get are consistent with the previous results in the literature. We also study the decay rates for the full Euler equations of stratified fluids, which were not studied before. For both models, the decay rates depend on the Richardson number in a very similar way. Besides, we also study the dispersive decay due to the exponential stratification when there is no shear

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian partial differential equations. The last one is to study the singular and nonresonant neutral modes using Sturm‐Liouville theory and hypergeometric functions.

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Consider the steady solution to the incompressible Euler equation ū = Ae 1 in the periodic tunnel Ω = T d−1 × (0, 1) in dimension d = 2, 3. Consider now the family of solutions u ν to the associated Navier-Stokes equation with the no-slip condition on the flat boundaries, for small viscosities ν = A/Re, and initial values close in L 2 to Ae 1 . Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that u ν converges to Ae 1 when the viscosity converges to 0 and the initial value converges to Ae 1 . It is still unknown whether this inviscid limit is unconditionally true. The convex integration method predicts the possibility of a layer separation with energy at time T up to u ν (T ) − Ae 1 2 L 2 (Ω) ≈ A 3 T. In this work, we prove that at the double limit for the inviscid asymptotic, where both the Reynolds number Re converges to infinity and the initial value u ν (0) converges to Ae 1 in L 2 , the energy of layer separation cannot be more than u ν (T ) − Ae 1 2A 3 T (log Re) N . This result holds unconditionally for any limit of Leray-Hopf solutions of Navier-Stokes equation. Especially, it shows that, even if the limit is not unique, the shear flow pattern is observable up to time A −1 (log Re) −N . This provides a notion of stability despite the possible non-uniqueness of the limit predicted by the convex integration theory. The result relies on a new boundary vorticity estimate for the Navier-Stokes equation. This new estimate, inspired by previous work on higher regularity estimates for Navier-Stokes, provides a non-linear control rescalable through the inviscid limit.

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A smart energy meter life prediction system is proposed to evaluate the actual state of the smart energy meter, predict life and complete quality control. A method combining single component modeling and partial module integral function simulation modeling is adopted. According to the sampling module and the metering module, a smart electric energy table simulation model is established by using a reasonable mathematical algorithm. The life influencing factors of each module of the smart energy meter are integrated and organized into a simulation model of the life prediction software of the smart energy meter. The smart meter measured data, the power meter measured data and the model simulation measured data are compared to verify the validity of the simulation model. The proposed method provides new technical support for the life prediction of smart energy meter.

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