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In this paper, we consider the convergence rates for the 2D stationary and non-stationary Navier–Stokes Equations over highly oscillating periodic bumpy John domains with C2 regularity in some neighborhood of the boundary point (0,0). For the stationary case, using the variational equation satisfied by the solution and the correctors for the bumpy John domains obtained by Higaki and Zhuge [Arch. Ration. Mech. Anal. 247(4), 66 (2023)] after correcting the values on the inflow/outflow boundaries ({0} ∪ {1}) × (0, 1), we can obtain an O(ɛ3/2) approximation in L2 for the velocity and an O(ɛ3/2) convergence rates in L2 approximated by the so called Navier's wall laws, which generalized the results obtained by Jäger and Mikelić [J. Differ. Equations 170(1), 96–122 (2001)]. Moreover, for the non-stationary case, using the energy method, we can obtain an O(ɛ3/2 + exp(−Ct)) convergence rate for the velocity in L x 2 . [ABSTRACT FROM AUTHOR]

The Cauchy problem of compressible quantum Navier–Stokes–Poisson equations in three-dimensional space is considered in this paper. Under some smallness conditions on the initial data, we derive the existence of the global classical solution near the non-constant steady state by using the energy method. Combining the linear decay rate and the energy method, we also prove the algebraic decay rate of the solution toward the non-constant steady state with a small doping profile. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for the incompressible resistive Hall-magnetohydrodynamic equations with horizontal dissipation. By using very delicate energy estimate, we establish a blow-up criterion for the classical solutions and show the global-in-time existence of the classical solution for small initial data. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of the Cauchy problem associated with an n-dimensional generalized Benney-Luke equation, utt-mΔu-Δutt+Δ2u+α(2∇u·∇ut+utΔu)+β∇(|∇u|p∇u)=0, where n=1,2,3,4. We prove the existence and the uniqueness of the global solution of the Cauchy problem for the β≼0 case by using energy conservation law and give the existence and the nonexistence of the global solution of the Cauchy problem for the β>0 case by constructing the stable set and the unstable set. [ABSTRACT FROM AUTHOR]