Motivated by the paper Grard-Varet and Dormy (2010) [6] [JAMS, 2010] about the linear ill-posedness for the Prandtl equations around a shear flow with exponential decay in normal variable, and the recent study of well-posedness on the Prandtl equations in Sobolev spaces, this paper aims to extend the result in [6] to the case when the shear flow has general decay. The key observation is to construct an approximate solution that captures the initial layer to the linearized problem motivated by the precise formulation of solutions to the inviscid Prandtl equations.

In the paper, we study the three-dimensional Prandtl equations, and prove that if one component of the tangential velocity field satisfies the monotonicity assumption in the normal direction, then the system is locally well-posed in the Gevrey function space with Gevrey index in ] 1, 2]. The proof relies on some new cancellation mechanism in the system in addition to those observed in the two-dimensional setting.

We consider the Cauchy problem for a second order weakly hyperbolic equation, with coefficients depending only on the time variable. We prove that if the coefficients of the equation belong to the Gevrey class $\gamma^{s_{0}}$ and the Cauchy data belong to $\gamma^{s_{1}}$ , then the Cauchy problem has a solution in $\gamma^{s_{0}}([0,T^{*}];\gamma^{s_{1}}(\mathbb{R}))$ for some$T$^{*}>0, provided 1≤$s$_{1}≤2−1/$s$_{0}. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤$s$_{1}≤$s$_{0}.

This is a continuation of the paper [15] on nonlinear boundary layers of the Boltzmann equation where the existence is established and shown to be strongly dependent on the Mach number <i>M</i> ^<i></i> of the Maxwellian state at far field. In this paper, when <i>M</i> ^<i></i>&lt;1, we will show that the linearized operator has the exponential decay in time property and therefore a bootstrapping argument yields nonlinear stability of the boundary layers.

In this paper, we consider the initial boundary value problem in a cylindrical domain to the three dimensional primitive equations with full eddy viscosity in the momentum equations but with only horizontal eddy diffusivity in the temperature equation. Global well-posedness of $z$-weak solution is established for any such initial datum that itself and its vertical derivative belong to $L^2$. This not only extends the results in \cite{Cao5} from the spatially periodic case to general cylindrical domains but also weakens the regularity assumptions on the initial data which are required to be $H^2$ there.