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In this paper we study some boundary operators of a class of Bessel-type Littlewood-Paley extensions whose prototype is $$ \alignat 2 \Delta_xu(x,y)+\frac{1-2s}{y}\frac{\partial u}{\partial y}(x,y)+\frac{\partial^2u}{\partial y^2}(x,y)&=0&&\qquad\text{for}\ x\in\Bbb{R}^dd,\ y>0,\\u(x,0)&=f(x)&&\qquad\text{for}\ x\in\Bbb{R}^d. \endalignat $$ In particular, we show that with a logarithmic scaling one can capture the failure of analyticity of these extensions in the limiting cases $s=k\in\Bbb{N}$.

In this paper, we investigate the boundary layer behavior of solutions to the one dimensional Broadwell model of the nonlinear Boltzmann equation for small mean free path. We consider the analogue of Maxwell's diffusive and the reflexive boundary conditions. It is found that even for such a simple model, there are boundary layers due to purely kinetic effects which cannot be detected by the corresponding Navier-Stokes system. It is also found numerically that a compressive boundary layer is not always stable in the sense that it may detach from the boundary and move into the interior of the gas as a shock layer.

In this paper, we give an instability criterion for the Prandtl equations in three-dimensional space, which shows that the monotonicity condition on tangential velocity fields is not sufficient for the well-posedness of the three-dimensional Prandtl equations, in contrast to the classical well-posedness theory of the two-dimensional Prandtl equations under the Oleinik monotonicity assumption. Both linear stability and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three-dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, and this result is an exact complement to our recent work (A well-posedness theory for the Prandtl equations in three space variables. arXiv:1405.5308 , 2014) on the well-posedness theory for the three-dimensional Prandtl

In this paper we study the zero-viscosity limit of 2-D Boussinesq equations with vertical viscosity and zero diffusivity, which is a nonlinear system with partial dissipation arising in atmospheric sciences and oceanic circulation. The domain is taken as R2+ with Navier-type boundary. We prove the nonlinear stability of the approximate solution constructed by boundary layer expansion in conormal Sobolev space. The optimal expansion order and convergence rates for the inviscid limit are also identified in this paper. Our paper extends the partial zero-dissipation limit results of Boussinesq system with full dissipation by Chae D. [Adv.Math.203,no.2,2006] in the whole space to the case with partial dissipation and Navier boundary in the half plane.