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.

We study the large time asymptotic speeds (turbulent flame speeds sT) of the simplified HamiltonJacobi (HJ) models arising in turbulent combustion. One HJ model is G-equation describing the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are HJ equations with convex (L1 type) but non-coercive Hamiltonians. The other is the quadratically nonlinear (L2 type) inviscid HJ model of MajdaSouganidis derived from the KolmogorovPetrovskyPiskunov reactive fronts. Motivated by a question posed by Embid, Majda and Souganidis (1995)[10], we compare the turbulent flame speeds sTs from these inviscid HJ models in two-dimensional cellular flows or a periodic array of steady vortices via sharp asymptotic estimates in the regime of large amplitude. The estimates are obtained by analyzing the action minimizing trajectories in the Lagrangian representation of solutions (Lax formula and its extension) in combination with delicate gradient bound of viscosity solutions to the associated cell problem of homogenization. Though the inviscid turbulent flame speeds share the same leading order asymptotics, their difference due to nonlinearities is identified as a subtle double logarithm in the large flow amplitude from the sharp growth laws. The turbulent flame speeds differ much more significantly in the corresponding viscous HJ models. 2013 Elsevier Masson SAS. All rights reserved.

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 $s=k\in\Bbb{N}$.

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}.

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