Stability Of Boundary Layer Solutions Research Articles

Boundary layer solutions to singularly perturbed quasilinear systems

<p style='text-indent:20px;'>We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon^2\left(A(x, u(x), \varepsilon)u'(x)\right)' = f(x, u(x), \varepsilon) $\end{document}</tex-math></inline-formula>. The new features are that we do not consider one scalar equation, but systems, that the systems are allowed to be quasilinear, and that the systems are spatially non-smooth. Although the results about existence, asymptotic behavior, local uniqueness and stability of boundary layer solutions are similar to those known for semilinear, scalar and smooth problems, there are at least three essential differences. First, the asymptotic convergence rates valid for smooth problems are not true anymore, in general, in the non-smooth case. Second, a specific local uniqueness condition from the scalar case is not sufficient anymore in the vectorial case. And third, the monotonicity condition, which is sufficient for stability of boundary layers in the scalar case, must be adjusted to the vectorial case.</p>

Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows

We investigate the stability of boundary layer solutions of the two-dimensional incompressible Navier-Stokes equations. We consider shear flow solutions of Prandtl type : $$ u^\nu(t,x,y) \, = \, \big (U^E(t,y) + U^{BL}(t,\frac{y}{\sqrt{\nu}})\,, \, 0 \big )\, , \quad 0<\nu \ll 1\,. $$ We show that if $U^{BL}$ is monotonic and concave in $Y = y /\sqrt{\nu}$ then $u^\nu$ is stable over some time interval $(0,T)$, $T$ independent of $\nu$, under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$. We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in $x$ and $y$). Moreover, in the case where $U^{BL}$ is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr-Sommerfeld operator.

Stability of Boundary Layers for a Viscous Hyperbolic System Arising from Chemotaxis: One-Dimensional Case

This paper is concerned with the stability of boundary layer solutions for a viscous hyperbolic system transformed via a Cole--Hopf transformation from a singular chemotactic system modeling the in...

ONE-DIMENSIONAL DAMPED WAVE EQUATION WITH LARGE INITIAL PERTURBATION

This paper is concerned with the global stability of boundary layer solutions, together with the corresponding convergence rates, to the initial-boundary value problem for one-dimensional damped wave equation with a nonlinear convection term in the half space R+.

Large-time behavior of solutions to an outflow problem for a shallow water model

We investigate the existence and the time-asymptotic stability of boundary layer solutions for a one-dimensional shallow water model. Under the subsonic condition on the far-field equilibrium state, we show that there exists a one-parameter family of boundary layer solutions satisfying the outflow constant-flux boundary condition. We also show that there is a unique time-asymptotic boundary layer profile for a given initial data, and prove its stability using standard energy methods, provided that the initial perturbation is sufficiently small. Our stability result is obtained without the zero mass condition nor “artificial” condition on the profile strength. Rather the time-asymptotic profile is determined alone by the initial perturbation from the end state so that it satisfies the zero mass condition and its amplitude is appropriately small simultaneously.

Energy estimate for a linear symmetric hyperbolic-parabolic system in half line

In the present paper, we study the initial boundary value problem for a linear symmetric hyperbolic-parabolic system in one-dimensional half space. We obtain a priori estimates by using an energy method developed by Matsumura--Nishida for half space problem under the assumption that a stability condition of Shizuta--Kawashima type holds. The method developed in the present paper is applicable to showing the nonlinear stability of boundary layer solutions for a system of viscous conservation laws in half space.

Nonlinear stability of boundary layer solution to the Boltzmann equation with diffusive effect at the boundary

In this paper, the stability of boundary layer solutions to the Boltzmann equation with diffusive effect at the boundary for hard sphere model is considered. When the Mach number of the far field is less than −1, the exponential decay in time is proven for linearized operator first. At last, based on this property, nonlinear stability of the boundary layers is obtained by bootstrap argument.

Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space

This paper is concerned with the initial-boundary value problem of the generalized Benjamin-Bona-Mahony-Burgers equation in the half space $ R_+$ $u_t-$u txx -u xx $+f(u)_{x}=0, t>0, x\in R_+, $ $u(0,x)=u_0(x)\to u_+, as x\to +\infty,$ $u(t,0)=u_b$. Here $u(t,x)$ is an unknown function of $t>0$ and $x\in R_+$, $u_+$≠$u_b$ are two given constant states and the nonlinear function $f(u)$ is a general smooth function. &nbsp Asymptotic stability and convergence rates (both algebraic and exponential) of global solution $u(t,x)$ to the above initial-boundary value problem toward the boundary layer solution $\phi(x)$ are established in [9] for both the non-degenerate case $f'(u_+)<0$ and the degenerate case $f'(u_+)=0$. We note, however, that the analysis in [9] relies heavily on the assumption that $f(u)$ is strictly convex. Moreover, for the non-degenerate case, if the boundary layer solution $\phi(x)$ is monotonically decreasing, only the stability of weak boundary layer solution is obtained in [9]. This manuscript is concerned with the non-degenerate case and our main purpose is two-fold: Firstly, for general smooth nonlinear function $f(u)$, we study the global stability of weak boundary layer solutions to the above initial-boundary value problem. Secondly, when $f(u)$ is convex and the corresponding boundary layer solutions are monotonically decreasing, we discuss the local nonlinear stability of strong boundary layer solution. In both cases, the corresponding decay rates are also obtained.

Nonlinear stability of boundary layers for the Boltzmann equation with cutoff soft potentials

The study on the boundary layer is important in both mathematics and physics. This paper considers the nonlinear stability of boundary layer solutions for the Boltzmann equation with cutoff soft potentials when the Mach number of the far field is less than −1. Unlike the collision frequency is strictly positive in the hard potential or hard sphere model, the collision frequency has no positive lower bound for the cutoff soft potentials, so the decay in time cannot be expected. Instead, the present paper proves that the solution will always be in a small region around the boundary layer by noticing the decay property of collision operator in velocity.

Nonlinear stability of boundary layers of the Boltzmann equation for cutoff hard potentials

Many physical models have boundaries. For the Boltzmann equation, the study on the boundary layer in the region of the width in the order of the Kundsen number along the boundary is important both in mathematics and physics. In this paper, we consider the nonlinear stability of boundary layer solutions to the Boltzmann equation for hard potentials with angular cut-off. The boundary condition is imposed on incoming particles of the Dirichlet type and the solution tends to a global Maxwellian in the far field. For the existence of the boundary layer solutions, it is proved by Chen et al. [Anal. Appl. 2, 337–363 (2004)] by introducing a weight function which is a function of both position and velocity to overcome the difficulty from the sublinear growth in the collision frequency. Unlike the hard sphere model, even for stability in the case when the Mach number of the far field is less than −1, exponential decay in time cannot be expected for the cutoff hard potentials. Instead, an algebraic decay in time to the boundary layer solution is proved in this paper by using some recursive weighted energy estimates and the bootstrap argument.