A note on boundary layer of a nonlinear evolution system with damping and diffusions

Boundary Layer Effects for the Nonlinear Evolution Equations with the Vanishing Diffusion Limit

“…System (1) has been extensively studied by several authors in different contexts [2,3,[5][6][7][8][9], such as the well-posedness problems, the nonlinear stability problem, and optimal decay rate of the solutions. However, all the results above need to assume that all parameters are fixed constants.…”

“…However, all the results above need to assume that all parameters are fixed constants. Another interesting problem is the zero diffusion limit: that is, consider the limit problem of solution sequences when one or more of parameters vanishes for the corresponding Cauchy problem or initial-boundary value problem [5][6][7]9]. It should be emphasized that flows often move in bounded domains with constraints from boundaries in real world; in the bounded domains case, the boundary effect requires a careful mathematical analysis.…”

“…Ruan and Zhu [5] consider the initialboundary value problem (1) with the zero Dirichlet boundary conditions when diffusion parameter → 0 + and they show that the boundary layer thickness ( ) is of the order ( ) with 0 < < 1/2. Subsequently, Peng et al [6] consider the initial-boundary value problem (1) with the Dirichlet-Neumann boundary conditions when diffusion parameter → 0 + and they show that the boundary layer thickness ( ) is of the order ( ) with 0 < < 3/4. For the parameter → 0 + , Chen and Zhu in [10] consider the Cauchy problem for (1); Peng in [7] study the initial-boundary value problem for (1) with the zero Dirichlet boundary conditions and obtain the thickness of layer of the order ( 1/2 ).…”

“…However, for the asymptotic limit → 0 + , all these results are restricted to the boundary conditions to the Dirichlet boundary conditions. By [6], which motivates our investigation in this paper, we concentrate our efforts to investigate the boundary layer effect and the convergence rates on the initial-boundary value (1)-(3) with the Dirichlet-Neumann boundary conditions when diffusion parameter → 0 + , and the solutions of the mixed boundary conditions usually exhibit different behaviors and much rich phenomena comparing with the general boundary conditions.…”

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