In this paper, we consider the vanishing viscosity limit problem for a system arising from the Keller‐Segel equations in three space dimensions. First, we construct an accurate approximate solution that incorporates the effects of boundary layers. Then, we prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero. Our approach is based on the method of matched asymptotic expansions of singular perturbation theory and the classical energy estimates.
In this paper, we study the approximation of π through the semiperimeter or area of a randomn-sided polygon inscribed in a unit circle in ℝ2. We show that, with probability 1, the approximation error goes to 0 as n → ∞, and is roughly sextupled when compared with the classical Archimedean approach of using a regular n-sided polygon. By combining both the semiperimeter and area of these random inscribed polygons, we also construct extrapolation improvements that can significantly speed up the convergence of these approximations.
We study the boundary layer problem of a Keller-Segel model in a domain of two space dimensions with vanishing chemical diffusion coefficient. By using the method of matched asymptotic expansions of singular perturbation theory, we construct an accurate approximate solution which incorporates the effects of boundary layers and then use the classical energy estimates to prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero.
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