On the rate of convergence and asymptotic profile of solutions to the viscous Burgers equation
On the rate of convergence and asymptotic profile of solutions to the viscous Burgers equation
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关于粘性 Burgers 方程解的收敛速度和渐近轮廓
DOI：
10.1512/iumj.2002.51.2247
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
W. Ni
中科院分区：
文献类型：
--
作者：
Yong;W. Ni
在本文中，我们控制初始近似的一阶矩，并通过两个显式正则近似（扩散 N 波解和扩散波解）获得收敛阶数和通解的渐近轮廓。当 t → ∞ 时，两种近似的收敛阶数在 L r 范数中为 O(t 1/(2r)-3/2 )，1 ≤ r ≤ ∞，这比无粘 Burgers 方程情况下众所周知的经典收敛阶数 O(t 1/(2r)-1/2 ) 更快。还进一步比较了这两种近似的收敛速度，并讨论了 Burgers 方程的亚稳态现象。这里设计的方法允许我们通过引入新的规范解并控制初始近似的更高矩来获得任意阶的收敛。
In this paper we control the first moment of the initial approximations and obtain the order of convergence and the asymptotic profile of a general solution by two explicit canonical approximations: a diffusive N-wave and a diffusion wave solution. The order of convergence of both approximations is O(t 1/(2r)-3/2 ) in L r norm, 1 ≤ r ≤ ∞, as t → ∞, which is faster than the well-known classical convergence order O(t 1/(2r)-1/2 ) for the inviscid Burgers equations case. A further comparison between the convergence rates of these two approximations and a discussion of the metastability phenomenon of the Burgers equation are also included. The method devised here allows us to obtain convergence up to any order by introducing new canonical solutions and controlling higher moments of the initial approximation.