粘性Burgers方程的交叉渐近性与尖锐限制速率

中文摘要 AI 辅助

我们研究区间$(0,L)$上具有保守边界条件$u_x=-u^2$的一维粘性Burgers方程。在半直线上，解趋近于一个非线性自相似剖面，而在有界区间上，解收敛到一个非恒定平衡态。我们明确描述了在临界扩散尺度$t=cL^2$下动力学如何在这些状态之间过渡。对于质量为$M$的紧支撑初始数据，当$L\to\infty$时，重标度后的解收敛到一个显式的交叉剖面$\Phi_c$。在相似变量中，当$c\downarrow0$时，$\Phi_c$收敛到半直线剖面$f_M$；在重标度回定义域变量后，当$c\to\infty$时，它收敛到区间平衡态。我们还确定了限制的尖锐起始点：\\[ \lim_{c\downarrow0}-c\log|\Phi_c(0)-f_M(0)|=1\qquad(M\ne0). \\]此外，在相似变量的紧集上，区间解与半直线解在$C^k$范数下的差异至多为$C_{k,\varepsilon}\exp(-(1-\varepsilon)L^2/t)$，且该界对$L$一致成立，并且指数常数是最优的。因此，我们不仅识别出过渡尺度$L^2$，还识别出控制交叉的剖面以及远端边界变得可见的尖锐速率。最后，我们讨论了对于空间依赖扩散系数仍然成立的结论，并通过数值方法展示了三种渐近区域。

英文摘要

We study the one-dimensional viscous Burgers equation on $(0,L)$ with the conservative boundary conditions $u_x=-u^2$. On the half-line, solutions approach a nonlinear self-similar profile, whereas on a bounded interval they converge to a nonconstant equilibrium. We describe explicitly how the dynamics passes between these states at the critical diffusive scale $t=cL^2$. For compactly supported initial data of mass $M$, the rescaled solution converges as $L\to\infty$ to an explicit crossover profile $Φ_c$. In similarity variables, $Φ_c$ converges to the half-line profile $f_M$ as $c\downarrow0$; after rescaling to domain variables, it converges to the interval equilibrium as $c\to\infty$. We also determine the sharp onset of confinement: \[ \lim_{c\downarrow0}-c\log|Φ_c(0)-f_M(0)|=1\qquad(M\ne0). \] Moreover, on compact sets in similarity variables, the interval and half-line solutions differ in $C^k$ by at most $C_{k,\varepsilon}\exp(-(1-\varepsilon)L^2/t)$, uniformly in $L$, and the exponential constant is optimal. Thus we identify not only the transition scale $L^2$, but also the profile governing the crossover and the sharp rate at which the remote boundary becomes visible. We finally discuss the conclusions that persist for space-dependent diffusivity and illustrate the three asymptotic regimes numerically.

发表机构

• Instituto de Matemática e Computação, Universidade Federal de Itajubá（伊塔朱巴联邦大学数学与计算研究所）
• Friedrich–Alexander-Universität Erlangen–Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）
• University of Deusto（德乌斯托大学）
• Universidad Autónoma de Madrid（马德里自治大学）

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