Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation
Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation
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Burgers-FKPP 方程的推、拉和 Pushmi-pullyu 前沿
DOI：
10.4171/jems/1407
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发表时间：
2021
影响因子：
2.6
通讯作者：
L. Ryzhik
中科院分区：
文献类型：
--
作者：
Jing An;Christopher Henderson;L. Ryzhik
我们考虑了Burgers-FKPP方程在平流强度为$\beta\in\mathbb{R}$时解的长时间性态。这个方程在$\beta_c=2$时表现出从拉力前锋到推力前锋的转变。我们证明了在以$m_\beta(T)$为中心的坐标系中行波解的收敛，并研究了前沿位置$m_\beta(T)$的渐近性。当$β<；2$时，它与由Bramson建立的标准Fisher-KPP方程的形式相同：$m_β(T)=2t-(3/2)\log(T)+x_inty+o(1)$as$t\to+\Infty$。这种形式是典型的拉前额。当$\beta>；2$时，正面位于位置$m_\beta(T)=c_*(\beta)t+x_\infty+o(1)$，其中$c_*(\beta)=\beta/2+2/\beta$，这是推前的典型形式。然而，在临界值$\beta_c=2$时，展开变为$m_\beta(T)=2t-(1/2)\log(T)+x_\inty+o(1)$，反映了锋面的“推-冲”性质。$\beta<；2$的变元依赖于一种新的加权Hopf-Cole变换，当与附加的陡度比较变元相结合时，该变换允许控制平流项。案例$\beta>；2$依赖于标准的前置推送技术。在案件$\beta=\beta_c$中的证明要复杂得多，并且涉及在研究Bramson校正时通常不会遇到的论点。它依赖于Burgers-FKPP方程在$\beta_c=2$处某种程度上隐藏的粘性守恒律结构，并利用了一个来自相对熵型计算的耗散不等式和一个涉及动态变权重的加权Nash不等式。
We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength $\beta\in\mathbb{R}$. This equation exhibits a transition from pulled to pushed front behavior at $\beta_c=2$. We prove convergence of the solutions to a traveling wave in a reference frame centered at a position $m_\beta(t)$ and study the asymptotics of the front location $m_\beta(t)$. When $\beta<2$, it has the same form as for the standard Fisher-KPP equation established by Bramson \cite{Bramson1,Bramson2}: $m_\beta(t) = 2t - (3/2)\log(t) + x_\infty + o(1)$ as $t\to+\infty$. This form is typical of pulled fronts. When $\beta>2$, the front is located at the position $m_\beta(t)=c_*(\beta)t+x_\infty+o(1)$ with $c_*(\beta)=\beta/2+2/\beta$, which is the typical form of pushed fronts. However, at the critical value $\beta_c = 2$, the expansion changes to $m_\beta(t) = 2t - (1/2)\log(t) + x_\infty + o(1)$, reflecting the"pushmi-pullyu"nature of the front. The arguments for $\beta<2$ rely on a new weighted Hopf-Cole transform that allows to control the advection term, when combined with additional steepness comparison arguments. The case $\beta>2$ relies on standard pushed front techniques. The proof in the case $\beta=\beta_c$ is much more intricate and involves arguments not usually encountered in the study of the Bramson correction. It relies on a somewhat hidden viscous conservation law structure of the Burgers-FKPP equation at $\beta_c=2$ and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights.
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DOI：
10.1016/j.matpur.2022.09.004
发表时间：
2021-02
期刊：
Journal de Mathématiques Pures et Appliquées
影响因子：
--
作者：
Christopher Henderson
通讯作者：
Christopher Henderson
影响因子：
2
作者：
L. Arguin;Anton Bovier;N. Kistler
通讯作者：
L. Arguin;Anton Bovier;N. Kistler
DOI：
10.1090/cams/8
发表时间：
2022
期刊：
Communications of the American Mathematical Society
影响因子：
--
作者：
Avery, Montie;Scheel, Arnd
通讯作者：
Scheel, Arnd
通过基本谱附近的分辨展开实现临界拉锋的渐近稳定性
影响因子：
2
作者：
Avery, Montie;Scheel, Arnd
通讯作者：
Scheel, Arnd
FKPP-Burgers 系统中的行波速度
影响因子：
2.5
作者：
Bramburger, Jason J.;Henderson, Christopher
通讯作者：
Henderson, Christopher