Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation

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.