Sharp large time behaviour in NN-dimensional reaction-diffusion equations of bistable type

We study the large time behaviour of the reaction-diffsuion equation $\partial_t u=\Delta u +f(u)$ in spatial dimension $N$, when the nonlinear term is bistable and the initial datum is compactly supported. We prove the existence of a Lipschitz function $s^\infty$ of the unit sphere, such that $u(t,x)$ converges uniformly in $\mathbb{R}^N$, as $t$ goes to infinity, to $U_{c_*}\bigg(|x|-c_*t + \frac{N-1}{c_*} \mathrm{ln}t + s^\infty\Big(\frac{x}{|x|}\Big)\bigg)$, where $U_{c*}$ is the unique 1D travelling profile. This extends earlier results that identified the locations of the level sets of the solutions with $o_{t\to+\infty}(t)$ precision, or identified precisely the level sets locations for almost radial initial data

Global Existence and Long-Time Asymptotics for Rotating Fluids in a 3D Layer

The Navier-Stokes-Coriolis system is a simple model for rotating fluids, which allows to study the influence of the Coriolis force on the dynamics of three-dimensional flows. In this paper, we consider the NSC system in an infinite three-dimensional layer delimited by two horizontal planes, with periodic boundary conditions in the vertical direction. If the angular velocity parameter is sufficiently large, depending on the initial data, we prove the existence of global, infinite-energy solutions with nonzero circulation number. We also show that these solutions converge toward two-dimensional Lamb-Oseen vortices as time goes to infinity.Comment: 26 pages, no figur

Sharp large time behaviour in n-dimensional Fisher-KPP equations

We study the large time behaviour of the Fisher-KPP equation ∂tu = ∆u+u−u2 in spatial dimension N, when the initial datum is compactly supported. We prove the existence of a Lipschitz function s∞ of the unit sphere, such that u(t, x) approaches, as t goes to infinity, the function Uc∗ ( |x| − c∗t + Nc+∗2 lnt + s∞(|xx| )) , where Uc∗ is the 1D travelling front with minimal speed c∗ = 2. This extends an earlier result of Gärtner

Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

We construct travelling wave graphs of the form $z=-ct+\phi(x)$, $\phi: x \in \mathbb{R}^{N-1} \mapsto \phi(x)\in \mathbb{R}$, $N \geq 2$, solutions to the $N$-dimensional forced mean curvature motion $V_n=-c_0+\kappa$ ($c\geq c_0$) with prescribed asymptotics. For any 1-homogeneous function $\phi_{\infty}$, viscosity solution to the eikonal equation $|D\phi_{\infty}|=\sqrt{(c/c_0)^2-1}$, we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by $\phi_{\infty}$. We also describe $\phi_{\infty}$ in terms of a probability measure on $\mathbb{S}^{N-2}$.Comment: 36 pages, 6 figure