A Hamilton-Jacobi approach to nonlocal kinetic equations

Abstract

Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, model of BGK type implementing a velocity-jump process. We study both a linear and a nonlinear case and describe the concentration profile. In particular, we analyse a hyperbolic (or high frequency) regime that can be interpreted both as a local (microscopic) or as a nonlocal (macroscopic) rescaling. We consider a Hopf-Cole transform and derive a Hamilton-Jacobi equation. The concentrations are then explained as a consequence of the stationary points of the Hamiltonian that is spatially heterogeneous like the velocity-jump process. After revising the classical hydrodynamic limits for the aggregate quantities and the eikonal equation that can be derived from those with a Hopf-Cole transform, we find that the Hamilton-Jacobi equation is a second order approximation of the eikonal equation in the limit of small diffusivity. For nonlinear turning kernels, the Hopf-Cole transform allows to study the stability of the possible homogeneous configurations and of patterns and the results of a linear stability analysis previously obtained are found and extended to a nonlinear regime. In particular, it is shown that instability (pattern formation) occurs when the Hamiltonian is convex-concave.

Figures (5)

• Figure 1: First line. First panel: $\mathcal{S}$ (red curve) given by \ref{['S1D']} and two different initial conditions $\rho^0$: the constant one (blue) and an asymmetric Gaussian (green). Second panel: temporal evolution of $\rho(t,x)$ in the case of constant $\rho^0$. Third panel: temporal evolution of $\rho(t,x)$ in the case of an asymmetric $\rho^0$. Second line. First panel: $H(x,p)$ in the local regime \ref{['def:hyp']}, second and third panel: $H(x,p)$ in the nonlocal regime \ref{['def:hyp_v']}. In the third panel the red dashed lines correspond to $\pm U|p|$, while the black horizontal dashed line indicates the level zero.
• Figure 2: Temporal evolution of $\rho(t,x)$ in case of $\mathcal{S}$ given by \ref{['S2D']}. Here $R=0.4$, the initial condition $\rho^0$ is constant. In (a): three different $\mathcal{S}$ as given by \ref{['S2D']} with $\sigma_1=\sigma_2=0.03$ and three different couples of $\bar{x}_2, \bar{x}_1$. In (b)-(c) temporal evolution of $\rho(t,x)$ for the two different $\mathcal{S}$: in (b) $\bar{x}_2-\bar{x}_1=R$, in (c) $\bar{x}_2-\bar{x}_1>R$. In (d) we plot the Hamiltonian for the case $\bar{x}_2-\bar{x}_1>R$.
• Figure 3: Temporal evolution of $\rho(t,x)$ in case of $\mathcal{S}$ given by \ref{['S2D']}. Here $R=0.4$ and $\sigma_1=0.06, \sigma_2=0.03$. In (a): two different $\mathcal{S}$ given by \ref{['S2D']} with two different couples $\bar{x}_2, \bar{x}_1$. In (b) and (c) the corresponding time evolution of the densities $\rho$.
• Figure 4: The convex-concave Hamiltonian $H$ in the one-dimensional case as computed explicitly in formula \ref{['eq:H1D']}. The values of $\mu, V, R$ are those corresponding to the example in section \ref{['sec:nonlin_ex']}. The red dashed lines correspond to the asymptotes $\pm U |p|$, the black horizontal dashed line is the zero-level. The values $\pm \bar{p} \neq 0$ with $H(\bar{p})=0$ determines the slopes of the saw tooth solutions in Fig. \ref{['fig:adh']}
• Figure 5: One dimensional example with parameters $V=1, \mu=100, R=5\cdot 10^{-2}$. In (a) the initial condition is the perturbation of the homogeneous steady profile, in (b) the initial condition is a bimodal gaussian centered in $\bar{x}_1=2.3, \bar{x}_2=2.7$, in (c) the initial condition is a bimodal gaussian (i.e. $\rho_0$ as in \ref{['S2D']}) centered in $\bar{x}_1=2.4, \bar{x}_2=2.6$. In the second line we report the corresponding profiles of $-\log(\rho(t,x))$. These saw tooth curves result from the convex-concave Hamiltonian in fig. \ref{['fig:H']}

Theorems & Definitions (2)

• Theorem 1
• proof