论文标题
非线性透射差异方程式在动脉壁上滚动的渐近极限
Asymptotic limits for a non-linear integro-differential equation modelling leukocytes' rolling on arterial walls
论文作者
论文摘要
我们考虑了一个描述$ z $的非线性间断模型，这是细胞中心在[Grec等人，J。Theo中介绍的真实线上的位置。生物。 2018]。我们介绍了一个新的$ \ varepsilon $尺度，当$ \ varepsilon $变为零时，我们严格地证明了渐近性。我们表明，这种缩放表征了电影制度中问题解决方案的长期行为（速度$ \ dot {z} $趋向于极限）。当$ψ$（与链接相关的弹性能量）为凸面和常规（$ψ$的二阶导数是有限的）时，收敛结果首先给出。在没有血流的情况下，当$ψ$是二次时，我们计算了最终位置$ z_ \ infty $，我们证明$ z $会趋向于。然后，我们为$ψ$构建了一个严格的数学框架是凸的，但只有Lipschitz。当$ψ'$接收有限的跳跃时，我们将相对于$ \ varepsilon $的收敛结果扩展到了这种情况。在最后一部分中，我们表明，在恒定的情况下（请参阅[Grec等人]中的模型3，$ψ$是绝对值），我们明确解决了问题并恢复上述渐近结果。
We consider a non-linear integro-differential model describing $z$, the position of the cell center on the real line presented in [Grec et al., J. Theo. Bio. 2018]. We introduce a new $\varepsilon$-scaling and we prove rigorously the asymptotics when $\varepsilon$ goes to zero. We show that this scaling characterizes the long-time behavior of the solutions of our problem in the cinematic regime (the velocity $\dot{z}$ tends to a limit). The convergence results are first given when $ψ$, the elastic energy associated to linkages, is convex and regular (the second order derivative of $ψ$ is bounded). In the absence of blood flow, when $ψ$, is quadratic, we compute the final position $z_\infty$ to which we prove that $z$ tends. We then build a rigorous mathematical framework for $ψ$ being convex but only Lipschitz. We extend convergence results with respect to $\varepsilon$ to this case when $ψ'$ admits a finite number of jumps. In the last part, we show that in the constant force case (see Model 3 in [Grec et al], $ψ$ is the absolute value), we solve explicitly the problem and recover the above asymptotic results.
