论文标题
量化vlasov型方程的流体动力极限,该方程与对齐力和非局部力
Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forces
论文作者
论文摘要
在本文中,我们量化了通过具有非局部相互作用力和比对的vlasov-type方程建模的数学生物学集体行为方程的渐近极限。更准确地说,我们研究了具有约束,非局部相互作用和局部比对力,线性阻尼和速度中扩散的动力学cucker-微型羊群模型的流体动力极限。我们首先讨论在强局部比对和扩散方面下,主要方程的流体动力极限,我们严格地以非本地力来得出等温欧拉方程。我们还分析了与没有扩散的强局局部比对相对应的流体动力极限。在这种情况下,限制系统是无压力的欧拉型方程。我们的分析包括病例的库仑相互作用潜力以及在限制流体动力方程的距离上的明确估计。相对熵方法是我们主要结果的关键技术,但是,对于没有扩散的情况,我们将调制的宏观动能与有界的Lipschitz距离相结合,以处理相互作用的非局部性。为了完整性,还建立了动力学方程的弱和强溶液。
In this paper, we quantify the asymptotic limit of collective behavior kinetic equations arising in mathematical biology modeled by Vlasov-type equations with nonlocal interaction forces and alignment. More precisely, we investigate the hydrodynamic limit of a kinetic Cucker--Smale flocking model with confinement, nonlocal interaction, and local alignment forces, linear damping and diffusion in velocity. We first discuss the hydrodynamic limit of our main equation under strong local alignment and diffusion regime, and we rigorously derive the isothermal Euler equations with nonlocal forces. We also analyze the hydrodynamic limit corresponding to strong local alignment without diffusion. In this case, the limiting system is pressureless Euler-type equations. Our analysis includes the Coulombian interaction potential for both cases and explicit estimates on the distance towards the limiting hydrodynamic equations. The relative entropy method is the crucial technology in our main results, however, for the case without diffusion, we combine a modulated macroscopic kinetic energy with the bounded Lipschitz distance to deal with the nonlocality in the interaction forces. For the sake of completeness, the existence of weak and strong solutions to the kinetic and fluid equations are also established.