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 Coulomb 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. The existence of weak and strong solutions to the kinetic and fluid equations is also obtained. We emphasize that the existence of global weak solution with the needed free energy dissipation for the kinetic model is established.

中文翻译：

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量化具有对齐和非局部力的 Vlasov 型方程的流体动力学极限

在本文中，我们量化了由具有非局部相互作用力和对齐的 Vlasov 型方程建模的数学生物学中出现的集体行为动力学方程的渐近极限。更准确地说，我们研究了具有约束、非局部相互作用和局部对齐力、线性阻尼和速度扩散的动力学 Cucker-Smale 植绒模型的流体动力学极限。我们首先讨论了在强局部对齐和扩散状态下我们的主方程的流体动力学极限，并且我们严格推导出具有非局部力的等温欧拉方程。我们还分析了与没有扩散的强局部对齐相对应的流体动力学极限。在这种情况下，限制系统是无压欧拉型方程。我们的分析包括两种情况下的库仑相互作用势和对极限流体动力学方程距离的明确估计。相对熵方法是我们主要结果中的关键技术，然而，对于没有扩散的情况，我们将调制的宏观动能与有界 Lipschitz 距离相结合来处理相互作用力的非定域性。还获得了动力学和流体方程的弱解和强解的存在性。我们强调，动力学模型需要自由能耗散的全局弱解的存在是成立的。我们将调制的宏观动能与有界 Lipschitz 距离相结合，以处理相互作用力中的非定域性。还获得了动力学和流体方程的弱解和强解的存在性。我们强调，动力学模型需要自由能耗散的全局弱解的存在是成立的。我们将调制的宏观动能与有界 Lipschitz 距离相结合，以处理相互作用力中的非定域性。还获得了动力学和流体方程的弱解和强解的存在性。我们强调，动力学模型需要自由能耗散的全局弱解的存在是成立的。