论文标题
生物运输网络作为自我调节过程的出现
Emergence of biological transportation networks as a self-regulated process
论文作者
论文摘要
我们研究了建模生物运输网络的自我调节过程。首先，我们为对称张量的扩散率$ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d。引入规定的电势会导致Fokker-Planck方程，我们还为其熵耗散了，我们还调查了正式的$ l^2 $差异流。我们得出了耗散功能第二个变化的积分公式，证明了二次熵密度建模焦耳加热的凸度（依赖扩散张量）。最后，我们将泊松方程式融为一体，以获取电势获得Poisson-Nernst-Planck系统。得出了相关熵损失功能的形式梯度流，从而为$ d $耦合的演变方程与两个辅助椭圆形PDE提供了进化方程。
We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal $L^2$-gradient flow for the symmetric tensor valued diffusivity $D$ of a broad class of entropy dissipations associated with a purely diffusive model. The introduction of a prescribed electric potential leads to the Fokker-Planck equation, for whose entropy dissipations we also investigate the formal $L^2$-gradient flow. We derive an integral formula for the second variation of the dissipation functional, proving convexity (in dependence of diffusivity tensor) for a quadratic entropy density modeling Joule heating. Finally, we couple in the Poisson equation for the electric potential obtaining the Poisson-Nernst-Planck system. The formal gradient flow of the associated entropy loss functional is derived, giving an evolution equation for $D$ coupled with two auxiliary elliptic PDEs.
