论文标题
部分可观测时空混沌系统的无模型预测
Hyperbolic-parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion
论文作者
论文摘要
我们研究了具有等级缺陷扩散矩阵的退化交叉扩散方程，这些矩阵被认为是为了避免空间拥挤而模拟的种群，并最近发现在相互作用的随机粒子系统的平均景观极限中出现。迄今为止，它们在多个空间维度中的分析已局限于具有相同的移动性系数的纯对流案例。在本文中，我们介绍了此类方程式的熵类别的正常形式，该形式揭示了它们的对称双曲线 - 寄生虫系统的结构。由于奇异扩散矩阵的范围和内核的状态依赖性，我们重写方程的方式与具有NullSpace不变性属性的对称二阶系统的经典使用的方式不同。通过这种变量的这种更改，我们在短时间内解决了库奇问题，并以$ h^s（\ mathbb {t}^d）$的阳性初始数据解决了$ s> d/2+1 $。
We investigate degenerate cross-diffusion equations with a rank-deficient diffusion matrix that are considered to model populations which move as to avoid spatial crowding and have recently been found to arise in a mean-field limit of interacting stochastic particle systems. To date, their analysis in multiple space dimensions has been confined to the purely convective case with equal mobility coefficients. In this article, we introduce a normal form for an entropic class of such equations which reveals their structure of a symmetric hyperbolic--parabolic system. Due to the state-dependence of the range and kernel of the singular diffusive matrix, our way of rewriting the equations is different from that classically used for symmetric second-order systems with a nullspace invariance property. By means of this change of variables, we solve the Cauchy problem for short times and positive initial data in $H^s(\mathbb{T}^d)$ for $s>d/2+1$.
