AI 中文总结
本文提出度量空间中由满足耗散条件的双函数驱动的演化变分不等式的一般框架，引入变分移动方案构造解，推广至多物种耦合梯度流与纳什均衡问题，证明离散解存在性。
AI 中文摘要
我们研究了一类由双函数 $\mathsf{b}:\mathrm{D} \times \mathrm{D} \to \mathbb{R}$ 驱动的新型演化变分不等式（EVI），该双函数定义在度量空间 $(\mathrm{X},\mathrm{d})$ 的子集 $\mathrm{D}$ 上，并满足自然耗散条件 $$\mathsf{b}(x,y)+\mathsf{b}(y,x)\le \eta \mathrm{d}^2(x,y).$$ 我们通过提出一个度量框架，推广了希尔伯特空间中由单调算子驱动的演化的经典结构，为解的存在性、稳定性、正则性、逼近性和渐近行为提供了一般条件。一个激励性的应用是最小最大和多物种耦合梯度流的情形，其中 $N$ 个物种中的每一个都沿其自身能量函数的最陡下降方向演化，而联合系统不是任何单一能量的梯度流。为了构造解，我们引入了一个变分移动方案（VMS），这是一种时间离散化方案，其中每次更新都是度量惩罚双函数的鞍点，推广了单物种梯度下降的经典最小化移动/JKO方案。在耦合多物种情形下，VMS 归结为纳什均衡问题。我们通过一种一般方法证明了 VMS 离散解的存在性，该方法将 $\mathsf{b}$ 的耗散性与沿合适的 $\textit{重心插值}$ 的凸性的新概念相结合（该概念受沿广义测地线的凸性启发，而广义测地线是梯度流的关键概念）。这些条件共同提供了一种零阶博弈论单调性概念，适用于没有线性结构的一般度量空间。
英文摘要
We study a novel class of evolution variational inequalities (EVI) driven by bifunctions $\mathsf{b}:\mathrm{D} \times \mathrm{D} \to \mathbb{R}$ defined on a subset $\mathrm{D}$ of a metric space $(\mathrm{X},\mathrm{d})$ and satisfying a natural dissipativity condition $$\mathsf{b}(x,y)+\mathsf{b}(y,x)\le η\mathrm{d}^2(x,y).$$ We provide general conditions for existence, stability, regularity, approximation and asymptotic behavior of solutions, by proposing a metric framework that generalizes the classical structure of evolutions driven by monotone operators in Hilbert spaces. A motivating application is the setting of minimax and multispecies coupled gradient flows, in which each of $N$ species evolves in the direction of steepest descent of its own energy functional and the joint system is not a gradient flow of any single energy. To construct solutions, we introduce a variational movement scheme (VMS), a time discretization scheme in which each update is a saddle point of a metrically penalized bifunction, generalizing the classical minimizing movement/ JKO scheme for single species gradient descent. In the coupled multispecies case, the VMS reduces to a Nash equilibrium problem. We prove existence of discrete solutions to the VMS via a general approach which combines dissipativity of $\mathsf{b}$ with a new notion of convexity along suitable $\textit{barycentric interpolations}$ (which is inspired by convexity along generalized geodesics, a crucial notion for gradient flows). Together, these conditions provide a zeroth-order notion of game-theoretic monotonicity applicable to general metric spaces without linear structure.
