We propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle. Our choice of discretization is motivated by a novel gradient flow formulation with respect to a metric that generalizes martingale transport. The discrete dynamics inherits this gradient flow structure, and in addition further properties, such as an alternative gradient flow formulation in the Wasserstein distance, contractivity in the Hellinger distance, and monotonicity of several Lypunov functionals. Our main result is the convergence in the limit of vanishing mesh size. The proof relies an a discrete version of a nonlinear functional inequality between integral expressions involving second order derivatives.

翻译：我们提出一种在圆环上对四阶非线性DLSS方程的空间离散化方法。该离散化选择的动机源于一种新型梯度流形式，该梯度流对应于一种推广鞅输运的度量。离散动力学继承了这种梯度流结构，并进一步保持了其他性质，例如Wasserstein距离下的交替梯度流形式、Hellinger距离下的压缩性以及多个Lyapunov泛函的单调性。我们的主要结果是当网格尺寸趋于零时的收敛性。该证明依赖于涉及二阶导数的积分表达式之间非线性函数不等式的离散版本。