In this paper we construct a unique global in time weak nonnegative solution to the corrected Derrida-Lebowitz-Speer-Spohn equation, which statistically describes the interface fluctuations between two phases in a certain spin system. The construction of the weak solution is based on the dissipation of a Lyapunov functional which equals to the square of the Hellinger distance between the solution and the constant steady state. Furthermore, it is shown that the weak solution converges at an exponential rate to the constant steady state in the Hellinger distance and thus also in the $ L^1 $-norm. Numerical scheme which preserves the variational structure of the equation is devised and its convergence in terms of a discrete Hellinger distance is demonstrated.

中文翻译：

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利用Hellinger距离校正的Derrida-Lebowitz-Speer-Spohn方程的数值格式的适定性和收敛性

在本文中，我们为校正后的德里达-莱博维茨-施佩尔-斯波恩方程构建了唯一的全局时弱弱非负解，该方程以统计方式描述了特定自旋系统中两相之间的界面波动。弱解的构造基于Lyapunov泛函的耗散，该泛函等于解与恒定稳态之间的Hellinger距离的平方。此外，还表明，在Hellinger距离中，因而在$ L ^ 1 $范数中，弱解以指数速率收敛到恒定的稳态。设计了保留方程变分结构的数值方案，并证明了其在离散Hellinger距离方面的收敛性。