A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport
CoRR(2023)
摘要
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.
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