August 1, 2025

On the decay of viscous surface waves in 2D

Key Points

The analysis reveals that the Lipschitz norm of the velocity at the boundary decays at a rate of (1 + t)−5/4, highlighting time-decay effects.
Research demonstrates the optimal time-decay rate of low-order energy for a viscous fluid solution, leading to significant findings.
The study utilizes Navier–Stokes equations to describe incompressible fluid behavior under a gravity-influenced environment.
Improving upon previous results, the findings stress the boundedness of high-order energy in dynamic fluid systems.

Abstract

We consider the free boundary problem for a layer of incompressible viscous fluid lying above a fixed rigid bottom and below the atmosphere of positive constant pressure in 2D. The fluid dynamics is governed by the incompressible Navier–Stokes equations with gravity, and the effect of surface tension is neglected on the upper free boundary. We prove the optimal time-decay rate of the low-order energy of the solution with minimal derivative count 3, which implies that on the upper boundary the Lipschitz norm of the velocity decays at the rate (1 + t)−5/4. This together with a time-weighted estimate for the highest order spatial derivatives of the free boundary function leads to the boundedness of the high-order energy, which in particular improves the result of Wang Adv. Math. 374, 107330 (2020).

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