[Paper Review] Decay of viscous surface waves without surface tension

Yan Guo, Ian Tice|arXiv (Cornell University)|Nov 23, 2010

Navier-Stokes equation solutions14 references16 citations

TL;DR

This paper establishes the long-time decay of viscous surface waves in a fluid with no surface tension, using a novel two-tier energy method and control of both positive and negative Sobolev norms. In the infinite domain case, the free surface decays algebraically; in the periodic case, it decays at an almost exponential rate due to enhanced control from data smallness.

ABSTRACT

Consider a viscous fluid of finite depth below the air. In the absence of the surface tension effect at the air-fluid interface, the long time behavior of a free surface with small amplitude has been an intriguing question since the work of Beale \cite{beale_1}. In this monograph, we develop a new mathematical framework to resolve this question. If the free interface is horizontally infinite, we establish that it decays to a flat surface at an algebraic rate. On the other hand, if the free interface is periodic, we establish that it decays at an almost exponential rate, i.e. at an arbitrarily fast algebraic rate determined by the smallness of the data. Our framework contains several novel techniques, which include: (1) a local well-posedness theory of the Navier-Stokes equations in the presence of a moving boundary; (2) a two-tier energy method that couples the boundedness of high-order energy to the decay of low-order energy, the latter of which is necessary to balance out the growth of the highest derivatives of the free interface; (3) control of both negative and positive Sobolev norms, which enhances interpolation estimates and allows for the decay of infinite surface waves; (4) a localization procedure that is compatible with the energy method and allows for curved lower surface geometry in the periodic case. Our decay results lead to the construction of global-in-time solutions to the surface wave problem.

Motivation & Objective

• To resolve the long-standing open problem of surface wave decay in viscous fluids without surface tension.
• To establish global-in-time existence and decay rates for small-amplitude free surface waves in the absence of surface tension.
• To develop a new mathematical framework capable of handling moving boundaries and high-order energy estimates in the absence of surface tension.
• To extend the theory to both infinite and periodic spatial domains, accounting for curved bottom geometries in the periodic case.
• To prove that decay occurs even when surface tension is absent, contradicting Beale's non-decay theorem in the absence of viscous effects.

Proposed method

• Develop a local well-posedness theory for the Navier-Stokes equations with a moving free boundary in time-dependent domains.
• Implement a two-tier energy method that couples boundedness of high-order energy to decay of low-order energy, counteracting growth in highest-order derivatives.
• Control both negative and positive Sobolev norms to enhance interpolation estimates and enable decay analysis for infinite surface waves.
• Introduce a localization procedure compatible with the energy method, allowing for curved lower boundary geometries in the periodic case.
• Use a perturbed linear form of the equations to derive energy evolution estimates, combined with geometric and nonlinear estimates.
• Apply Riesz potential and Poisson integral techniques to control nonlocal terms and ensure regularity propagation.

Experimental results

Research questions

• RQ1 Can viscous surface waves decay over time in the absence of surface tension, despite Beale’s non-decay result for the inviscid case?
• RQ2 What decay rate is achievable for small-amplitude surface waves in a viscous fluid with no surface tension?
• RQ3 How can a two-tier energy method effectively balance the growth of highest-order derivatives in the presence of moving boundaries?
• RQ4 To what extent can negative Sobolev norms improve interpolation and decay estimates in the infinite domain case?
• RQ5 Can the energy method be adapted to handle curved bottom geometries in the periodic setting?

Key findings

• In the infinite, flat bottom case, the free surface decays algebraically to a flat state, with decay rate determined by the initial data size.
• In the periodic, curved bottom case, the surface decays at an almost exponential rate—arbitrarily fast algebraic decay—due to the smallness of initial data.
• The two-tier energy method successfully controls the growth of high-order derivatives by linking them to the decay of low-order energy.
• Control of both positive and negative Sobolev norms enables improved interpolation estimates, essential for proving decay in the infinite domain.
• The localization procedure allows for curved lower boundary geometries while preserving compatibility with the energy method in the periodic case.
• Global-in-time solutions exist and decay to equilibrium, resolving the long-standing question of decay without surface tension.