Decay of Solutions to the Linearized Free Surface Navier–Stokes Equations with Fractional Boundary Operators

Abstract

In this paper we consider a slab of viscous incompressible fluid bounded above by a free boundary, bounded below by a flat rigid interface, and acted on by gravity. The unique equilibrium is a flat slab of quiescent fluid. It is well-known that equilibria are asymptotically stable but that the rate of decay to equilibrium depends heavily on whether or not surface tension forces are accounted for at the free interface. The aim of the paper is to better understand the decay rate by studying a generalization of the linearized dynamics in which the surface tension operator is replaced by a more general fractional-order differential operator, which allows us to continuously interpolate between the case without surface tension and the case with surface tension. We study the decay of the linearized problem in terms of the choice of the generalized operator and in terms of the horizontal cross-section. In the case of a periodic cross-section we identify a critical order of the differential operator at which the decay rate transitions from almost exponential to exponential.

Appendix A: Some Useful Analytic Facts

Here we have compiled some analytic facts that we use in the paper. The forms given here are the forms we use, and no attempt has been made to state them in any additional generality.

First we record a simple version of the trace theorem.

Theorem A.1

(Trace). If $\Omega =(0,\ell )$ or $\Omega =\mathbb {R}^{N-1}\times (0,\ell )$ or $\Omega =\mathbb {T}^{N-1}\times (0,\ell )$, then there is a continuous linear map $Tr :H^1(\Omega )\rightarrow L^2(\partial \Omega )$ so that for $f\in C^\infty (\bar{\Omega })$, we have $Tr f=f\upharpoonright \partial \Omega$ and so that $f, Tr f$ satisfy the integration by parts formula for all $f\in H^1(\Omega )$.

Proof

This is a special case of Theorem 3 in Chapter 5.9 of [4]. $\square$

Next we record a version of the Poincaré inequality.

Theorem A.2

(Poincaré inequality.) If $\Omega =(0,\ell )$ or $\Omega =\mathbb {R}^{N-1}\times (0,\ell )$ or $\Omega =\mathbb {T}^{N-1}\times (0,\ell )$, then there is some constant $C>0$ so that for all $f\in H^1(\Omega )$ satisfying $f=0$ on $\{y=0\}$,

$$\begin{aligned} ||f||_{H^1}\lesssim ||Df||_{L^2}. \end{aligned}$$

(A.1)

Proof

For $\mathbb {R}^{N-1}\times (0,\ell )$ and $\mathbb {T}^{N-1}\times (0,\ell )$, it follows from the Poincaré inequality for $(0,\ell )$, which in turn follows from integration and Minkowski’s inequality. $\square$

Next we record a version of Korn’s inequality.

Theorem A.3

(Korn’s inequality). If $\Omega =\mathbb {R}^{N-1}\times (0,\ell )$ or $\Omega =\mathbb {T}^{N-1}\times (0,\ell )$, then there is some constant $C>0$ so that for all $f\in H^1(\Omega )$ satisfying $f=0$ on $\{y=0\}$, we have

$$\begin{aligned} ||Df||_{L^2}\lesssim ||\mathbb {D}f||_{L^2}. \end{aligned}$$

(A.2)

Proof

For a proof, see [1], Lemma 2.7. $\square$

Finally, we record a result about time derivatives.

Theorem A.4

If v is a complex-valued function satisfying $v\in L^2_T({H^1_\xi }(0,\ell ))$ and $\partial _t v\in L^2_T({H^1_\xi }^*(0,\ell ))$, then $v\in C([0,T];L^2(0,\ell ))$ and

$$\begin{aligned} \frac{d}{dt}||v||_{L^2(0,\ell )}^2=[\partial _t v,v]_{{H^1_\xi }^*,{H^1_\xi }}+\overline{[\partial _t v,v]_{({H^1_\xi })^*,{H^1_\xi }}}. \end{aligned}$$

(A.3)

The same holds if we replace ${H^1_\xi }$ with ${H^1_{\xi ,s}}$. Also, if v is a complex-valued function satisfying $v\in L^2_T({{{}_0{H}^1}}(\Omega ))$ and $\partial _t v\in L^2_T({{{}_0{H}^1}}^*(\Omega ))$, then $v\in C([0,T];L^2(\Omega ))$ and

$$\begin{aligned} \frac{d}{dt}||v||_{L^2(\Omega )}^2=[\partial _t v,v]_{({{{}_0{H}^1}})^*,{{{}_0{H}^1}}}+\overline{[\partial _t v,v]_{({{{}_0{H}^1}})^*,{{{}_0{H}^1}}}}. \end{aligned}$$

(A.4)

The same holds if we replace ${{{}_0{H}^1}}$ with ${{{}_0{H}^1_{\mathrm{sol}}}}$.

Proof

This is proved in the same manner as Theorem 3 in Chapter 5.9 of [4]. $\square$