Amplitude Equations for Weakly Nonlinear Surface Waves in Variational Problems

Abstract

Among hyperbolic Initial Boundary Value Problems (IBVP), those coming from a variational principle ‘generically’ admit linear surface waves, as was shown by Serre (J Funct Anal 236(2):409–446, 2006). At the weakly nonlinear level, the behavior of surface waves is expected to be governed by an amplitude equation that can be derived by means of a formal asymptotic expansion. Amplitude equations for weakly nonlinear surface waves were introduced by Lardner (Int J Eng Sci 21(11):1331–1342, 1983), Parker and co-workers (J Elast 15(4):389–426, 1985) in the framework of elasticity, and by Hunter (Nonlinear surface waves. In: Current progress in hyberbolic systems: Riemann problems and computations (Brunswick, 1988). Contemporary mathematics, vol 100. American Mathematical Society, pp 185–202, 1989) for abstract hyperbolic problems. They consist of nonlocal evolution equations involving a complicated, bilinear Fourier multiplier in the direction of propagation along the boundary. It was shown by the authors in an earlier work (Benzoni-Gavage and Coulombel Arch Ration Mech Anal 205(3):871–925, 2012) that this multiplier, or kernel, inherits some algebraic properties from the original IBVP. These properties are crucial for the (local) well-posedness of the amplitude equation, as shown together with Tzvetkov (Adv Math, 2011). Properties of amplitude equations are revisited here in a somehow simpler way, for surface waves in a variational setting. Applications include various physical models, from elasticity of course to the director-field system for liquid crystals introduced by Saxton (Dynamic instability of the liquid crystal director. In: Current progress in hyperbolic systems: Riemann problems and computations (Brunswick, 1988). Contemporary mathematics, vol 100. American Mathematical Society, Providence, pp 325–330, 1989) and studied by Austria and Hunter (Commun Inf Syst 13(1):3–43, 2013). Similar properties are eventually shown for the amplitude equation associated with surface waves at reversible phase boundaries in compressible fluids, thus completing a work initiated by Benzoni-Gavage and Rosini (Comput Math Appl 57(3–4):1463–1484, 2009).

Appendix

Proposition A.1

Let $b: \mathbb{R}^{3} \rightarrow \mathbb{C}$ be a symmetric, continuous function outside Z:={ (k,ℓ,m) ; kℓm = 0} and such that, for all $(k,\ell,m) \in \mathbb{R}^{3}\setminus Z$,

$$\displaystyle{ b(-k,-\ell,-m) = \overline{b(k,\ell,m)}, }$$

$$\displaystyle{ \vert b(k,\ell,m)\vert \,\leq \, C(1 + k^{2} +\ell ^{2} + m^{2}), }$$

with C a positive constant. Then the functional

$$\displaystyle{ \mathcal{T} [w] = \frac{1} {3}\,\iint b(-k - m,k,m)\,\widehat{w}(-k - m)\,\widehat{w}(k)\,\widehat{w}(m)\,\mathrm{d}k\,\mathrm{d}m\, }$$

is well-defined on $H^{1}(\mathbb{R})$, and its variational derivative $\delta \mathcal{T}$ is given by

$$\displaystyle{ \widehat{\delta \mathcal{T} [w]}(k) = 2\pi \,\int b(-k,k - m,m)\,\widehat{w}(k - m)\,\widehat{w}(m)\,\mathrm{d}m,\;\forall w \in H^{2}(\mathbb{R}; \mathbb{R})\,. }$$

Proof

By the Cauchy–Schwarz inequality, $\widehat{w} \in L^{1}(\mathbb{R})$ for all $w \in H^{1}(\mathbb{R})$, and

$$\displaystyle{ \|\widehat{w}\|_{L^{1}}\, \leq \,\sqrt{\pi }\,\|w\|_{H^{1}}\,. }$$

Therefore, using that

$$\displaystyle{ \vert b(-k,k - m,m)\vert \,\leq \, C(1 + \vert k\vert ^{2} + \vert k - m\vert ^{2} + \vert m\vert ^{2})\, }$$

$$\displaystyle{ \leq \, C(1 + 2\vert k\vert \vert k - m\vert + 2\vert k - m\vert \vert m\vert + 2\vert m\vert \vert k\vert ), }$$

we see that the trilinear mapping $\mathcal{T}$ is continuous on $H^{1}(\mathbb{R})$, with

$$\displaystyle{ \vert \mathcal{T} [w]\vert \leq 8C\,\pi ^{2}\,\|\widehat{w}\|_{ L^{1}}\,\|w\|_{H^{1}}^{2}\, \leq 8C\,\pi ^{3}\,\|w\|_{ H^{1}}^{3}, }$$

by the Fubini, Cauchy–Schwarz, and Plancherel theorems. Furthermore, for all w, $v \in H^{1}(\mathbb{R}; \mathbb{R})$, we have

$$\displaystyle\begin{array}{rcl} \frac{\mathrm{d}} {\mathrm{d}\theta }\mathcal{T} [w +\theta v]_{\vert \theta =0}\,& =& \,\frac{1} {3}\,\iint b(-k - m,k,m)\,\widehat{v}(-k - m)\,\widehat{w}(k)\,\widehat{w}(m)\,\mathrm{d}k\,\mathrm{d}m {}\\ & & +\frac{1} {3}\,\iint b(-k - m,k,m)\,\widehat{w}(-k - m)\,\widehat{v}(k)\,\widehat{w}(m)\,\mathrm{d}k\,\mathrm{d}m {}\\ & & +\frac{1} {3}\,\iint b(-k - m,k,m)\,\widehat{w}(-k - m)\,\widehat{w}(k)\,\widehat{v}(m)\,\mathrm{d}k\,\mathrm{d}m {}\\ \,& =& \,\iint b(-k - m,k,m)\,\widehat{w}(-k - m)\,\widehat{w}(k)\,\widehat{v}(m)\,\mathrm{d}k\,\mathrm{d}m {}\\ \end{array}$$

by the symmetry of b and obvious changes of variables. The integral above is well defined for all w, $v \in H^{1}(\mathbb{R}; \mathbb{R})$, as expected from the fact that $\mathcal{T}$ is differentiable since it is trilinear continuous. However, the definition of its variational derivative $\delta \mathcal{T} [w]$ is more demanding on w. It amounts to rewriting

$$\displaystyle{ \frac{\mathrm{d}} {\mathrm{d}\theta }\mathcal{T} [w +\theta v]_{\vert \theta =0}\, =\,\int \delta \mathcal{T} [w]\;v(y)\,\mathrm{d}y, }$$

so that $\delta \mathcal{T} [w]$ bears all the derivatives. In view of the large frequency behavior of the kernel b, it turns out that this is possible as soon as w belongs to H ², as we show below.

Let us recall that, by the Plancherel theorem,

$$\displaystyle{ \int \widehat{f}(-m)\,\widehat{v}(m)\,\mathrm{d}m\, =\, 2\pi \,\int f(y)\,v(y)\,\mathrm{d}y }$$

for all real valued, square integrable functions f and v. We claim that for $w \in H^{2}(\mathbb{R}; \mathbb{R})$, we can define a real valued f ∈ L ² by

$$\displaystyle{ \widehat{f}(m)\, =\,\int b(-m,m - k,k)\,\widehat{w}(m - k)\,\widehat{w}(k)\,\mathrm{d}k\,. }$$

Indeed, using that

$$\displaystyle{ \vert b(-m,m - k,k)\vert \,\leq \, C(1 + m^{2} + (m - k)^{2} + k^{2})\, \leq \, C(1 + 3(m - k)^{2} + 3k^{2}), }$$

we find that

$$\displaystyle{ \left \vert \int b(-m,m - k,k)\,\widehat{w}(m - k)\,\widehat{w}(k)\,\mathrm{d}k\right \vert \leq C\big((\vert \widehat{w}\vert {\ast}\vert \widehat{w}\vert )(m) + 6(\vert \widehat{w}\vert {\ast}\vert \widehat{w_{yy}}\vert )(m)\big), }$$

and

$$\displaystyle{ \|\vert \widehat{w}\vert {\ast}\vert \widehat{w}\vert \|_{L^{2}} \leq \|\widehat{ w}\|_{L^{1}}\|\widehat{w}\|_{L^{2}} = 2\pi \|\widehat{w}\|_{L^{1}}\|w\|_{L^{2}}, }$$

$$\displaystyle{ \|\vert \widehat{w}\vert {\ast}\vert \widehat{w_{yy}}\vert \|_{L^{2}} \leq \|\widehat{ w}\|_{L^{1}}\|\widehat{w_{yy}}\|_{L^{2}} \leq 2\pi \|\widehat{w}\|_{L^{1}}\|w\|_{H^{2}}\,. }$$

This shows that

$$\displaystyle{ \int \left (\int b(m,-m - k,k)\,\widehat{w}(-m - k)\,\widehat{w}(k)\,\mathrm{d}k\right )\,\widehat{v}(m)\,\mathrm{d}m\, =\, 2\pi \,\int f(y)\,v(y)\,\mathrm{d}y }$$

for all $w \in H^{2}(\mathbb{R}; \mathbb{R})$. By the Fubini theorem and the symmetry of b, the left-hand side is exactly what we have found for the directional derivative of $\mathcal{T}$. We thus have

$$\displaystyle{ \frac{\mathrm{d}} {\mathrm{d}\theta }\mathcal{T} [w +\theta v]_{\vert \theta =0}\, =\, 2\pi \,\int f(y)\,v(y)\,\mathrm{d}y, }$$

which means that $\delta \mathcal{T} [w] = 2\pi f$.

Proposition A.2

Let us consider the functional $\mathcal{M}$ defined by

$$\displaystyle{ \mathcal{M}[w] = \frac{1} {2}\,\iint \vert k\vert \,\vert \widehat{w}(k)\vert ^{2}\,\mathrm{d}k\, }$$

for all $w \in H^{1/2}(\mathbb{R})$ . Then its variational derivative $\delta \mathcal{M}$ is such that

$$\displaystyle{ \partial _{y}w = -\tfrac{1} {2\pi }\,\mathcal{H}(\delta \mathcal{M}[w]),\;\forall w \in H^{1}(\mathbb{R}; \mathbb{R})\,. }$$

Proof

The computations are similar to, and simpler than in the previous proposition. We have

$$\displaystyle{ \frac{\mathrm{d}} {\mathrm{d}\theta }\mathcal{M}[w +\theta v]_{\vert \theta =0}\, =\,\iint \vert k\vert \,\widehat{w}(-k)\,\widehat{v}(k)\,\mathrm{d}k\, =\, 2\pi \,\iint u(y)\,v(y)\,\mathrm{d}y, }$$

provided that $\widehat{u}(-k) = \vert k\vert \widehat{w}(-k)$ a.e, or equivalently, $ik\widehat{w}(k) = i\mathsf{sgn}(k)\widehat{u}(k)$, that is, $\partial _{y}w = -\mathcal{H}(u) = -\tfrac{1} {2\pi }\,\mathcal{H}(\delta \mathcal{M}[w])$.