Validity of the nonlinear Schrödinger approximation for the two-dimensional water wave problem with and without surface tension in the arc length formulation (Q2220645)

The author considers the two-dimensional water wave problem in an infinitely long canal of finite depth, with and without surface tension. The fluid flows in the 2D domain \(\Omega (t)=\{(x,y)\in \mathbb{R}^{2}:x\in \mathbb{R}\) , \(-h < y<\eta (x,t)\}\), between the bottom \(B=\{(x,y)\in \mathbb{R}^{2}:x\in \mathbb{R}\), \(y=-h\}\) and the free top surface \(\Gamma (t)=\{(x,y)\in \mathbb{R}^{2}:x\in \mathbb{R},y=\eta (x,t)\}\). The fluid velocity satisfies the incompressible Euler equations \(V_{t}+(V\cdot \nabla )V=-\nabla p+g\left( \begin{array}{c} 0 \\ -1 \end{array} \right) \), \(\nabla \cdot V=0\) in \(\Omega (t)\). The boundary conditions \( \eta _{t}=V\cdot \left( \begin{array}{c} -\eta _{x} \\ 1 \end{array} \right) \), \(p=-bgh^{2}\kappa \) at \(\Gamma (t)\), and \(v_{2}=0\) at \(B\) are added. Here \(b\geq 0\) is the Bond number, which is proportional to the strength of the surface tension and \(\kappa \) is the curvature of \(\Gamma (t) \). Assuming that the flow is irrotational and introducing the velocity potential \(\varphi \) (such that \(V=\nabla \varphi \)) which is harmonic in \( \Omega (t)\) and has a vanishing normal derivative at \(B\), the above problem may be written as \(\eta _{t}=\mathcal{K}v_{1}-v_{1}\eta _{x}\), \( (v_{1})_{t}=-g\eta _{x}-\frac{1}{2}(v_{1}^{2}+(\mathcal{K} v_{1})^{2})_{x}+bgh^{2}\left( \frac{\eta _{x}}{\sqrt{1+\eta _{x}^{2}}} \right) _{xx}\) at \(\Gamma (t)\). The author recalls the nonlinear Schrödinger (NLS) approximation \(\left( \begin{array}{c} \eta \\ v_{1} \end{array} \right) (x,t)=\varepsilon A(\varepsilon (x-c_{g}t),\varepsilon ^{2}t)e^{i(k_{0}x-\omega _{0}t)}\varphi (k_{0},b)+\mathcal{O}(\varepsilon ^{2})+c.c.\), where \(\omega _{0}>0\) is the basic temporal wave number associated to the basic spatial wave number \(k_{0}>0\) of the underlying carrier wave \(e^{i(k_{0}x-\omega _{0}t)}\). The amplitude \(A\) has to satisfy the NLS equation \(A_{\tau }=i\frac{\partial _{k}^{2}\omega (k_{0},b)}{2} A_{\xi \xi }+i\nu (k_{0},b)A\left\vert A\right\vert ^{2}\), where \(\tau =\varepsilon ^{2}t\), \(\xi =\varepsilon (x-c_{g}t)\) and \(\nu (k_{0},b)\in \mathbb{R}\). The purpose of the paper is to analyze the validity of this NLS approximation by proving error estimates. The main result of the paper proves that for all Bond numbers \(b\) satisfying some constraints and all solutions \(A\in C^{0}([0,\tau _{0}],H^{s}(\mathbb{R},\mathbb{C}))\) of the previous NLS equation satisfying \(\sup_{\tau \in \lbrack 0,\tau _{0}]}\left\Vert A(\text{.},\tau )\right\Vert _{H^{s}(\mathbb{R},\mathbb{C} )}\leq C_{0}\), and all \(\varepsilon \in (0,\varepsilon _{0})\), there exists a solution \((\eta ,v_{1})\in C^{0}([0,\tau _{0}\varepsilon ^{-2}],(H^{s}( \mathbb{R},\mathbb{R}))^{2})\) of the above system, which satisfies \( \sup_{\tau \in \lbrack 0,\tau _{0}\varepsilon ^{-2}]}\left\Vert \left( \begin{array}{c} \eta \\ v_{1} \end{array} \right) (.,t)-\Psi _{NLS}(.,t)\varphi (k_{0},b)\right\Vert _{(H^{s}(\mathbb{R },\mathbb{R}))^{2}}\leq C(b)\varepsilon ^{3/2}\), where \(\Psi _{NLS}(x,t)=A(\varepsilon (x-\partial _{k}\omega (k_{0},b)t),\varepsilon ^{2}t)e^{i(k_{0}x-\omega (k_{0},b)t)}+c.c.\), and \(\varphi (k_{0},b)\in \mathbb{R}^{2}\) is an explicitly computable vector. For the proof, the author assumes that \(s\) is an integer. He uses the arc length formulation of the 2D water wave problem and he proves properties of the operator obtained by linearizing \(\mathcal{K=K}(\eta )\) defined through \(\varphi _{y}=\mathcal{ K}(\eta )\varphi _{x}\) around the trivial solution \((\eta ,\phi _{x})=(0,0)\). He then formally derives the NLS approximation. He proves the main result, essentially using Fourier transform, a modified energy method, and an appropriate weight function to circumvent the problems with the resonances at \(\pm k_{0}\).