Abstract In this paper, we give the first rigorous justification of the Benjamin-Ono equation: $$\begin{aligned} \hspace{3cm} \partial _t \zeta + (1 - \frac{\gamma }{2}\sqrt{\mu }|\textrm{D}|)\partial _x \zeta + \frac{3{\varepsilon }}{2}\zeta \partial _x\zeta =0, \hspace{2cm} \text {(BO)} \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mspace/> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mo>+</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> </mml:mrow> <mml:mfrac> <mml:mi>γ</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:msqrt> <mml:mi>μ</mml:mi> </mml:msqrt> <mml:mrow> <mml:mo>|</mml:mo> <mml:mtext>D</mml:mtext> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mi>ζ</mml:mi> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>ε</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mi>ζ</mml:mi> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mi>ζ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>BO</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> as an internal water wave model on the physical time scale. Here, $${\varepsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> is a small parameter measuring the weak nonlinearity of the waves, $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> is the shallowness parameter, and $$\gamma \in (0,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is the ratio between the densities of the two fluids. To be precise, we first prove the existence of a solution to the internal water wave equations for a two-layer fluid with surface tension, where one layer is of shallow depth and the other is of infinite depth. The existence time is of order $${\mathcal {O}}(\frac{1}{{\varepsilon }})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>ε</mml:mi> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for a small amount of surface tension such that $${\varepsilon }^2 \le \textrm{bo}^{-1} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>≤</mml:mo> <mml:msup> <mml:mtext>bo</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> where $$\textrm{bo}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>bo</mml:mtext> </mml:math> is the Bond number. Then, we show that these solutions are close, on the same time scale, to the solutions of the BO equation with a precision of order $${\mathcal {O}}(\mu + \textrm{bo}^{-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mtext>bo</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In addition, we provide the justification of new equations with improved dispersive properties, the Benjamin equation, and the Intermediate Long Wave (ILW) equation in the deep-water limit. The long-time well-posedness of the two-layer fluid problem was first studied by Lannes [Arch. Ration. Mech. Anal., 208(2):481-567, 2013] in the case where both fluids have finite depth. Here, we adapt this work to the case where one of the fluid domains is of finite depth, and the other one is of infinite depth. The novelties of the proof are related to the geometry of the problem, where the difference in domains alters the functional setting for the Dirichlet-Neumann operators involved. In particular, we study the various compositions of these operators that require a refined symbolic analysis of the Dirichlet-Neumann operator on infinite depth and derive new pseudo-differential estimates that might be of independent interest.

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Justification of the Benjamin–Ono equation as an internal water waves model

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