On the Zero Capillarity Limit for the Euler–Korteweg System

We study the Euler–Korteweg equations with a weak capillarity tensor. It formally converges to the Euler equations in the zero capillarity limit. Our aim is twofold: first, we prove rigorously this limit in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:math> , and obtain a more precise WKB expansion of the solution, second, we initiate the study of the problem on the half-space. In this case, we obtain a priori estimates for the solutions that degenerate as the capillary coefficient converges to zero, and we explain this degeneracy with the construction of a (formal) WKB expansion that exhibits boundary layers. The results on the full space extend and improve the classical theory on the semiclassical limit of nonlinear Schrödinger equations. The analysis of the half-space is restricted to the case of quantum fluids with irrotational velocity.