2025/02/14 by Audiard, Corentin, Vassenet, Marc-Antoine · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2502.10034
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 two-fold : first we prove rigorously this limit in R d , d ≥ 1, and obtain a more precise BKW 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) BKW expansion that exhibits boundary layers. The results on the full space extend and improve a classical result of Grenier (1998) on the semi-classical limit of nonlinear Schrödinger equations. The analysis on the half space is restricted to the case of quantum fluids with irrotational velocity.