Global well-posedness of the Euler-Korteweg system for small irrotational data

Abstract

The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schrödinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni-Danchin-Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well-posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension $d\ge 3$ for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if $d\ge 5$, and a careful study of the nonlinear structure of the quadratic terms in dimension $3$ and $4$ involving the theory of space time resonance.

arXiv comment45 pages

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• The Quantum Hydrodynamics system in two space dimensions
• Madelung, Gross-Pitaevskii and Korteweg
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Cited by in corpus (6)

• Global Existence of Finite Energy Weak Solutions of Quantum Navier-Stokes Equations
• On the compactness of weak solutions to the Navier-Stokes-Korteweg equations for capillary fluids
• Small energy traveling waves for the Euler-Korteweg system
• Long-time stability of the quantum hydrodynamic system on irrational tori
• Rigidity results in generalized isothermal fluids
• Global-in-time Well-posedness of the One-dimensional Hydrodynamic Gross-Pitaevskii Equations without Vacuum