Weak solutions to problems involving inviscid fluids

We consider an abstract functional-differential equation derived from the pressure-less Euler system with variable coefficients that includes several systems of partial differential equations arising in the fluid mechanics. Using the method of convex integration we show the existence of infinitely many weak solutions for prescribed initial data and kinetic energy

Madelung, Gross-Pitaevskii and Korteweg

This paper surveys various aspects of the hydrodynamic formulation of the nonlinear Schrodinger equation obtained via the Madelung transform in connexion to models of quantum hydrodynamics and to compressible fluids of the Korteweg type.Comment: 32 page

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

45 pagesInternational audienceThe 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 ≥ 3 for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if d ≥ 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.Les equations d'Euler-Korteweg sont une modification des equations d'Euler prenant en compte l'effet de la capillarité. Dans le cas général elles forment un syst eme quasi-linéaire qui peut se reformuler comme uné equation de Schrödinger dégénérée. L'existence locale de solutions fortes a ´ eté obtenue par Benzoni-Danchin-Descombes en toute dimension, mais sauf cas tr es particuliers il n'existe pas de résultat d'existence globale. En dimension au moins 3, et sous une condition naturelle de stabilité sur la pression on prouve que pour toute donnée initiale irrotationnelle petite, la solution est globale. La preuve s'appuie sur une estimation d'´ energie modifiée. En dimension au moins 5 les propriétés standard de dispersion suffisent pour conclure tandis que les dimensions 3 et 4requì erent uné etude précise de la structure des nonlinéarités quadratiques pour utiliser la méthode des résonances temps espaces

DISPERSIVE SMOOTHING FOR THE EULER–KORTEWEG MODEL ∗

Abstract. The Euler–Korteweg system consists of a quasi-linear, dispersive perturbation of the Euler equations. The Cauchy problem has been studied in any dimension d ≥ 1 by Benzoni, Danchin, and Descombes, who obtained local well-posedness results when the velocity is in H s for s>d/2+1. They noticed that one may expect to find some smoothing effect due to the dispersive effects, but there was no proof so far. Our aim here is to give such results in any dimension under their local existence assumptions. In the simpler case of dimension 1 we obtain unconditionnal Kato smoothing (local smoothing of 1/2 derivative). In higher dimensions a few additional hypotheses must be made to get smoothing and we briefly discuss the pertinence of these restrictions