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 wellposedness 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. RésuméLeséquations d'Euler-Korteweg sont une modification deséquations d'Euler prenant en compte l'effet de la capillarité. Dans le cas général elles forment un système quasi-linéaire qui peut se reformuler comme uneéquation de Schrödinger dégénérée. L'existence locale de solutions fortes aété obtenue par Benzoni-Danchin-Descombes en toute dimension, mais sauf cas très 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'énergie modifiée. En dimension au moins 5 les propriétés standard de dispersion suffisent pour conclure tandis que les dimensions 3 et 4 requièrent uneétude précise de la structure des nonlinéarités quadratiques pour utiliser la méthode des résonances temps espaces.

Global strong solution for the Korteweg system with quantum pressure in dimension $$N\ge 2$$ N ≥ 2

“…When µ = 0 the system corresponds to the Euler system with quantum pressure. In [2], the authors prove the existence of global weak solution for irrotational initial data, in [4,5] we prove with Audiard the existence of global strong solution for small irrotational initial data.…”

On Navier–Stokes–Korteweg and Euler–Korteweg Systems: Application to Quantum Fluids Models

“…Following the framework of the paper, we first present the Euler-Kortewg system and then the Navier-Stokes Korteweg one. Note that in all the paper, the systems are supplemented with the following initial conditions (1) ρ| t=0 = ρ 0 , (ρ u)| t=0 = ρ 0 u 0 for x ∈ Ω.…”

“…Let ρ 0 and u 0 smooth enough. Let (ρ ν , u ν ) be a global weak solution to the quantum Navier-Stokes system (28)-(29) with initial conditions (1). Let (ρ, u) be the weak limit of (ρ ν , u ν ) when ν tends to 0 in the sense…”

Small energy traveling waves for the Euler–Korteweg system

“…Such results were used to construct global unique solutions of (1.1) for small irrotational data by the author and B.Haspot in [2]. The result was later extended by the same authors in [3] for general K, g and d ≥ 3: for initial data near the constant state (ρ 0 , 0) with the stability condition g ′ (ρ 0 ) > 0, the solution is global and converges to a solution of the linearized equation near (ρ 0 , 0) (in other words it scatters). The price to pay for this generalization is the necessity to work with much smoother functions, basically: ρ − ρ 0 ∈ H 50 .…”

“…1. their speed is bounded by the sound speed c s = ρ 0 g ′ (ρ 0 ) for (1.1), 2ρ 0 g ′ (ρ 0 ) for NLS, 2. if there exists a traveling wave of speed c 0 < c s , there exists a local branch of traveling waves parametrized by their speed as ψ c or (ρ c , φ c ), 3. the stability criterion is dP N LS (ψ c )/dc < 0, resp.…”