Global weak solutions for quantum isothermal fluids

Carles, Rémi; Carrapatoso, Kleber; Hillairet, Matthieu

Annales de l'Institut Fourier, Tome 72 (2022) no. 6, pp. 2241-2298.

Nous construisons des solutions faibles globales pour l’équation de Navier-Stokes quantique isotherme, avec ou sans terme de Korteweg, dans tout l’espace en dimension au plus trois. Au lieu de travailler sur les inconnues originales, nous considérons une reformulation équivalente basée sur un changement d’échelle dépendant du temps, introduit précédemment pour étudier le comportement en temps grand, et qui fournit des estimations a priori convenables, par opposition à la formulation originale dans laquelle l’énergie potentielle n’a pas de signe. Nous travaillons sur des tores dont la taille tend vers l’infini. Sur chacun des tores, nous considérons les équations en présence d’une force de traînée. Ces équations sont résolues par régularisation, et on traite la limite où la force de traînée devient nulle en reprenant la notion de solution renormalisée développée par I. Lacroix-Violet et A. Vasseur. Nous montrons également l’existence globale de solutions faibles pour l’équation de Korteweg isotherme (sans viscosité) lorsque les données initiales sont bien préparées, au sens où elles proviennent d’une transformée de Madelung.

We construct global weak solutions to isothermal quantum Navier–Stokes equations, with or without Korteweg term, in the whole space of dimension at most three. Instead of working on the initial set of unknown functions, we consider an equivalent reformulation, based on a time-dependent rescaling, that we introduced in a previous paper to study the large time behavior, and which provides suitable a priori estimates, as opposed to the initial formulation where the potential energy is not signed. We proceed by working on tori whose size eventually becomes infinite. On each fixed torus, we consider the equations in the presence of drag force terms. Such equations are solved by regularization, and the limit where the drag force terms vanish is treated by resuming the notion of renormalized solution developed by I. Lacroix-Violet and A. Vasseur. We also establish global existence of weak solutions for the isothermal Korteweg equation (no viscosity), when initial data are well-prepared, in the sense that they stem from a Madelung transform.

Classification : 35D30, 35Q30, 35Q35, 35Q40, 76N15, 76Y05
Keywords: Weak solutions, Renormalized solutions, Quantum isothermal fluids, Navier–Stokes equation, Korteweg equation
Mot clés : Solutions faibles, Solutions renormalisées, Fluides quantiques isothermes, équation de Navier-Stokes, équation de Korteweg

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