Global dissipative solutions of the defocusing isothermal Euler–Langevin–Korteweg equations

Chauleur, Quentin

doi:10.3233/asy-211681

Cited by 5 publications

(6 citation statements)

References 22 publications

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“…On the other hand, having proven Theorem 1.3, one may ask if the solutions from Proposition 1.7 can be obtained through the inviscid limit ν → 0. Such a convergence has been proven in [8] for the barotropic case, and [17] for the (damped) isothermal case, both times in a periodic setting x ∈ T d . The damping in [17] can easily be removed, but in order to consider the case x ∈ R d , the order of the limits → ∞ and ν → 0 is certainly a delicate issue, which we leave out at this stage.…”

Section: Annales De L'institut Fouriermentioning

confidence: 79%

“…Such a convergence has been proven in [8] for the barotropic case, and [17] for the (damped) isothermal case, both times in a periodic setting x ∈ T d . The damping in [17] can easily be removed, but in order to consider the case x ∈ R d , the order of the limits → ∞ and ν → 0 is certainly a delicate issue, which we leave out at this stage. Finally, both limits γ → 1 and ν → 0 seem highly singular when = 0 (or goes simultaneously to 0) even in terms of (R, U ).…”

Section: Annales De L'institut Fouriermentioning

confidence: 79%

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Global weak solutions for quantum isothermal fluids

Carles¹,

Carrapatoso²,

Hillairet³

2022

Annales De l'Institut Fourier

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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.Résumé. -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.

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Section: Annales De L'institut Fouriermentioning

confidence: 79%

Section: Annales De L'institut Fouriermentioning

confidence: 79%

Global weak solutions for quantum isothermal fluids

Carles¹,

Carrapatoso²,

Hillairet³

2022

Annales De l'Institut Fourier

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“…We now state our second result, which analyzes the case µ = 0, that it the logarithmic Schrödinger equation (2). Recall that for m ∈ Z d , plane wave solutions of (2) are explicitly given by (12) ν m (t, x) = ρe iθ 0 e im•x e −i(|m| 2 +2λ log ρ)t for all t ≥ 0 and x ∈ T d , and…”

Section: Algebraic Context and Main Resultsmentioning

confidence: 99%

“…However, few results are known about these equations on the d-dimensional torus geometry. The existence and uniqueness of global weak solutions to the logarithmic Schrödinger equation (2) are given in [4], and the existence of global dissipative solutions to the Euler-Langevin-Korteweg equations, which is the fluid counterpart of the Schrödinger-Langevin equation through the Madelung transform ψ = √ ρe iS , is established in [12]. As no behaviour properties or asymptotic features are currently known up to the authors knowledge, this paper is a first step in order to give some qualitative description of the solutions of these equations on T d .…”

Section: Introductionmentioning

confidence: 99%

Around plane waves solutions of the Schr{ö}dinger-Langevin equation

Chauleur¹,

Faou²

2021

Preprint

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We consider the logarithmic Schrödinger equations with damping, also called Schrödinger-Langevin equation. On a periodic domain, this equation possesses plane wave solutions that are explicit. We prove that these solutions are asymptotically stable in Sobolev regularity. In the case without damping, we prove that for almost all value of the nonlinear parameter, these solutions are stable in high Sobolev regularity for arbitrary long times when the solution is close to a plane wave. We also show and discuss numerical experiments illustrating our results.

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“…Meanwhile, there are a lot of results on existence of weak solutions to these equations. Readers can refer to [14][15][16][17][18][19][20] if they have interest.…”

Section: Introductionmentioning

confidence: 99%

Blow-up of Classical Solutions to the Isentropic Compressible Barotropic Navier-Stokes-Langevin-Korteweg Equations

Hu¹

2022

JPDE

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In this paper, we study the barotropic Navier-Stokes-Langevin-Korteweg system in R 3 . Assuming the derivatives of the square root of the density and the velocity field decay to zero at infinity, we can prove the classical solutions blow up in finite time when the initial energy has a certain upper bound. We obtain this blow up result by a contradiction argument based on the conservation of the total mass and the total quasi momentum.

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Global dissipative solutions of the defocusing isothermal Euler–Langevin–Korteweg equations

Cited by 5 publications

References 22 publications

Global weak solutions for quantum isothermal fluids

Global weak solutions for quantum isothermal fluids

Around plane waves solutions of the Schr{ö}dinger-Langevin equation

Blow-up of Classical Solutions to the Isentropic Compressible Barotropic Navier-Stokes-Langevin-Korteweg Equations

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