On the well-posedness for the Euler-Korteweg model in several space dimensions

Abstract

International audienceThe Euler-Korreweg model results from a modification of the standard Euler equations governing the motion of compressible inviscid fluids through the adjunction of the Korteweg stress tensor, which takes into account capillarity effects in regions where the density experiences large variations, typically across interfaces for fluids exhibiting phase changes. One of the main difficulties in the analysis of the Cauchy problem for this model, a third order system of conservation laws, is the absence of dissipative regularization, since viscosity is neglected. The Cauchy problem for isothermal fluids in one space dimension has been addressed by the authors in an earlier paper, using Lagrangian coordinates. Here the Cauchy problem is investigated in arbitrary space dimension N, still for isothermal fluids, and a variable capillarity coefficient. A local well-posedness result is obtained in Sobolev spaces as though the density gradient and the velocity field were solutions of a symmetrizable hyperbolic system. More precisely, well-posedness is shown for Hs+1 x H-s (s > N/2 + 1) perturbations of smooth global solutions, either constant states or traveling profiles. In addition, almost-global existence is proved for small enough perturbations, and a blow-up criterion is shown. Proofs rely on a suitable extended formulation of the system, which turns out to amount to a nonlinear degenerate Schrodinger equation coupled with a transport equation, and on a priori estimates without loss of derivatives for the extended system, which necessitate various 'gauge' functions to cancel out bad commutators

Co-periodic stability of periodic waves in some Hamiltonian PDEs

“…Finally, in view of earlier work [BGDD06,BGDD07] on the Cauchy problem for (EKL) and (EKE) in one space dimension, we claim that (H3) is satisfied in H s+1 × H s for s > 3/2. We thus have all the ingredients to apply Theorem 3 to (EKL) and (EKE).…”

On the linear wave regime of the Gross-Pitaevskii equation

“…However for the full system, in the computation of the energy estimates, the higher order derivatives are difficult to control, both by themselves and by their interaction with the previously mentioned singular terms. A similar difficulty in a related context was overcome by S. Benzoni-Gavage, the second author and S. Descombes in [4]. The crucial point there, inspired by earlier works by F. Coquel [13], is to consider an augmented system, adding the equation for ∇(log ρ 2 ε ).…”

“…In the existing mathematical literature, the Gross-Pitaevskii equation is sometimes considered in its semi-classical form (4) iε ∂Ψ ε ∂t + ε 2 ∆Ψ ε = Ψ ε (|Ψ ε | 2 − 1). One can easily recover the original equation (GP ) by mean of the hyperbolic scaling Ψ ε (x, t) = Ψ( x ε , t ε ).…”

Diffusive semiconductor moment equations using Fermi–Dirac statistics

“…has been shown in [14] and for µ = νn, λ = 0 in [9]. More general Korteweg stress tensors have been considered in [2,9,24]. In particular, the existence of solutions to the one-dimensional problem with the term div K = n∇(σ ′ (n)∆σ(n)), suggested by [5], was proved in [26].…”

“…For instance, a variable related to the effective velocity w has been employed in the analysis of the interfacial tension in the mixture of incompressible liquids [27, formula (3.6)]. Furthermore, an Euler-Korteweg model has been reformulated in [2] by using the complex variable w = u + iκ∇ log n, where i 2 = −1 and κ = κ(n) is the capillary function. It turns out that in the new variable, the momentum equation becomes a variable-coefficient Schrödinger equation.…”