IUMJ

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

Authors: S. Benzoni-Gavage, R. Danchin and S. Descombes

Issue: Volume 56 (2007), Issue 4, 1499-1579

Abstract:

The Euler-Korteweg 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 $H^{s+1} \times H^s$ ($s > N/2+1$) 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 Schr\"odinger 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.