Electron. J. Diff. Eqns., Vol. 2006(2006), No. 59, pp. 1-35.

Well-posedness of one-dimensional Korteweg models

Sylvie Benzoni-Gavage, Raphael Danchin, Stephane Descombes

We investigate the initial-value problem for one-dimensional compressible fluids endowed with internal capillarity. We focus on the isothermal inviscid case with variable capillarity. The resulting equations for the density and the velocity, consisting of the mass conservation law and the momentum conservation with Korteweg stress, are a system of third order nonlinear dispersive partial differential equations. Additionally, this system is Hamiltonian and admits travelling solutions, representing propagating phase boundaries with internal structure. By change of unknown, it roughly reduces to a quasilinear Schrodinger equation. This new formulation enables us to prove local well-posedness for smooth perturbations of travelling profiles and almost-global existence for small enough perturbations. A blow-up criterion is also derived.

Submitted June 14, 2004. Published May 2, 2006.
Math Subject Classifications: 76N10, 76T10.
Key Words: Capillarity; Korteweg stress; local well-posedness; Schrodinger equation.

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Sylvie Benzoni-Gavage
Institut Camille Jordan, Université Claude Bernard Lyon I
21, avenue Claude Bernard, F-69622 Villeurbanne cedex, France
email: benzoni@math.univ-lyon1.fr
Raphaël Danchin
Centre de Mathématiques, Université Paris XII Val de Marne
61, avenue du Géenéral de Gaulle, F-94010 Créteil cedex, France
email: danchin@univ-paris12.fr
Stephane Descombes
46, allée d'Italie, F-69364 Lyon cedex 07, France
email: sdescomb@umpa.ens-lyon.fr

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