Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 59, pp. 1-35.

Title: Well-posedness of one-dimensional Korteweg models
       
Authors: Sylvie Benzoni-Gavage (Univ. Claude Bernard Lyon I, France)
         Raphael Danchin (Univ. Paris XII, France)
	 Stephane Descombes (UMPA, ENS Lyon, France)

Abstract: 
  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.