Well-posedness of one-dimensional Korteweg models
Date
2006-05-02
Authors
Benzoni-Gavage, Sylvie
Danchin, Raphael
Descombes, Stephane
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University-San Marcos, Department of Mathematics
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.
Description
Keywords
Capillarity, Korteweg stress, Local well-posedness, Schrodinger equation
Citation
Benzoni-Gavage, S., Danchin, R., & Descombes, S. (2006). Well-posedness of one-dimensional Korteweg models. Electronic Journal of Differential Equations, 2006(59), pp. 1-35.
Rights
Attribution 4.0 International