Geodesic
Well-posedness of one-dimensional Korteweg models
Electronic journal of differential equations, Tome 2006 (2006)
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.
Classification : 76N10, 76T10
Keywords: capillarity, Korteweg stress, local well-posedness, Schrödinger equation 
@article{EJDE_2006__2006__a133,
     author = {Benzoni-Gavage,  Sylvie and Danchin,  Rapha\"el and Descombes,  St\'ephane},
     title = {Well-posedness of one-dimensional {Korteweg} models},
     journal = {Electronic journal of differential equations},
     year = {2006},
     volume = {2006},
     zbl = {1114.76058},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a133/}
}
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%A Descombes,  Stéphane
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%F EJDE_2006__2006__a133
Benzoni-Gavage,  Sylvie; Danchin,  Raphaël; Descombes,  Stéphane. Well-posedness of one-dimensional Korteweg models. Electronic journal of differential equations, Tome 2006 (2006). http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a133/