Differential and Integral Equations

Planar traveling waves in capillary fluids

Sylvie Benzoni-Gavage

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By capillary fluids we mean compressible, inviscid fluids whose energy depends not only on their density but also on their density gradient. Their motion is thus governed by systems of conservation laws, either in Eulerian coordinates or in Lagrangian coordinates, that are a higher-order modification of the usual equations of gas dynamics. In both settings, we receive models that also arise in other fields, in particular in water-waves theory and quantum hydrodynamics. Those Hamiltonian systems typically admit three types of planar traveling waves, namely, heteroclinic, homoclinic, and periodic ones. The purpose here is to review the main tools and results regarding the stability of those waves, under most general assumptions on the energy law. Special attention is devoted to the correspondence between traveling waves in Eulerian coordinates and those in Lagrangian coordinates.

Article information

Differential Integral Equations, Volume 26, Number 3/4 (2013), 439-485.

First available in Project Euclid: 5 February 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B35: Stability 35B10: Periodic solutions 35Q35: PDEs in connection with fluid mechanics 35Q40: PDEs in connection with quantum mechanics 35Q51: Soliton-like equations [See also 37K40] 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 37K45: Stability problems


Benzoni-Gavage, Sylvie. Planar traveling waves in capillary fluids. Differential Integral Equations 26 (2013), no. 3/4, 439--485. https://projecteuclid.org/euclid.die/1360092830

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