Structure of Korteweg models and stability of diffuse interfaces

• DOI: 10.4171/ifb/130
• Publication Date: 2009-12-23T00:14:17Z
• Authors: Sylvie Benzoni-Ga...; Raphaël Danchin; Stéphane Descombes; Didier Jamet
• Keywords: Hamiltonian (control theory); Convexity

The models considered are supposed to govern the motion of compressible fluids such as liquidvapor mixtures endowed with a variable internal capillarity.Several formulations and simplifications discussed, from full multi-dimensional equations for non-isothermal motions in Eulerian coordinates one-dimensional isothermal Lagrangian coordinates.Hamiltonian structures pointed out each case, they used study stability two kinds non-linear waves: solitary, or homoclinic waves, heteroclinic which correspond propagating phase boundaries non-zero thickness, also called diffuse interfaces.It is known an earlier work by Benzoni-Gavage [Phys.D, 2001] that latter (weakly) spectrally stable.Here, interfaces shown be orbitally stable.The proof relies on their interpretation critical points Hamiltonian under constraints, whose justification requires some care because different endstates at infinity.Another difficulty comes higher order derivatives not controlled Hamiltonian.In case capillarity, our result unfortunately does imply global existence.As regards solitary come families parametrized wave speed, stable variational point view.However, using method due Grillakis, Shatah Strauss [J.Funct.Anal., 1987], it possible show depending stable.Namely, convexity function speed moment instability determines waves.This approach, already Bona Sachs [Comm.Math.Phys., 1988] Boussinesq equation, here adapted waves Korteweg models, first classified according endstate structures.The corresponding moments computed quadrature.They exhibit both concavity regions.