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Large amplitude variations for the density of a viscous compressible fluid. (Variations de grande amplitude pour la densité d’un fluide visqueux compressible.) (French) Zbl 0739.35071
Summary: The flow of a compressible viscous fluid is governed by the Navier-Stokes equations. This system is of mixed parabolic-hyperbolic type. The hyperbolic part is associated with a linear degeneracy so that initial large-amplitude high-frequency waves can propagate along the particle paths. However, the parabolic part kills the oscillations of the velocity field. We give a formal relaxed (i.e. homogenized) system in any dimension for the Eulerian formulation. In the \(1-d\) case, we prove the relevance of this system in the equivalent Lagrangian formulation.

MSC:
35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35M10 PDEs of mixed type
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[1] Hoff, D., Global existence for 1D, compressible isentropic Navier-Stokes equations with large initial data, Trans. AMS, 303, 169-181, (1987) · Zbl 0656.76064
[2] D. Hoff and D. Serre, The failure of continuous dependence on initial data for the Navier-Stokes equations for compressible flow, Indiana University at Bloomington, Report #9005, submitted to SIAM J. Math. Anal. · Zbl 0741.35057
[3] Hoff, D.; Smoller, J., Solutions in the large for certain nonlinear parabolic systems, Ann. IHP, analyse non linéaire, 2, 213-235, (1985) · Zbl 0578.35044
[4] Kazhikhov, A.V., Cauchy problem for viscous gas equations, Sibirskii mat. Z., 23, 60-64, (1982) · Zbl 0519.35065
[5] Kazhikhov, A.V.; Shelukin, V.V., Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, Pmm, 41, 282-291, (1977)
[6] Lax, P.D., Hyperbolic systems of conservation laws II, Comm. pure appl. math., 10, 537-566, (1957) · Zbl 0081.08803
[7] Matsumura, A.; Nishida, T., The initial value problem for the equations of motion compressible viscous and heat-conductive fluids, Proc. Japan acad. ser A, 55, 337-342, (1979) · Zbl 0447.76053
[8] Murat, F., Compacité par compensation, Ann. scuola norm. sup. Pisa, ser IV, 5, 489-507, (1978) · Zbl 0399.46022
[9] Padula, M.; Padula, M., Existence of global solutions for 2-dimensional viscous compressible flows, J. func. anal., J. func. anal., 77, 232 (E)-20, (1988) · Zbl 0641.76015
[10] Serre, D., Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, (), 639-642, (1986) · Zbl 0597.76067
[11] Tartar, L., Cours peccot, (1977)
[12] W.E., Propagation of oscillations in the solutions of 1 - d compressible fluid equations, preprint.
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