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Elastic wave propagation in fluid-saturated porous media. I. The existence and uniqueness theorems. (English) Zbl 0616.76104

The existence and uniqueness theorem for Biot’s dynamic equations, describing elastic wave propagation in a system composed of a bounded solid saturated by a compressible viscous fluid is proved, using in a classical manner the Galerkin method.
Reviewer: G.Pasa

MSC:

76S05 Flows in porous media; filtration; seepage
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76T99 Multiphase and multicomponent flows
76M99 Basic methods in fluid mechanics

Citations:

Zbl 0616.76105
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References:

[1] M.A. BIOT, General Theory of Three-Dimensional Consolidation, Journal of Applied Physics, Vol. 12 (1941), pp. 155-165. Zbl67.0837.01 JFM67.0837.01 · JFM 67.0837.01
[2] M. A. BIOT, Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range, Journal of the Acoustical Society of America, Vol. 28, Number 2 (1965), pp. 168-178. MR134056
[3] M. A. BIOT and D. G. WILLIS, The Elastic Coefficient of the Theory of Consolidation, Journal of Applied Mechanics, Vol. 24, Trans. Asme, Vol. 79 (1957), pp. 594-601. MR92472
[4] G. DUVAUT and J. L. LIONS, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. Zbl0331.35002 MR521262 · Zbl 0331.35002
[5] I. FATT, The Biot-Willis Elastic Coefficients for a Sandstone, Journal of Applied Mechanics, Vol. 26 (1959), pp. 296-297.
[6] G. FICHERA, Existence Theorems in Elasticity-Boundary Value Problems of Elasticity with Unilateral Constrains, Encyclopedia of Physics, S. Flüge, Ed., Vol. VI a/2 : Mechanics of Solids II, C. Truesdell, Ed., Springer-Verlag, Berlin, 1972, pp. 347-424.
[7] V. GIRAULT and P. A. RAVIART, Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, Berlin, 1981. Zbl0441.65081 MR548867 · Zbl 0441.65081
[8] J. L. LIONS, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. Zbl0189.40603 MR259693 · Zbl 0189.40603
[9] J. A. NITSCHE, On Korn’s Second Inequality, preprint, Institute für Angenwandte Mathematik, Albert Ludwig Universitat, Herman-Herder Str. 10, 7800, Freiburg i, Br., West Germany.
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