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We study the Euler equations for slightly compressible fluids, that is, after rescaling, the limits of the Euler equations of fluid dynamics as the Mach number tends to zero. In this paper, we consider the general non-isentropic equations and general data. We first prove the existence of classical solutions for a time independent of the small parameter. Then, on the whole space R d , we prove that the solution converges to the solution of the incompressible Euler equations.

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This paper is concerned with a priori C ∞ regularity for threedimensional doubly periodic travelling gravity waves whose fundamental domain is a symmetric diamond. The existence of such waves was a long standing open problem solved recently by Iooss and Plotnikov. The main difficulty is that, unlike conventional free boundary problems, the reduced boundary system is not elliptic for three-dimensional pure gravity waves, which leads to small divisors problems. Our main result asserts that sufficiently smooth diamond waves which satisfy a diophantine condition are automatically C ∞ . In particular, we prove that the solutions defined by Iooss and Plotnikov are C ∞ . Two notable technical aspects are that (i) no smallness condition is required and (ii) we obtain an exact paralinearization formula for the Dirichlet to Neumann operator.

We extend the Kreiss-Majda theory of stability of hyperbolic initial-boundary-value and shock problems to a class of systems, notably including the equations of magnetohydrodynamics (MHD), for which Majda's block structure condition does not hold: namely, simultaneously symmetrizable systems with characteristics of variable multiplicity, satisfying at points of variable multiplicity either a ''totally nonglancing'' or a ''nonglancing and linearly splitting'' condition. At the same time, we give a simple characterization of the block structure condition as ''geometric regularity'' of characteristics, defined as analyticity of associated eigenprojections. The totally nonglancing or nonglancing and linearly splitting conditions are generically satisfied in the simplest case of crossings of two characteristics, and likewise for our main physical examples of MHD or Maxwell equations for a crystal. Together with previous analyses of spectral stability carried out by Gardner-Kruskal and Blokhin-Trakhinin, this yields immediately a number of new results of nonlinear inviscid stability of shock waves in MHD in the cases of parallel or transverse magnetic field, and recovers the sole previous nonlinear result, obtained by Blokhin-Trakhinin by direct ''dissipative integral'' methods, of stability in the zero-magnetic field limit. We also discuss extensions to the viscous case. r 2004 Published by Elsevier Inc.