LEADER 03522nam 22004575a 450 001 9910151936903321 005 20091109150325.0 010 $a3-03719-531-2 024 70$a10.4171/031 035 $a(CKB)3710000000953806 035 $a(CH-001817-3)51-091109 035 $a(PPN)178155160 035 $a(EXLCZ)993710000000953806 100 $a20091109j20070110 fy 0 101 0 $aeng 135 $aurnn|mmmmamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aThe Formation of Shocks in 3-Dimensional Fluids$b[electronic resource] /$fDemetrios Christodoulou 210 3 $aZuerich, Switzerland $cEuropean Mathematical Society Publishing House$d2007 215 $a1 online resource (1000 pages) 225 0 $aEMS Monographs in Mathematics (EMM) ;$x2523-5192 330 $aThe equations describing the motion of a perfect fluid were first formulated by Euler in 1752. These equations were among the first partial differential equations to be written down, but, after a lapse of two and a half centuries, we are still far from adequately understanding the observed phenomena which are supposed to lie within their domain of validity. These phenomena include the formation and evolution of shocks in compressible fluids, the subject of the present monograph. The first work on shock formation was done by Riemann in 1858. However, his analysis was limited to the simplified case of one space dimension. Since then, several deep physical insights have been attained and new methods of mathematical analysis invented. Nevertheless, the theory of the formation and evolution of shocks in real three-dimensional fluids has remained up to this day fundamentally incomplete. This monograph considers the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. We consider initial data for these equations which outside a sphere coincide with the data corresponding to a constant state. Under suitable restriction on the size of the initial departure from the constant state, we establish theorems that give a complete description of the maximal classical development. In particular, it is shown that the boundary of the domain of the maximal classical development has a singular part where the inverse density of the wave fronts vanishes, signalling shock formation. The theorems give a detailed description of the geometry of this singular boundary and a detailed analysis of the behavior of the solution there. A complete picture of shock formation in three-dimensional fluids is thereby obtained. The approach is geometric, the central concept being that of the acoustical spacetime manifold. The monograph will be of interest to people working in partial differential equations in general and in fluid mechanics... 606 $aDifferential equations$2bicssc 606 $aFluid mechanics$2bicssc 606 $aPartial differential equations$2msc 606 $aGlobal analysis, analysis on manifolds$2msc 606 $aFluid mechanics$2msc 615 07$aDifferential equations 615 07$aFluid mechanics 615 07$aPartial differential equations 615 07$aGlobal analysis, analysis on manifolds 615 07$aFluid mechanics 686 $a35-xx$a58-xx$a76-xx$2msc 700 $aChristodoulou$b Demetrios$0320209 801 0$bch0018173 906 $aBOOK 912 $a9910151936903321 996 $aFormation of shocks in 3-dimensional fluids$91228818 997 $aUNINA