02937nam a22003495a 4500001001100000003001200011005001700023006001900040007001500059008004100074020001800115024002100133040001400154072001700168084002200185100003900207245008400246260008200330300003400412336002600446337002600472338003600498347002400534490005300558506006500611520170100676650003502377650004002412650002502452856003202477856007802509238-181122CH-001817-320181122233002.0a fot ||| 0|cr nn mmmmamaa181122e20190131sz fot ||| 0|eng d a978303719692270a10.4171/1922doi ach0018173 7aPBKJ2bicssc a35-xxa76-xx2msc1 aChristodoulou, Demetrios,eauthor.10aThe Shock Development Problemh[electronic resource] /cDemetrios Christodoulou3 aZuerich, Switzerland :bEuropean Mathematical Society Publishing House,c2019 a1 online resource (932 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aEMS Monographs in Mathematics (EMM) ;x2523-51921 aRestricted to subscribers:uhttp://www.ems-ph.org/ebooks.php aThis monograph addresses the problem of the development of shocks in the context of the Eulerian equations of the mechanics of compressible fluids. The mathematical problem is that of an initial-boundary value problem for a nonlinear hyperbolic system of partial differential equations with a free boundary and singular initial conditions.
The free boundary is the shock hypersurface and the boundary conditions are jump conditions relative to a prior solution, conditions following from the integral form of the mass, momentum and energy conservation laws. The prior solution is provided by the authorâ€˜s previous work which studies the maximal classical development of smooth initial data. New geometric and analytic methods are introduced to solve the problem. Geometry enters as the acoustical structure, a Lorentzian metric structure defined on the spacetime manifold by the fluid. This acoustical structure interacts with the background spacetime structure. Reformulating the equations as two coupled first order systems, the characteristic system, which is fully nonlinear, and the wave system, which is quasilinear, a complete regularization of the problem is achieved.
Geometric methods also arise from the need to treat the free boundary. These methods involve the concepts of bi-variational stress and of variation fields. The main new analytic method arises from the need to handle the singular integrals appearing in the energy identities. Shocks being an ubiquitous phenomenon, occuring also in magnetohydrodynamics, nonlinear elasticity, and the electrodynamics of noninear media, the methods developed in this monograph are likely to be found relevant in these fields as well.07aDifferential equations2bicssc07aPartial differential equations2msc07aFluid mechanics2msc40uhttps://doi.org/10.4171/192423cover imageuhttp://www.ems-ph.org/img/books/christodoulou_shock_mini.jpg