The Shock Development Problem

Description

This 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 author‘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. Keywords: Nonlinear hyperbolic partial differential equations, free boundary problems, mechanics of compressible fluids, development of shocks Prologue......Page 14 General equations of motion......Page 56 The irrotational case and the nonlinear wave equation......Page 67 The non-relativistic limit......Page 71 Jump conditions......Page 86 The shock development problem......Page 100 The restricted shock development problem......Page 110 General construction in Lorentzian geometry......Page 114 The characteristic system......Page 120 The wave system......Page 127 Variations by translations and the wave equation for the rectangular components of $\beta$......Page 129 Geometric construction for the shock development problem......Page 138 Connection coefficients and the first variation equations......Page 142 Structure functions and the formulas for the torsion forms......Page 148 Propagation equations for $\lambda$ and $\underline{\lambda}$......Page 160 Second variation and cross variation equations......Page 161 The case $n=2$......Page 166 The Codazzi and Gauss equations ($n>2$)......Page 171 Analysis of the boundary conditions......Page 178 Transformation functions and identification equations......Page 184 Regularization of identification equations......Page 206 Propagation equations for $\underline{\lambda}$ and $s_{NL}$ on $\underline{\cal C}$......Page 212 Propagation equations for higher-order derived data $T^m \underline{\lambda}$ and $T^m s_\{NL}$ on $\underline{\cal C}$......Page 221 Boundary conditions for higher-order derived data and determination of the $T$-derivatives of the transformation functions on $\partial_{-}{\cal B}$......Page 224 Bi-variational stress......Page 242 Variation fields $V$ and associated $1$-forms $\theta^\mu$......Page 246 Fundamental energy identities......Page 252 Boundary condition on ${\cal K}$ for the $1$-forms $^{(V)}\xi$......Page 259 Coercivity at the boundary. Choice of multiplier field......Page 280 Deformation tensor of the multiplier field. Error integral associated to $^{(V)}Q_1$......Page 290 Error integral associated to $ ^{(V)}Q_2$......Page 299 Commutation fields and higher-order variations......Page 310 Recursion formulas for source functions......Page 311 Deformation tensors of the commutation fields......Page 318 Principal acoustical error terms......Page 322 Setup of the truncated power series......Page 332 Estimates for the quantities by which the $N$th approximants fail to satisfy the characteristic and wave systems......Page 335 Estimates for the quantities by which the $N$th approximants fail to satisfy the boundary conditions......Page 350 Estimates for the quantities by which the $N$th approximants fail to satisfy the identification equations......Page 358 Estimates for the quantity by which the $\beta_{\mu,N}$ fail to satisfy the wave equation relative to $\tilde{h}_N$ and to $\tilde{h}^\prime_N$......Page 361 Variation differences $^{(m,l)}\check{\dot{\phi}}_\mu$ and rescaled source differences $^{(m,l)}\check{\tilde{\rho}}_\mu$......Page 375 Difference $1$-forms $^{(V;m,l)}\check{\xi}$. Difference energies and difference energy identities......Page 381 Regularization of the propagation equations for $\tilde{\chi}$ and $\underline{\tilde{\chi}}$......Page 386 Regularization of the propagation equations for $E^2 \lambda$ and $E^2 \underline{\lambda}$......Page 392 Structure equations for the $N$th approximants......Page 403 Propagation equations for $\check{\theta}_l$, $\underline{\check{\theta}}_l$ and for $\check{\nu}_{m,l}$, $\underline{\check{\nu}}_{m,l}$......Page 421 Estimates for $\check{\theta}_l$......Page 439 Estimates for $\check{\nu}_{m-1,l+1}$......Page 461 Estimates for $\underline{\check{\theta}}_l$ and $\underline{\check{\nu}}_{m-1,l+1}$ in terms of their boundary values on $K$......Page 484 Boundary conditions on $K$ and preliminary estimates for $\underline{\check{\theta}}_l$ and $\underline{\check{\nu}}_{m-1,l+1}$ on $K$......Page 511 Outline of top-order acoustical estimates for more than 2 spatial dimensions......Page 526 Propagation equations for the next-to-top-order acoustical difference quantities $(^(n-1)\check{\tilde{\chi}}, ^(n-1)\check{\tilde{\underline{\chi}}})$ and $(^(m,n-m)\check{\lambda}, ^(m,n-m)\check{\underline{\lambda}})$ : $m=0,…,n$......Page 556 Estimates for $(^(n-1)\check{\tilde{\chi}}, ^(n-1)\check{\tilde{\underline{\chi}}})$ and $(^(0,n)\check{\lambda}, ^(0,n)\check{\underline{\lambda}})$......Page 579 Estimates for $(T\Omega^n \check{\hat{f}},T\Omega^n \check{v},T\Omega^n \check{\gamma})$......Page 648 Estimates for $(^(m,n-m)\check{\lambda},^(m,n-m)\check{\underline{\lambda}}) : m=1,…,n$......Page 662 Estimates for $(T^(m+1)\Omega^(n-m)\check{\hat{f}}, T^(m+1)\Omega^(n-m)\check{v}, T^(m+1)\Omega^(n-m)\check{\gamma}) : m=1,…,n$......Page 719 Estimates for $(\Omega^(n+1)\check{\hat{f}}, \Omega^(n+1)\check{v}, \Omega^(n+1)\check{\gamma})$......Page 733 Estimates for $^(V;m,n-m)\check{b}$......Page 754 Borderline error integrals contributed by $^(V;m,n-m)\check{Q}_1$, $^(V;m,n-m)\check{Q}_2$......Page 767 Borderline error integrals associated to $\check{\theta}_n$ and to $\check{\nu}_{m-1,n-m+1} : m=1,…,n$......Page 768 Borderline error integrals associated to $\check{\underline{\theta}}_n$ and to $\check{\underline{\nu}}_{m-1,n-m+1} : m=1,…,n$......Page 778 Top-order energy estimates......Page 790 Summary of the preceding, outline of the following, and statement of the theorem......Page 804 Bootstrap assumptions needed......Page 810 $L^2(S_{\underline{u},u})$ estimates for $^{(n-1)}\check{\tilde{\chi}}$ and for $^{(m,n-m)}\check{\lambda} : m=0,…,n-1$......Page 819 $L^2(\cal{K}_ igma^\tau)$ estimates for the $n$th-order acoustical differences......Page 832 $L^2(S_{\underline{u},u})$ estimates for the $n$th-order variation differences......Page 841 $L^2(S_{\underline{u},u})$ estimates for the $(n-1)$th-order acoustical differences......Page 855 $L^2(S_{\underline{u},u})$ estimates for all $n$th-order derivatives of the $\beta_\mu$......Page 864 $L^2(S_{\underline{u},u})$ estimates for $\Omega^{n-1}log\cross{h}$ and $\Omega^{n-1}b$......Page 868 Lower-order $L^2(S_{\underline{u},u})$ estimates......Page 873 Pointwise estimates and recovery of the bootstrap assumptions......Page 876 Completion of the argument......Page 882 Bibliography......Page 928 Index......Page 930