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
Author(s): Demetrios Christodoulou
Series: EMS Monographs in Mathematics
Publisher: European Mathematical Society
Year: 2019
Language: English
Pages: 934
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
