Benzoni-Gavage S., Serre D. Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications
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Benzoni-Gavage and Serre: Multi-dimensional hyperbolic partial differential equations
Multidimensional Hyperbolic
Partial Differential Equations
First-order Systems and Applications
Sylvie Benzoni-Gavage
Universit´e Claude Bernard Lyon I
Lyon, France
Denis Serre
ENS de Lyon
Lyon, France
CLARENDON PRESS • OXFORD
2007
3
Great Clarendon Street, Oxford ox2 6dp
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c
Sylvie Benzoni-Gavage and Denis Serre, 2007
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PREFACE
Hyperbolic Partial Differential Equations (PDEs), and in particular first-order
systems of conservation laws, have been a fashionable topic for over half a
century. Many books have been written, but few of them deal with genuinely
multidimensional hyperbolic problems: in this respect the most classical, though
not so well-known, references are the books by Reiko Sakamoto, by Jacques
Chazarain and Alain Piriou, and by Andrew Majda. Quoting Majda from his
1984 book, “the rigorous theory of multi-D conservation laws is a field in its
infancy”. We dare say it is still the case today. However, some advances have been
made by various authors. To speak only of the stability of shock waves, we may
think in particular of: M´etivier and coworkers, who continued Majda’s work in
several interesting directions – weak shocks, lessening the regularity of the data,
elucidation of the ‘block structure’ assumption in the case of characteristics with
constant multiplicities (we shall speak here of constantly hyperbolic operators);
Freist¨uhler, who extended Majda’s approach to undercompressive shocks, of
which an important example is given by phase boundaries in van der Waals fluids,
as treated by Benzoni-Gavage; Coulombel and Secchi, who dealt very recently
with neutrally stable discontinuities (2D-vortex sheets), thanks to Nash–Moser
techniques.
Even though it does not pretend to cover the most recent results, this book
aims at presenting a comprehensive view of the state-of-the-art, with particular
emphasis on problems in which modern tools of analysis have proved useful.
A large part of the book is indeed devoted to initial boundary value problems
(IBVP), which can only be dealt with by using symbolic symmetrizers, and thus
necessitate pseudo-differential calculus (for smooth, non-constant coefficients)
or even para-differential calculus (for rough coefficients and therefore also non-
linear problems). In addition, the construction of symbolic symmetrizers conceals
intriguing questions related to algebraic geometry, which were somewhat hidden
in Kreiss’ original paper and in the book by Chazarain and Piriou. In this respect
we propose here new insight, in connection with constant coefficient IBVPs.
Furthermore, the analysis of (linear) IBVPs, which are important in themselves,
enables us to prepare the way for the (non-linear) stability analysis of shock
waves. In the matter of complexity, stability of shocks is the culminating topic
in this book, which we hope will contribute to make more accessible some of
the finest results currently known on multi-D conservation laws. Finally, quoting
Constantin Dafermos from his 2000 book, ‘hyperbolic conservation laws and gas
dynamics have been traveling hand-by-hand over the past one hundred and fifty
years’. Therefore it is not a surprise that we devote a significant part of this book
vi Preface
to that specific and still important application. The idea of dealing with ‘real’
gases was inspired by the PhD thesis of St´ephane Jaouen after Sylvie Benzoni-
Gavage was asked by his advisor, Pierre-Arnaud Raviart to act as a referee in
the defense.
This volume contains enough material for several graduate courses – which
were actually taught by either one of the authors in the past few years – depend-
ing on the topic one is willing to emphasize: hyperbolic Cauchy problem and
IBVP, non-linear waves, or gas dynamics. It provides an extensive bibliography,
including classical papers and very recent ones, both in PDE analysis and in
applications (mainly to gas dynamics). From place to place, we have adopted an
original approach compared to the existing literature, proposed new results, and
filled gaps in proofs of important theorems. For some highly technical results,
we have preferred to point out the main tools and ideas, together with precise
references to original papers, rather than giving extended proofs.
We hope that this book will fulfill the expectations of researchers in both
hyperbolic PDEs and compressible fluid dynamics, while being accessible to
beginners in those fields. We have tried our best to make it self-contained and
to proceed as gradually as possible (at the price of some repetition), so that the
reader should not be discouraged by her/his first reading.
We warmly thank Jean-Fran¸cois Coulombel, whose PhD thesis (under the
supervision of Benzoni-Gavage and with the kind help of Guy M´etivier) provided
the energy necessary to complete the writing of the most technical parts, for his
careful reading of the manuscript and numerous useful suggestions. We also thank
our respective families for their patience and support.
Lyon, April 2006 Sylvie Benzoni-Gavage
Denis Serre
CONTENTS
Preface v
Introduction xiii
Notations xxi
PART I. THE LINEAR CAUCHY PROBLEM
1 Linear Cauchy Problem with Constant Coefficients 3
1.1 Very weak well-posedness 4
1.2 Strong well-posedness 7
1.2.1 Hyperbolicity 7
1.2.2 Distributional solutions 9
1.2.3 The Kreiss’ matrix Theorem 10
1.2.4 Two important classes of hyperbolic systems 13
1.2.5 The adjoint operator 15
1.2.6 Classical solutions 15
1.2.7 Well-posedness in Lebesgue spaces 16
1.3 Friedrichs-symmetrizable systems 17
1.3.1 Dependence and influence cone 18
1.3.2 Non-decaying data 20
1.3.3 Uniqueness for non-decaying data 21
1.4 Directions of hyperbolicity 23
1.4.1 Properties of the eigenvalues 23
1.4.2 The characteristic and forward cones 26
1.4.3 Change of variables 27
1.4.4 Homogeneous hyperbolic polynomials 30
1.5 Miscellaneous 32
1.5.1 Hyperbolicity of subsystems 32
1.5.2 Strichartz estimates 36
1.5.3 Systems with differential constraints 41
1.5.4 Splitting of the characteristic polynomial 45
1.5.5 Dimensional restrictions for strictly hyperbolic systems 47
1.5.6 Realization of hyperbolic polynomial 48
viii Contents
2 Linear Cauchy problem with variable coefficients 50
2.1 Well-posedness in Sobolev spaces 51
2.1.1 Energy estimates in the scalar case 51
2.1.2 Symmetrizers and energy estimates 52
2.1.3 Energy estimates for less-smooth coefficients 58
2.1.4 How energy estimates imply well-posedness 63
2.2 Local uniqueness and finite-speed propagation 72
2.3 Non-decaying infinitely smooth data 80
2.4 Weighted in time estimates 81
PART II. THE LINEAR INITIAL BOUNDARY
VA L U E P RO B L E M
3 Friedrichs-symmetric dissipative IBVPs 85
3.1 The weakly dissipative case 85
3.1.1 Traces 88
3.1.2 Monotonicity of A 89
3.1.3 Maximality of A 90
3.2 Strictly dissipative symmetric IBVPs 93
3.2.1 The a priori estimate 95
3.2.2 Construction of ˆu and u 96
4 Initial boundary value problem in a half-space with constant
coefficients 99
4.1 Position of the problem 99
4.1.1 The number of scalar boundary conditions 100
4.1.2 Normal IBVP 102
4.2 The Kreiss–Lopatinski˘ı condition 102
4.2.1 The non-characteristic case 103
4.2.2 Well-posedness in Sobolev spaces 106
4.2.3 The characteristic case 107
4.3 The uniform Kreiss–Lopatinski˘ı condition 109
4.3.1 A necessary condition for strong well-posedness 109
4.3.2 The characteristic IBVP 111
4.3.3 An equivalent formulation of (UKL) 112
4.3.4 Example: The dissipative symmetric case 113
4.4 The adjoint IBVP 114
4.5 Main results in the non-characteristic case 118
4.5.1 Kreiss’ symmetrizers 119
4.5.2 Fundamental estimates 120
4.5.3 Existence and uniqueness for the boundary value
problem in L
2
γ
123
4.5.4 Improved estimates 125
Contents ix
4.5.5 Existence for the initial boundary value problem 126
4.5.6 Proof of Theorem 4.3 128
4.5.7 Summary 129
4.5.8 Comments 129
4.6 A practical tool 130
4.6.1 The Lopatinski˘ı determinant 130
4.6.2 ‘Algebraicity’ of the Lopatinski˘ı determinant 133
4.6.3 A geometrical view of (UKL) condition 136
4.6.4 The Lopatinski˘ı determinant of the adjoint IBVP 137
5 Construction of a symmetrizer under (UKL) 139
5.1 The block structure at boundary points 139
5.1.1 Proof of Lemma 4.5 139
5.1.2 The block structure 141
5.2 Construction of a Kreiss symmetrizer under (UKL) 144
6 The characteristic IBVP 158
6.1 Facts about the characteristic case 158
6.1.1 A necessary condition for strong well-posedness 159
6.1.2 The case of a linear eigenvalue 162
6.1.3 Facts in two space dimensions 167
6.1.4 The space E
−
(0,η) 169
6.1.5 Conclusion 174
6.1.6 Ohkubo’s case 175
6.2 Construction of the symmetrizer; characteristic case 176
7 The homogeneous IBVP 182
7.1 Necessary conditions for strong well-posedness 184
7.1.1 An illustration: the wave equation 189
7.2 Weakly dissipative symmetrizer 191
7.3 Surface waves of finite energy 196
8 A classification of linear IBVPs 201
8.1 Some obvious robust classes 202
8.2 Frequency boundary points 202
8.2.1 Hyperbolic boundary points 203
8.2.2 On the continuation of E
−
(τ,η) 205
8.2.3 Glancing points 207
8.2.4 The Lopatinski˘ı determinant along the boundary 208
8.3 Weakly well-posed IBVPs of real type 208
8.3.1 The adjoint problem of a BVP of class WR 210
8.4 Well-posedness of unsual type for BVPs of class WR 211
x Contents
8.4.1 A priori estimates (I) 211
8.4.2 A priori estimates (II) 214
8.4.3 The estimate for the adjoint BVP 216
8.4.4 Existence result for the BVP 217
8.4.5 Propagation property 218
9 Variable-coefficients initial boundary value problems 220
9.1 Energy estimates 222
9.1.1 Functional boundary symmetrizers 225
9.1.2 Local/global Kreiss’ symmetrizers 229
9.1.3 Construction of local Kreiss’ symmetrizers 233
9.1.4 Non-planar boundaries 242
9.1.5 Less-smooth coefficients 245
9.2 How energy estimates imply well-posedness 255
9.2.1 The Boundary Value Problem 255
9.2.2 The homogeneous IBVP 264
9.2.3 The general IBVP (smooth coefficients) 267
9.2.4 Rough coefficients 271
9.2.5 Coefficients of limited regularity 281
PART III. NON-LINEAR PROBLEMS
10 The Cauchy problem for quasilinear systems 291
10.1 Smooth solutions 292
10.1.1 Local well-posedness 292
10.1.2 Continuation of solutions 302
10.2 Weak solutions 304
10.2.1 Entropy solutions 305
10.2.2 Piecewise smooth solutions 311
11 The mixed problem for quasilinear systems 315
11.1 Main results 316
11.1.1 Structural and stability assumptions 316
11.1.2 Conditions on the data 318
11.1.3 Local solutions of the mixed problem 319
11.1.4 Well-posedness of the mixed problem 320
11.2 Proofs 321
11.2.1 Technical material 321
11.2.2 Proof of Theorem 11.1 326
12 Persistence of multidimensional shocks 329
12.1 From FBP to IBVP 331