H ANDBOOK
OF D IFFERENTIAL E QUATIONS
E VOLUTIONARY E QUATIONS
VOLUME II


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H ANDBOOK
OF D IFFERENTIAL E QUATIONS
E VOLUTIONARY E QUATIONS
Volume II

Edited by

C.M. DAFERMOS
Division of Applied Mathematics
Brown University
Providence, USA

E. FEIREISL
Mathematical Institute AS CR
Praha, Czech Republic

2005
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Preface
This is the second volume in the series Evolutionary Equations, part of the Handbook of
Differential Equations project. Whereas Volume I was intended to provide an overview
of diverse abstract approaches, the guiding philosophy of the present volume is to offer
a representative sample of the most challenging specific equations and systems arising in
scientific applications.
Three chapters are devoted to the modern mathematical theory of fluid dynamics: Chapter 1 deals with the Euler equations, Chapter 5 provides a general introduction to the theory
of incompressible viscous fluids, and Chapter 3 discusses the asymptotic limits of discrete
mechanical systems described by the Boltzmann equation.
In a different direction, Chapter 2 introduces the blow-up phenomena of solutions of
general parabolic equations and systems.
Chapters 4 and 6 are closely related and deal with mathematical problems arising in
materials science.
Finally, Chapter 7 explores the topic of nonlinear wave equations.
We have deliberately chosen diverse topics as well as styles of presentation in order to
expose the reader to the enormous variety of problems, methodology and potential applications.
We should like to express our thanks to the authors who have contributed to the present
volume, to the referees who have generously spent time reading the papers, and to the
editors and staff of Elsevier.
Constantine Dafermos
Eduard Feireisl

v


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List of Contributors
Chen, G.-Q., Northwestern University, Evanston, IL, USA (Ch. 1)
Fila, M., Comenius University, Bratislava, Slovakia (Ch. 2)
Golse, F., Institut Universitaire de France and Université Paris 7, Paris, France (Ch. 3)
Krejˇcí, P., Weierstrass-Institute for Applied Analysis and Stochastics, Berlin, Germany
(Ch. 4)
Málek, J., Charles University, Praha, Czech Republic (Ch. 5)
Mielke, A., Weierstraß-Institut für Angewandte Analysis und Stochastik and HumboldtUniversität zu Berlin, Berlin, Germany (Ch. 6)
Rajagopal, K. R., Texas A&M University, College Station, TX, USA (Ch. 5)
Zhang, P., Academy of Mathematics & Systems Science, Beijing, China (Ch. 7)
Zheng, Y., Penn State University, University Park, PA, USA (Ch. 7)

vii


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Contents
Preface
List of Contributors
Contents of Volume I

v
vii
xi

1. Euler Equations and Related Hyperbolic Conservation Laws

G.-Q. Chen
2. Blow-up of Solutions of Supercritical Parabolic Equations
M. Fila
3. The Boltzmann Equation and Its Hydrodynamic Limits
F. Golse
4. Long-Time Behavior of Solutions to Hyperbolic Equations with Hysteresis
P. Krejˇcí
5. Mathematical Issues Concerning the Navier–Stokes Equations and Some
of Its Generalizations
J. Málek and K.R. Rajagopal
6. Evolution of Rate-Independent Systems
A. Mielke
7. On the Global Weak Solutions to a Variational Wave Equation
P. Zhang and Y. Zheng

1
105
159
303

371
461
561

Author Index

649

Subject Index


659

ix


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Contents of Volume I
Preface
List of Contributors

v
vii

1. Semigroups and Evolution Equations: Functional Calculus, Regularity and
Kernel Estimates
W. Arendt
2. Front Tracking Method for Systems of Conservation Laws
A. Bressan
3. Current Issues on Singular and Degenerate Evolution Equations
E. DiBenedetto, J.M. Urbano and V. Vespri
4. Nonlinear Hyperbolic–Parabolic Coupled Systems
L. Hsiao and S. Jiang
5. Nonlinear Parabolic Equations and Systems
A. Lunardi
6. Kinetic Formulations of Parabolic and Hyperbolic PDEs: From Theory
to Numerics
B. Perthame
7. L1 -stability of Nonlinear Waves in Scalar Conservation Laws

D. Serre

1
87
169
287
385

437
473

Author Index

555

Subject Index

565

xi


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CHAPTER 1

Euler Equations and Related Hyperbolic
Conservation Laws


Gui-Qiang Chen
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA
E-mail:

Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . .
2. Basic features and phenomena . . . . . . . . . . . . .
2.1. Convex entropy and symmetrization . . . . . . .
2.2. Hyperbolicity . . . . . . . . . . . . . . . . . . .
2.3. Genuine nonlinearity . . . . . . . . . . . . . . .
2.4. Singularities . . . . . . . . . . . . . . . . . . . .
2.5. BV bound . . . . . . . . . . . . . . . . . . . . .
3. One-dimensional Euler equations . . . . . . . . . . .
3.1. Isentropic Euler equations . . . . . . . . . . . .
3.2. Isothermal Euler equations . . . . . . . . . . . .
3.3. Adiabatic Euler equations . . . . . . . . . . . .
4. Multidimensional Euler equations and related models
4.1. The potential flow equation . . . . . . . . . . . .
4.2. Incompressible Euler equations . . . . . . . . .
4.3. The transonic small disturbance equation . . . .
4.4. Pressure-gradient equations . . . . . . . . . . . .
4.5. Pressureless Euler equations . . . . . . . . . . .
4.6. Euler equations in nonlinear elastodynamics . .
4.7. The Born–Infeld system in electromagnetism . .
4.8. Lax systems . . . . . . . . . . . . . . . . . . . .
5. Multidimensional steady supersonic problems . . . .
5.1. Wedge problems involving supersonic shocks . .
5.2. Stability of supersonic vortex sheets . . . . . . .
6. Multidimensional steady transonic problems . . . . .
6.1. Transonic shock problems in Rd . . . . . . . . .

6.2. Nozzle problems involving transonic shocks . .
6.3. Free boundary approaches . . . . . . . . . . . .
HANDBOOK OF DIFFERENTIAL EQUATIONS
Evolutionary Equations, volume 2
Edited by C.M. Dafermos and E. Feireisl
© 2005 Elsevier B.V. All rights reserved
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3
6
6
14
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21
24
26
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37
38
39
40
40
42
43

44
45
46
47
51
53
56
58
61


2

G.-Q. Chen

7. Multidimensional unsteady problems . . . . . . . . . . . . .
7.1. Spherically symmetric solutions . . . . . . . . . . . . .
7.2. Self-similar solutions . . . . . . . . . . . . . . . . . . .
7.3. Global solutions with special Cauchy data . . . . . . .
8. Divergence-measure fields and hyperbolic conservation laws
8.1. Connections . . . . . . . . . . . . . . . . . . . . . . . .
8.2. Basic properties of divergence-measure fields . . . . . .
8.3. Normal traces and the Gauss–Green formula . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract
Some aspects of recent developments in the study of the Euler equations for compressible
fluids and related hyperbolic conservation laws are analyzed and surveyed. Basic features and
phenomena including convex entropy, symmetrization, hyperbolicity, genuine nonlinearity,
singularities, BV bound, concentration and cavitation are exhibited. Global well-posedness for
discontinuous solutions, including the BV theory and the L∞ theory, for the one-dimensional
Euler equations and related hyperbolic systems of conservation laws is described. Some analytical approaches including techniques, methods and ideas, developed recently, for solving
multidimensional steady problems are presented. Some multidimensional unsteady problems
are analyzed. Connections between entropy solutions of hyperbolic conservation laws and
divergence-measure fields, as well as the theory of divergence-measure fields, are discussed.

Some further trends and open problems on the Euler equations and related multidimensional
conservation laws are also addressed.
Keywords: Adiabatic, Clausius–Duhem inequality, Compensated compactness, Compressible fluids, Conservation laws, Divergence-measure fields, Entropy solutions, Euler equations,
Finite difference schemes, Free boundary approaches, Gauss–Green formula, Genuine nonlinearity, Geometric fluids, Glimm scheme, Hyperbolicity, Ill-posedness, Isentropic, Isothermal,
Lax entropy inequality, Potential flow, Self-similar, Singularity, Supersonic shocks, Supersonic vortex sheets, Traces, Transonic shocks, Multidimension, Well-posedness
MSC: Primary 00-02, 76-02, 35A05, 35L65, 35L67, 65M06, 35L40, 35L45, 35Q30, 35Q35,
76N15, 76H05, 76J20; secondary 35L80, 65M06, 76L05, 35A35, 35M10, 35M20, 35L50

64
64
75
77
79
79
81
83
91
91


Euler equations and related hyperbolic conservation laws

3

1. Introduction
Hyperbolic conservation laws, quasilinear hyperbolic systems in divergence form, are one
of the most important classes of nonlinear partial differential equations, which typically
take the following form:
∂t u + ∇x · f(u) = 0,


u ∈ Rn , x ∈ Rd ,

(1.1)

where ∇x = (∂x1 , . . . , ∂xd ) and
f = (f1 , . . . , fd ) : Rn → Rn

d

is a nonlinear mapping with fi : Rn → Rn for i = 1, . . . , d.
Consider plane wave solutions
u(t, x) = w(t, x · ω) for ω ∈ S d−1 .
Then w(t, ξ ) satisfies
∂t w + ∇f(w) · ω ∂ξ w = 0,
where ∇ = (∂w1 , . . . , ∂wn ).
In order that there is a stable plane wave solution, it requires that, for any ω ∈ S d−1 ,
∇f(w) · ω
1

i

n×n

have n real eigenvalues λi (w; ω) and be diagonalizable,

n.

(1.2)

Based on this, we say that system (1.1) is hyperbolic in a state domain D if condition (1.2)

holds for any w ∈ D and ω ∈ S d−1 .
The simplest example for multidimensional hyperbolic conservation laws is the following scalar conservation law
∂t u + divx f(u) = 0,

u ∈ R, x ∈ Rd ,

(1.3)

with f : R → Rd nonlinear. Then
λ(u, ω) = f ′ (u) · ω.
Therefore, any scalar conservation law is hyperbolic.
As is well known, the study of the Euler equations in gas dynamics gave birth to the
theory of hyperbolic conservation laws so that the system of Euler equations is an archetype


4

G.-Q. Chen

of this class of nonlinear partial differential equations. In general, the Euler equations for
compressible fluids in Rd are a system of d + 2 conservation laws

∂ ρ + ∇x · m = 0
(Euler 1755–1759),

 t
∂t m + ∇x · m⊗m
+
∇
p

=
0
(Cauchy
1827–1829),
x
ρ


∂t E + ∇x · m
(Kirchhoff 1868)
ρ (E + p) = 0

(1.4)

d+1
d+1
for (t, x) ∈ R+
, R+
= R+ × Rd := (0, ∞) × Rd . System (1.4) is closed by the constitutive relations

p = p(ρ, e),

E=

1 |m|2
+ ρe.
2 ρ

(1.5)


In (1.4) and (1.5), τ = 1/ρ is the deformation gradient (specific volume for fluids, strain
for solids), v = (v1 , . . . , vd )⊤ is the fluid velocity with ρv = m the momentum vector,
p is the scalar pressure and E is the total energy with e the internal energy which is a given
function of (τ, p) or (ρ, p) defined through thermodynamical relations. The notation a ⊗ b
denotes the tensor product of the vectors a and b. The other two thermodynamic variables
are temperature θ and entropy S. If (ρ, S) are chosen as the independent variables, then
the constitutive relations can be written as
(e, p, θ) = e(ρ, S), p(ρ, S), θ(ρ, S)

(1.6)

governed by
θ dS = de + p dτ = de −

p
dρ.
ρ2

(1.7)

For a polytropic gas,
p = Rρθ,

e = cv θ,

γ =1+

R
cv


(1.8)

and
p = p(ρ, S) = κρ γ eS/cv ,

e=

κ
ρ γ −1 eS/cv ,
γ −1

(1.9)

where R > 0 may be taken to be the universal gas constant divided by the effective molecular weight of the particular gas, cv > 0 is the specific heat at constant volume, γ > 1 is
the adiabatic exponent and κ > 0 can be any positive constant by scaling.
As shown in Section 2.4, no matter how smooth the initial data is, the solution of (1.4)
generally develops singularities in a finite time. Then system (1.4) is complemented by the
Clausius–Duhem inequality
∂t (ρS) + ∇x · (mS)

0

(Clausius 1854, Duhem 1901)

(1.10)


Euler equations and related hyperbolic conservation laws

5


in the sense of distributions in order to single out physical discontinuous solutions, socalled entropy solutions.
When a flow is isentropic, that is, entropy S is a uniform constant S0 in the flow, then
the Euler equations for the flow take the following simpler form
∂t ρ + ∇x · m = 0,
∂t m + ∇x · m⊗m
+ ∇p = 0,
ρ

(1.11)

where the pressure is regarded as a function of density, p = p(ρ, S0 ), with constant S0 . For
a polytropic gas,
p(ρ) = κρ γ ,

γ > 1,

(1.12)

where κ > 0 can be any positive constant under scaling. This system can be derived
from (1.4) as follows. It is well known that, for smooth solutions of (1.4), entropy
S(ρ, m, E) is conserved along fluid particle trajectories, i.e.,
∂t (ρS) + ∇x · (mS) = 0.

(1.13)

If the entropy is initially a uniform constant and the solution remains smooth, then (1.13)
implies that the energy equation can be eliminated, and entropy S keeps the same constant
in later time. Thus, under the constant initial entropy, a smooth solution of (1.4) satisfies
the equations in (1.11). Furthermore, it should be observed that solutions of system (1.11)

are also a good approximation to solutions of system (1.4) even after shocks form, since
the entropy increases across a shock to third order in wave strength for solutions of (1.4),
while in (1.11) the entropy is constant. Moreover, system (1.11) is an excellent model for
the isothermal fluid flow with γ = 1 and for the shallow water flow with γ = 2.
In the one-dimensional case, system (1.4) in Eulerian coordinates is
 ∂ ρ + ∂ m = 0,
t
x


2
∂t m + ∂x mρ + p = 0,


∂t E + ∂x m
ρ (E + p) = 0

(1.14)

2

with E = 21 mρ + ρe. The system above can be rewritten in Lagrangian coordinates in oneto-one correspondence as long as the fluid flow stays away from vacuum ρ = 0,

∂ τ − ∂x v = 0,

 t
∂t v + ∂x p = 0,

2


∂t e + v2 + ∂x (pv) = 0

(1.15)


6

G.-Q. Chen

with v = m/ρ, where the coordinates (t, x) are the Lagrangian coordinates, which are
different from the Eulerian coordinates for (1.14); for simplicity of notation, we do not
distinguish them. For the isentropic case, systems (1.14) and (1.15) reduce to
∂t ρ + ∂x m = 0,

(1.16)

∂t τ − ∂x v = 0,

(1.17)

∂t m + ∂x

m2
ρ

+p =0

and
∂t v + ∂x p = 0,


respectively, where pressure p is determined by (1.12) for the polytropic case, p = p(ρ) =
p(τ
˜ ) with τ = 1/ρ. The solutions of (1.16) and (1.17), even for entropy solutions, are
equivalent (see [52,332]).
This chapter is organized as follows. In Section 2 we exhibit some basic features and
phenomena of the Euler equations and related hyperbolic conservation laws such as convex
entropy, symmetrization, hyperbolicity, genuine nonlinearity, singularities and BV bound.
In Section 3 we describe some aspects of a well-posedness theory and related results for
the one-dimensional isentropic, isothermal and adiabatic Euler equations, respectively. In
Sections 4–7 we discuss some samples of multidimensional models and problems for the
Euler equations with emphasis on the prototype models and problems that have been solved
or expected to be solved rigorously at least for some cases. In Section 8 we discuss connections between entropy solutions of hyperbolic conservation laws and divergence-measure
fields, as well as the theory of divergence-measure fields to construct a good framework
for studying entropy solutions. Some analytical approaches including techniques, methods,
and ideas, developed recently, for solving multidimensional problems are also presented.

2. Basic features and phenomena
In this section we exhibit some basic features and phenomena of the Euler equations and
related hyperbolic conservation laws.

2.1. Convex entropy and symmetrization
A function η : D → R is called an entropy of system (1.1) if there exists a vector function
q : D → Rd , q = (q1 , . . . , qd ), satisfying
∇qi (u) = ∇η(u)∇fi (u),

i = 1, . . . , d.

An entropy η(u) is called a convex entropy in D if
∇ 2 η(u)


0

for any u ∈ D

(2.1)


Euler equations and related hyperbolic conservation laws

7

and a strictly convex entropy in D if
∇ 2 η(u)

c0 I

with a constant c0 > 0 uniform for u ∈ D1 for any D1 ⊂ D1 ⋐ D, where I is the n × n identity matrix. Then the correspondence of (1.10) in the context of hyperbolic conservation
laws is the Lax entropy inequality
∂t η(u) + ∇x · q(u)

0

(2.2)

in the sense of distributions for any C 2 convex entropy–entropy flux pair (η, q).
T HEOREM 2.1. A system in (1.1) endowed with a strictly convex entropy η in a state
domain D must be symmetrizable and hence hyperbolic in D.
P ROOF. Taking ∇ of both sides of the equations in (2.1) with respect to u, we have
∇ 2 η(u)∇fi (u) + ∇η(u)∇ 2 fi (u) = ∇ 2 qi (u),


i = 1, . . . , d.

Using the symmetry of the matrices
∇η(u)∇ 2 fi (u)

and ∇ 2 qi (u)

for fixed i = 1, 2, . . . , d, we find that
∇ 2 η(u)∇fi (u) is symmetric.

(2.3)

Multiplying (1.1) by ∇ 2 η(u), we get
d
2

∇ η(u) ∂t u +

i=1

∇ 2 η(u)∇fi (u)∇xi u = 0.

(2.4)

The fact that the matrices ∇ 2 η(u) > 0 and ∇ 2 η(u)∇fi (u), i = 1, 2, . . . , d, are symmetric
implies that system (1.1) is symmetrizable. Notice that any symmetrizable system must be
hyperbolic, which can be seen as follows.
Since ∇ 2 η(u) > 0 for u ∈ D, then the hyperbolicity of (1.1) is equivalent to the hyperbolicity of (2.4), while the hyperbolicity of (2.4) is equivalent to that, for any ω ∈ S d−1 ,
all zeros of the determinant λ∇ 2 η(u) − ∇ 2 η(u)∇f(u) · ω are real.


(2.5)

Since ∇ 2 η(u) is real symmetric and positive definite, there exists a matrix C(u) such
that
∇ 2 η(u) = C(u)C(u)⊤ .


8

G.-Q. Chen

Then the hyperbolicity is equivalent to that, for any ω ∈ S d−1 , the eigenvalues of the following matrix
C(u)−1 ∇ 2 η(u)∇f(u) · ω C(u)−1

⊤

(2.6)

are real, which is true since the matrix is real and symmetric. This completes the
proof.
R EMARK 2.1. This theorem is particularly useful to determine whether a large physical
system is symmetrizable and hence hyperbolic, since most of physical systems from continuum physics are endowed with a strictly convex entropy. In particular, for system (1.4),
(η∗ , q∗ ) = (−ρS, −mS)

(2.7)

is a strictly convex entropy–entropy flux pair when ρ > 0 and p > 0; while, for system (1.11), the mechanical energy and energy flux
(η∗ , q∗ ) =

m 1 |m|2

1 |m|2
+ ρe(ρ),
+ ρe(ρ) + p(ρ)
2 ρ
ρ 2 ρ

(2.8)

is a strictly convex entropy–entropy flux pair when ρ > 0 for polytropic gases. For multidimensional hyperbolic systems of conservation laws without a strictly convex entropy, it
is possible to enlarge the system so that the enlarged system is endowed with a globally
defined, strictly convex entropy. See [29,111,113,275,295].
R EMARK 2.2. The observation that systems of conservation laws endowed with a strictly
convex entropy must be symmetrizable is due to Godunov [155–157], Friedrich and Lax
[140] and Boillat [22]. See also [284].
R EMARK 2.3. This theorem has many important applications in the energy estimates. Basically, the symmetry plays an essential role in the following situation: For any u, v ∈ Rn ,
2u⊤ ∇ 2 η(v)∇fk (v) ∂xk u

= ∂xk u⊤ ∇ 2 η(v)∇fk (v)u − u⊤ ∂xk ∇ 2 η(v)∇fk (v) u

(2.9)

for k = 1, 2, . . . , d. This is very useful to make energy estimates for various problems.
There are several direct, important applications of Theorem 2.1 based on the symmetry
property of system (1.1) endowed with a strictly convex entropy such as (2.9). We list three
of them below.
2.1.1. Local existence of classical solutions. Consider the Cauchy problem for a general
hyperbolic system (1.1) with a strictly convex entropy η whose Cauchy data is
u|t=0 = u0 .

(2.10)



Euler equations and related hyperbolic conservation laws

9

T HEOREM 2.2. Assume that u0 : Rd → D is in H s ∩ L∞ with s > d/2 + 1. Then, for
the Cauchy problem (1.1) and (2.10), there exists a finite time T = T ( u0 s , u0 L∞ ) ∈
(0, ∞) such that there is a unique bounded classical solution u ∈ C 1 ([0, T ] × Rd ) with
u(t, x) ∈ D

for (t, x) ∈ [0, T ] × Rd

and
u ∈ C [0, T ]; H s ∩ C 1 [0, T ]; H s−1 .
Kato [184,185] first formulated and applied a basic idea in the semigroup theory to yield
the local existence of smooth solutions to (1.1).
The proof of this theorem in [241] relies solely on the elementary linear existence theory
for symmetric hyperbolic systems with smooth coefficients via a classical iteration scheme
(cf. [101]) by using the symmetry of system (1.1), especially (2.9). In particular, for all
u ∈ D, there is a positive definite symmetric matrix A0 (u) = ∇ 2 η(u) that is smooth in u
and satisfies
c0 I

A0 (u)

c0−1 I

(2.11)


with a constant c0 > 0 uniform for u ∈ D1 , for any D1 ⊂ D1 ⋐ D, such that Ai (u) =
A0 (u)∇fi (u) is symmetric. Moreover, a sharp continuation principle was also provided:
For u0 ∈ H s with s > d/2 + 1, the interval [0, T ) with T < ∞ is the maximal interval of
the classical H s existence for (1.1) if and only if either
(ut , Du)(t, ·)

L∞

→ ∞ as t → T ,

or
u(t, x) escapes every compact subset K ⋐ D

as t → T .

The first catastrophe in this principle is associated with the formation of shock waves and
vorticity waves, among others, in the smooth solutions, and the second is associated with
a blow-up phenomenon such as focusing and concentration.
In [246], Makino, Ukai and Kawashima established the local existence of classical solutions of the Cauchy problem with compactly supported initial data for the multidimensional
Euler equations, with the aid of the theory of quasilinear symmetric hyperbolic systems;
in particular, they introduced a symmetrization which works for initial data having either
compact support or vanishing at infinity. There are also discussions in [48] on the local
existence of smooth solutions of the three-dimensional Euler equations (1.4) by using an
identity to deduce a time decay of the internal energy and the Mach number.
The local existence and stability of classical solutions of the initial–boundary value problem for the multidimensional Euler equations can be found in [182,189,191] and the references cited therein.


10

G.-Q. Chen


2.1.2. Stability of Lipschitz solutions, rarefaction waves, and vacuum states in the class of
entropy solutions in L∞
T HEOREM 2.3. Assume that system (1.1) is endowed with a strictly convex entropy η on
compact subsets of D. Suppose that v is a Lipschitz solution of (1.1) on [0, T ), taking
values in a convex compact subset K of D, with initial data v0 . Let u be any entropy
solution of (1.1) on [0, T ), taking values in K, with initial data u0 . Then
2

|x|
u(t, x) − v(t, x) dx

2

C(T )
|x|
u0 (x) − v0 (x) dx

holds for any R > 0 and t ∈ [0, T ), with L > 0 depending solely on K and the Lipschitz
constant of v.
The main point for the proof of Theorem 2.3 is to use the relative entropy–entropy flux
pair (cf. [105])
α(u, v) = η(u) − η(v) − ∇η(v)(u − v),

(2.12)

β(u, v) = q(u) − q(v) − ∇η(v) f(u) − f(v)

(2.13)

and to calculate and find
∂t α(u, v) + ∇x · β(u, v)

− ∂t ∇η(v) (u − v) + ∇x ∇η(v) f(u) − f(v) .

Since v is a classical solution, we use the symmetry property of system (1.1) with the
strictly convex entropy η to have
∂t ∇η(v) = (∂t v)⊤ ∇ 2 η(v)
d

=−

k=1

(∂xk v)⊤ ∇fk (v)

⊤

∇ 2 η(v)

d

=−

k=1

(∂xk v)⊤ ∇ 2 η(v)∇fk (v).

Therefore, we have
d

∂t α(u, v) + ∇x · β(u, v)

−

k=1

(∂xk v)⊤ ∇ 2 η(v)Qfk (u, v),

where
Qfk (u, v) = fk (u) − fk (v) − ∇η(v)(u − v).


Euler equations and related hyperbolic conservation laws

11

Integrating over a set
(τ, x): 0

τ

T , |x|

t

R + L(t − τ )

with the aid of the Gauss–Green formula in Section 8 and choosing L > 0 large enough
yields the expected result.
Some further ideas have been developed to show the stability of planar rarefaction waves
and vacuum states in the class of entropy solutions in L∞ for the multidimensional Euler
equations by using the Gauss–Green formula in Section 8.
T HEOREM 2.4. Let ω ∈ S d−1 . Let
ˆ
R(t, x) = (ρ,
ˆ m)

x·ω
t

be a planar solution, consisting of planar rarefaction waves and possible vacuum states,
of the Riemann problem
R|t=0 =

ˆ − ),
(ρ− , m
ˆ + ),
(ρ+ , m

x · ω < 0,
x · ω > 0,

ˆ ± ). Suppose u(t, x) = (ρ, m)(t, x) is an entropy solution in L∞
with constant states (ρ± , m
of (1.11) that may contain vacuum. Then, for any R > 0 and t ∈ [0, ∞),
α(u, R)(t, x) dx

α(u, R)(0, x) dx,

|x|
|x|
where L > 0 depends solely on the bounds of the solutions u and R, and
α(u, R) = (u − R)⊤
with η∗ (u) = E ≡

1 |m|2
2 ρ

1
0

∇ 2 η∗ R + τ (u − R) dτ (u − R)

+ ρe(ρ).

R EMARK 2.4. Theorem 2.3 is due to Dafermos [110] (also see [111]). Theorem 2.4 is
due to Chen and Chen [56], where a similar theorem was also established for the adiabatic
Euler equations (1.4) with appropriate chosen entropy; also see [55] and [70].
R EMARK 2.5. For multidimensional hyperbolic systems of conservation laws with partially convex entropies and involutions, see [111]; also see [24,106].
R EMARK 2.6. For distributional solutions to the Euler equations (1.4) for polytropic
gases, it is observed in Perthame [269] that, under the basic integrability condition
ρ, E, ρv · x, |v|E ∈ L1loc R+ ; L1 Rd


12

G.-Q. Chen

and the condition that entropy S(t, x) has an upper bound, the internal energy decays in
time and, furthermore, the only time-decay on the internal energy suffices to yield the
time-decay of the density. Also see [48].
2.1.3. Local existence of shock front solutions. Shock front solutions, the simplest type of
discontinuous solutions, are the most important discontinuous nonlinear progressing wave
solutions in compressible Euler flows and other systems of conservation laws. For a general
multidimensional hyperbolic system of conservation laws (1.1), shock front solutions are
discontinuous piecewise smooth entropy solutions with the following structure:
T
(i) there exist a C 2 time–space hypersurface S(t) defined in (t, x) for 0 t
with time–space normal (nt , nx ) = (nt , n1 , . . . , nd ) and two C 1 vector-valued functions,
u+ (t, x) and u− (t, x), defined on respective domains D+ and D− on either side of the
hypersurface S(t), and satisfying
∂t u± + ∇ · f u± = 0

in D± ;

(2.14)

(ii) the jump across the hypersurface S(t) satisfies the Rankine–Hugoniot condition
nt u + − u − + n x · f u + − f u −

S

= 0.

(2.15)

For the quasilinear system (1.1), the surface S is not known in advance and must be determined as a part of the solution of the problem; thus the equations in (2.14) and (2.15)
describe a multidimensional, highly nonlinear, free-boundary value problem for the quasilinear system of conservation laws.
The initial data yielding shock front solutions is defined as follows. Let S0 be a smooth
hypersurface parametrized by α, and let n(α) = (n1 , . . . , nd )(α) be a unit normal to S0 .
Define the piecewise smooth initial data for respective domains D0+ and D0− on either side
of the hypersurface S0 as
u0 (x) =

u−
0 (x),
u+
0 (x),

x ∈ D0− ,

(2.16)

x ∈ D0+ .

It is assumed that the initial jump in (2.16) satisfies the Rankine–Hugoniot condition, i.e.,
there is a smooth scalar function σ (α) so that
−
+
−
−σ (α) u+
0 (α) − u0 (α) + n(α) · f u0 (α) − f u0 (α)

= 0,

(2.17)

and that σ (α) does not define a characteristic direction, i.e.,
σ (α) = λi u±
0 ,

α ∈ S0 , 1

i

n,

(2.18)

where λi , i = 1, . . . , n, are the eigenvalues of (1.1). It is natural to require that S(0) = S0 .