Benzoni-Gavage S., Serre D. Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications

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Weak solutions 305

10.2.1 Entropy solutions

A well-known drawback of weak solutions is their non-uniqueness, and an appro-

priate notion that is likely to restore uniqueness is the one of entropy solutions.

To deﬁne entropy solutions we need mathematical entropies, which are deﬁned

as follows.

Deﬁnition 10.3 Afunctionη ∈ C

(U ; R), U ⊂ R

, is called a (mathematical)

entropy of the system of balance laws (10.0.1) if and only if there exists q ∈

(U ; R

) such that

(u)=dη(u) · A

(u) , ∀ u ∈ U ,j=1,...,d,

where

(u)=df

(u)

−1

(u).

In this case, (η, q) is called an entropy–entropy ﬂux pair.

Equivalently, (η, q) is an entropy–entropy ﬂux pair if any smooth solution u

of (10.0.1) satisﬁes the additional balance law

∂

η(u)+



j=1

∂

(u)=dη(u) · b(u) ,b(u)=df

(u)

−1

c(u) .

Deﬁnition 10.4 A weak solution u of (10.0.1) is called an entropy solution if

it satisﬁes the inequality

∂

η(u)+



j=1

∂

(u) ≤ dη(u) · b(u)

in the sense of distributions for any entropy–entropy ﬂux pair (η, q) with η

convex.

Of course, the convexity restriction plays an essential role here. If it were

omitted, the requested inequality could only be an equality, which would be too

restrictive – smooth solutions do satisfy the equality but discontinuous solutions

in general do not.

Also note that Deﬁnition 10.4 might be helpless if no convex entropy exists,

which is ‘generically’ the case for n ≥ 2andnd ≥ 3 – there are too many

equations for two unknowns. However, it appears that physics is not generic.

For physical systems like the Euler equations considered in Chapter 13, there

does exist at least one entropy, in connection with the thermodynamical entropy,

and in standard cases it is convex

. But there is generally hardly more than one

(nontrivial) convex entropy. So one may wonder whether a single one is suﬃcient

to select a unique – supposedly good – solution. The answer is unfortunately no,

in general. However, it appears that a single strictly convex entropy η is able to

For examples of physical systems with only partially convex entropies, see [46], Section 5.3.

306 The Cauchy problem for quasilinear systems

select the smooth solution if and as long as it exists. This important result is due

to Dafermos. We reproduce it here for completeness. We shall use the shortcut

η-entropy solution for a function u satisfying











∂

(u)+



j=1

∂

(u)=c(u) ,

∂

η(u)+



j=1

∂

(u) ≤ dη(u) · df

(u)

−1

· c(u)

in the sense of distributions.

Theorem 10.4 (Dafermos) Let us assume that (10.0.1) has an entropy–entropy

ﬂux pair (η, q) with η ∈ C

(U ; R) and d

η positive-deﬁnite everywhere. Assume

that ¯u ∈ C

× [0,T]) is a smooth solution of (10.0.1) and u ∈ L

∞

× [0,T])

is a η-entropy solution, both taking values in the same compact convex subset

K of U , such that u(0) − ¯u(0) belongs to L

). Then there exists C =

C(T,K , ∇

¯u

∞

) such that

u − ¯u

∞

(0,T ;L

)

≤ C u(0) − ¯u(0)

Proof Interestingly, the proof of this ‘energy estimate’ involves new ingredients

compared to the ones performed previously in this book, and in particular it

incorporates, to some extent, Krushkov’s idea of doubled unknowns. It also

requires some information on the regularity in time of weak solutions, which

we merely recall without proof from [46], p. 50.

Lemma 10.1 For any weak solution u ∈ L

∞

× [0,T]) of (10.0.1), for all

t ∈ [0,T), the mean value



t+ε

u(τ)dτ

has a limit in the weak-∗ topology of L

∞

) when ε goes to 0, and this limit is

u(t) for almost all t in [0,T].

Admitting this lemma, we can thus choose a representative of the weak solution

u in the statement of the theorem such that u(t) is the limit everywhere.This

will be implicitly assumed in the computations below. In view of Lemma 10.1,

we have in particular

lim

ε0



t+ε



u(x, τ)dx dτ =



u(x, t)dx ∀t ∈ [0,T),

for any compact set P ⊂ R

, and using that

η(u(x, τ)) ≥ η(u(x, t)) + dη(u(x, t)) · (u(x, τ ) − u(x, t))

Weak solutions 307

by the convexity of η, we easily get

lim inf

ε0



t+ε



(τ)

η(u(x, τ)) dx dτ ≥



(t)

η(u(x, t)) dx ∀t ∈ [0,T),

if the set P

(τ) depends continuously on (ε, τ).

Having these preliminary results at hand, we can now give a complete proof

of Theorem 10.4, which is a slightly more detailed version of what can be found

in [46], p. 66–68. Despite its technicality, it is interesting and easy to understand.

We assume for simplicity that f

(u)=u – in this case it is sometimes said

that (10.0.1) is in canonical form – and deﬁne

h(v, w)=η(v) − η(w) − dη(w) · (v − w),

(v, w)=q

(v) − q

(w) − dη(w) · (f

(v) − f

(w)),

(v, w)=f

(v) − f

(w) − df

(w) · (v − w),

ζ(v, w)=dη(v) − dη(w) − d

η(w) · (v − w)

for all v, w ∈ K . By the strict convexity of η and Taylor expansions, we have

the following uniform estimates on K

β v − w

≤ h(v,w) ≤ β

−1

v − w

g(v,w) :=



(v, w)|≤K v − w

z(v, w) :=



z

(v, w)≤K v − w

ζ(v, w)≤K v − w

with β>0andK>0.

Considering a Lipschitz continuous test function ϕ, we substract the equality



×[0,T )



η(¯u) ∂

ϕ +



(¯u) ∂

ϕ + ϕd(¯u)





η(¯u(0)) ϕ(0) = 0,

to the inequality



×[0,T )



η(u) ∂

ϕ +



(u) ∂

ϕ + ϕd(u)





η(u(0)) ϕ(0) ≥ 0

wherewehavedenotedd(u)=dη(u) · b(u). Using the equalities



×[0,T )



¯u∂

ψ +



(¯u) ∂

+ ψ

b(¯u)





¯u(0) ψ

(0) = 0,



×[0,T )



u∂



(u) ∂

+ ψ

b(u)





u(0) ψ

(0) = 0

308 The Cauchy problem for quasilinear systems

for the family of test functions

= ϕ

∂η

∂u

(¯u) ,j=1,...,n,

we obtain the inequality



×[0,T )



h(u, ¯u) ∂

ϕ +



(u, ¯u) ∂

ϕ + ϕ (dη(u) − dη(¯u))· b(u)





h(u(0), ¯u(0)) ϕ(0)

≥



×[0,T )



∂

(dη(¯u)) · (u − ¯u)+



∂

(dη(¯u)) · (f

(u) − f

(¯u))



Using the fact that ¯u is smooth and satisﬁes

∂

¯u +



j=1

(¯u) · ∂

¯u = b(¯u)

in the classical sense and that the Jacobian matrices df

(¯u)ared

η(¯u)-

symmetric, we may rewrite the factor of ϕ in the integrand of the right-hand

side above as

η(¯u) · (z

(u, ¯u),∂

¯u)+d

η(¯u) · (b(¯u),u− ¯u).

Therefore, we are left with the inequality



×[0,T )



h(u, ¯u) ∂

ϕ +



(u, ¯u) ∂





h(u(0), ¯u(0)) ϕ(0)

≥



×[0,T )



η(¯u) · (z

(u, ¯u),∂

¯u) − ζ(u, ¯u) · b(u)

−d

η(¯u) · (b(u) − b(¯u),u− ¯u)



, (10.2.22)

where the right-hand side is quadratic, i.e. bounded in absolute value by

C ϕ

∞



Supp ϕ

h(u, ¯u),

where C = C(K, β, u

∞

, ∇¯u

∞

, db

∞

(K )

The rest of the proof is based on a suitable choice of ϕ that will cancel out

the left-hand side terms



×[0,T )

(u, ¯u) ∂

ϕ.

Weak solutions 309

In fact, we shall use a family of test functions ϕ

, whose deﬁnition depends on

the rate λ =2K/β in the inequality

g(v,w)≤

h(v, w).

(Observe that λ should be homogeneous to a velocity.)

We ﬁx R>0, t

∈ (0,T), 0 <ε<T− t

,0≤ s ≤ t

, and deﬁne ϕ

ε,s

(x, t)=











0ift ≥ s + ε

or x≥R + λ(s + ε − t) ,

1ift ≤ s

and x≤R + λ(s − t),

ε,s

(t)ifs ≤ t<s+ ε

and x≤R + λ(s − t),

ε,s

(t) χ

ε,s

(x, t)ifs ≤ t<s+ ε

and R + λ(s − t) ≤x <R+ λ(s + ε − t),

ε,s

(x, t)ift ≤ s

and R + λ(s − t) ≤x <R+ λ(s + ε − t),

where

ε,s

(t)=

s + ε − t

,χ

ε,s

(x, t)=

R + λ(s + ε − t) −x

λε

We ﬁrst observe that



R + λs ≤x <R+ λ(s+ε)

h(u(0), ¯u(0)) χ

ε,s

(x, 0) = O(ε)

(uniformly for s ∈ [0,t

]) hence



h(u(0), ¯u(0)) ϕ

ε,t

(0) =



x≤R + λs

h(u(0), ¯u(0)) + O(ε).

Next, we also have



s+ε



R + λ(s−t) ≤x <R+ λ(s+ε−t)

h(u, ¯u) ∂

(θ

ε,s

)=O(ε),

310 The Cauchy problem for quasilinear systems

and therefore



×[0,T ]

h(u, ¯u) ∂

ε,s

= O(ε)+



s+ε



x≤R + λ(s−t)

h(u, ¯u) ∂

ε,s



R + λ(s−t) ≤x <R+ λ(s+ε−t)

h(u, ¯u) ∂

ε,s

= O(ε) −



s+ε



x≤R + λ(s−t)

h(u, ¯u)

−



R + λ(s−t) ≤x <R+ λ(s+ε−t)

h(u, ¯u) .

Similarly, we ﬁnd that



×[0,T ]

(u, ¯u) ∂

ε,s

= O(ε) −

λε



R + λ(s−t) ≤x <R+ λ(s+ε−t)

(u, ¯u)

x

Now, using the fact that

λh(u, ¯u)+



x

(u, ¯u) ≥ λh(u, ¯u) −



(u, ¯u)|≥

h(u, ¯u)

by deﬁnition of λ, and substituting the expansions obtained hereabove into the

inequality (10.2.22) we arrive at



s+ε



x≤R + λ(s−t)

h(u, ¯u)+

2 ε



R + λ(s−t) ≤x <R+ λ(s+ε−t)

h(u, ¯u)

≤ C

ε +



x≤R + λs

h(u(0), ¯u(0)) + C



s+ε



x≤R+λ(s+ε−t)

h(u, ¯u)

(10.2.23)

for all ε ∈ (0,T − t

)ands ∈ [0,t

]. We claim this implies an estimate of the

form



x≤R

h(u(t), ¯u(t)) ≤





x≤R + λt

h(u(0), ¯u(0))

for all t ∈ [0,t

Indeed, setting

a(s)=



x≤R+λ(s−t)

h(u(x, t), ¯u(x, t)) dx dt,

Weak solutions 311

we have

a(s + ε) − a(s)=



s+ε



x≤R + λ(s+ε−t)

h(u, ¯u)



R + λ(s−t) ≤x <R+ λ(s+ε−t)

h(u, ¯u)

= O(ε



s+ε



x≤R + λ(s−t)

h(u, ¯u)



R + λ(s−t) ≤x <R+ λ(s+ε−t)

h(u, ¯u),

so that (10.2.23) implies

a(s + ε) − a(s)

≤



ε +



x≤R + λs

h(u(0), ¯u(0)) + Ca(s + ε),

where



is the constant C

plus an upper bound of O(ε

)/ε

. Applying our

discrete Gronwall Lemma A.4, this yields

a(s) ≤ e

2Cs



x≤R + λs

h(u(0), ¯u(0))

for all s. Returning to the original inequality (10.2.23), this shows in particular

that



s+ε



x≤R + λ(s−t)

h(u, ¯u) ≤ C

ε +



x≤R + λs

h(u(0), ¯u(0))

+ C e

2C(s+ε)



x≤R + λ (s+ε)

h(u(0), ¯u(0)) .

Finally, using our preliminary remark to deal with the left-hand side when ε goes

to 0 we obtain



x≤R

h(u(s), ¯u(s)) ≤ (1 + Ce

2Cs

)



x≤R + λs

h(u(0), ¯u(0)).

Together with the known two-sided inequalities for h, this completes the proof.



This is all we shall say about weak/entropy solutions in general, mainly

because we do not know much more in several space dimensions! For recent,

important results in one space dimension, see [24].

10.2.2 Piecewise smooth solutions

An amenable, and nonetheless interesting class of weak/entropy solutions is given

by piecewise smooth solutions. They consist of smooth solutions separated by

312 The Cauchy problem for quasilinear systems

fronts of discontinuity, across which the so-called Rankine–Hugoniot condition

is satisﬁed. We recall this condition below without justiﬁcation, which can be

found in many textbooks (see, for instance, [46], p. 10–11 or [184], p. 88–89).

We also summarize hereafter the material needed for our subsequent analysis in

Chapter 12.

Deﬁnition 10.5 Suppose that Σ is a codimension one surface in R

× R

and

that (R

× R

)\Σ has two connected components Ω

−

and Ω

. Assume that u

∈

(Ω

) solve the system of balance laws (10.0.1) on either side of Σ. Then the

function

u :(R

× R

)\Σ → R

(x, t) → u(x, t)=u

(x, t) for (x, t) ∈ Ω

is said to satisfy the Rankine–Hugoniot condition across Σ if and only if

(u)] +



j=1

(u)]=0, (10.2.24)

where N

, N

, ..., N

denote the components in the directions t, x

,..., x

respectively, of a vector N orthogonal to Σ. Here above, if N is pointing to Ω

the brackets [·] stand for

(u)](x, t) = lim

ε0

( f

((x, t)+εN(x, t))) − f

−

((x, t) − εN (x, t))) )

at each point (x, t) ∈ Σ.

The Rankine–Hugoniot condition is known to be necessary and suﬃcient for u

to be a weak solution of (10.0.1). Needless to say, the conservative, or divergence

form of the right-hand side in (10.0.1) is crucial for this statement to make sense.

For completeness, let us recall the following basic characterization of piecewise

smooth entropy solutions.

Proposition 10.2 Suppose that Σ is a codimension one surface in R

× R

and that (R

× R

)\Σ has two connected components Ω

−

and Ω

. A function

u :(R

× R

)\Σ → R

(x, t) → u(x, t)=u

(x, t) for (x, t) ∈ Ω

with u

∈ C

(Ω

) is a η-entropy solution of the system of conservation laws

∂f

(u)

∂t



j=1

∂f

(u)

∂x

= 0 (10.2.25)

Weak solutions 313

if and only if u

satisfy (10.2.25) in Ω

, the Rankine–Hugoniot (10.2.24) holds

true across Σ and, moreover,

[η(u)] +



j=1

(u)] ≤ 0, (10.2.26)

where N

, N

, ..., N

denote the components in the directions t, x

,..., x

respectively, of a vector N orthogonal to Σ pointing to Ω

Remark 10.2 This characterization of course does not depend on the choice

of Ω

−

and Ω

: if these parts of the space R

× R

are exchanged, N is changed

to −N and the jumps are also changed into their opposite.

Many solutions of the type described in Proposition 10.2 have been observed

experimentally (in gas dynamics, for instance) and numerically. But there is no

global-in-time existence result for arbitrary jump front Σ.

The special case of a planar Σ reduces the problem to one space dimen-

sion. Solutions consisting of a planar front propagating with constant speed in

constant, homogeneous states are easily found by examining the one-dimensional

Rankine–Hugoniot condition. These solutions are called planar shock waves when

(10.2.26) is a strict inequality. There are also planar contact discontinuities, for

which (10.2.26) is in fact an equality. Planar fronts in gas dynamics are discussed

in detail in Chapter 13.

Spherical shocks are already much more complicated, as they involve a non-

constant speed and non-homogeneous states in general. As regards gas dynamics,

for instance, spherical shocks have received attention for decades: this goes

back to World War II and the atomic bomb research [73, 75, 106, 180, 204], in

which basic solutions were obtained by means of similarity/dimensional analysis

[181]; more recently the interest in the ﬁeld has been renewed by (hopefully)

more peaceful and nonetheless fascinating phenomena (e.g. sonoluminescence,

cavitation) and various (potential) applications of shock focusing (extracorporeal

therapies, nuclear fusion, etc.); current research concerns complex ﬂuids (van

der Waals or dusty gases, superﬂuids, etc.) and has incorporated group-analysis

techniques. The interested reader may refer, for instance, to the collected papers

in [198].

Regarding ‘arbitrarily’ curved fronts, as far as gas dynamics is concerned

there is a wide literature on transonic (stationary) shocks (see, for instance,

[30, 34, 220]) or other patterns, in particular those related to the reﬂexion of

shocks (see, for instance, [182,225,226]). In more general, abstract settings, there

are far fewer results on curved shocks. The main existence results are due to

Majda [124–126], M´etivier [133,136], Blokhin [17,18] and are based on a stability

analysis of reference fronts (known to exist, such as planar ones, or assumed) and

thus are local in time. The description of those results will be the main purpose

of Chapter 12.

314 The Cauchy problem for quasilinear systems

Even though they are beyond the scope this book, other interesting results on

(abstract) multidimensional weak solutions are worth mentioning. In fact, they

date back to the late 1980s! One of them is due to M´etiver and deals with the

interaction of two shocks [131]. Another study is due to Alinhac [3,4], and answers

a question raised by Majda in [126] (p. 153–154) on the existence and structure

of multidimensional rarefaction waves (initially discontinuous, but smoother for

positive times). One of the main diﬃculties in constructing those waves is, ‘the

dominant signals in rarefaction fronts move at characteristic speeds’, Majda

said. Alinhac overcome it by using a Nash–Moser iterative scheme and ad hoc

functional spaces (based on the Littlewood–Paley decomposition). It is notable

that a similar diﬃculty arises for weak shocks. This case is paradoxical because,

when the initial shock strength goes to zero we expect to recover a smooth

solution, but if we apply Majda’s result with brute force the existence time of the

shock-front solution shrinks to 0, as the shock tends to be characteristic. It was

resolved rather recently by M´etivier and his coworker Francheteau (see [56,133],

and also the lecture notes [136]).