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

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How energy estimates imply well-posedness 285

The proof of (9.2.60) proceeds from the reﬁned L

estimate in (9.2.54) applied

to derivatives of w, as in the proof of (9.2.58) from the ‘regular’ L

estimate

(9.2.51), which is itself similar to the proof of Theorem 9.7. There are two main

steps: 1) estimate tangential derivatives and 2) estimate ‘normal’ derivatives.

There is basically no novelty in the estimate of tangential derivatives: we apply

(9.2.54) to u

= ∂

u,where∂

is a diﬀerential operator in the (y,t) directions

only, and the d-uple is of length |α|≤m, and show that the right-hand side is

bounded by

c



f

(Γ

))

+ g

(Γ

)

u

(Γ

))

u



(Γ

)



for some positive c depending only on ˘v

∞

and ˘v

(Γ

))

, hence



|α|≤m

−2γt

2(m−|α|)

∂

u(t)

d−1

×R

)

+ γ u

(Γ

))

+ u



(Γ

)

≤ 2c



L

u

(Γ

))

+ B



(Γ

)



for γ large enough (by absorption of the error terms in the left-hand side). More

care is needed in the estimate of ∂

u(t)

d−1

×R

)

when the operator ∂

contains a derivative in the normal variable x

(in this case β is a (d +1)-

uple). Let us look at the case of a single derivative ∂

. We take again a

p-uple α, of length |α|≤m − 1 this time. As in the proof of Theorem 9.7 we can

write

∂

u =(A

)

−1



− ∂

∂

u +

d−1



j=1

∂



with f

:= L

∂

u. The novelty here is that we need a timewise estimate of f

The trick is to use the L

estimates of both f

and ∂

. Indeed, by the same

computation as in the proof of Theorem 9.7 we have

m−|α|

e

−γt



)

 f

(Γ

))

+ u

(Γ

))

Furthermore,

∂

= f



j=1

∂

) ∂

286 Variable-coeﬃcients initial boundary value problems

with β =(α

+1,α



)ifα =(α

,α



), and f

can be estimated as above, so we

have

m−|α|−1

e

−γt

∂



)

 f

(Γ

))



j=1

∂

)

∞

e

−γt

∂

,

where each term in the sum can be estimated thanks to part 1) on tangential

derivatives. Therefore

m−|α|−1

e

−γt

∂



)

 f

(Γ

))

+ g

(Γ

)

Consequently, the L

bounds for f

and ∂

imply the timewise bound

m−|α|−1/2

−γτ

f

(τ)

(

Ω)

 f

(Γ

))

+ g

(Γ

)

and ﬁnally, estimating the other terms in ∂

∂

u thanks to part 1), we get

m−|α|−1/2

−γτ

∂

∂

u(τ)

(

Ω)

 f

(Γ

))

+ g

(Γ

)

hence

2(m−|α|−1)

−2γτ

∂

∂

u(τ)

(

Ω)



f

(Γ

))

+ g

(Γ

)

for γ large enough (≥ 1, which has been used for the g-term). The case of higher-

order derivatives in x

can be dealt with in a similar way, by induction (the details

are left to the reader). This in turns provides estimates of ∂

u(t)

d−1

×R

)

for all (d + 1)-uple β of length |β|≤m. Altogether these estimates prove (9.2.60).

Once we have (9.2.60), the usual weak=strong argument shows there

is a sequence (u

) such that for all p ∈{0,...,m}, ∂

goes to ∂

u in

C ([0,T]; H

m−p

d−1

× R

)), where u is the solution of IBVP with zero initial

data given by Theorem 9.21.

Step 2) The crucial tool here will be the following lifting result.

Lemma 9.6 Assume W and W

are both convex and contain zero, and

consider v ∈ D



d−1

× R

× R) is such that v

|t∈[0,T ]

∈ CH

, f ∈ H

d−1

× [0,T]),andg ∈ H

d−1

× [0,T]) with m>(d +1)/2+1.

Then for any u

∈ H

m+1/2

d−1

× R

) taking values in W with a trace on

=0 taking values in W

, compatible with f and g up to order m − 1, there

exists u

∈ H

m+1

d−1

× R

× R), also taking values in W with a trace on x

0 taking values in W

, vanishing for |t|≥ε>0, and such that

)

|t=0

= u

,∂



f − L



|t=0

≡ 0 , ∂



g − (B

)



|t=0

≡ 0

for all p ∈{0,...,m− 1}. Furthermore, f

:= f − L

belongs to H

d−1

× [0,T]) and g

:= g − (B

)

belongs to H

d−1

× [0,T]).

How energy estimates imply well-posedness 287

Proof We ﬁrst need to check that the induction formula in the compatibility

conditions (CC

m−1

(x)=

q−1



=0



q − 1







q−1−

(x)+∂

q−1

f(x, 0)

enables us to construct a sequence (u

,...,u

m−1

) such that for all q ∈

{0,...,m− 1}, u

belongs to H

m+1/2−q

d−1

× R

). Here the operators



:= −



j=1

(∂



◦ v)(x, 0) ∂

have, by assumption on v, coeﬃcients in H

m−

d−1

× R

)for ≥ 1, and this

is also true for P

up to a constant-coeﬃcients operator. Therefore, thanks to

Theorem C.10, the induction formula above does deﬁne u

in H

m+1/2−q

d−1

)ifu

∈ H

m+1/2−q

d−1

× R

Then by trace lifting (see, for instance, [1], pp. 216–217), we ﬁnd u

∈

m+1

d−1

× R

× R) such that u



m+1

d−1

×R

×R)

 u



m+1/2

d−1

×R)

and

∂

)

|t=0

= u

for all q ∈{0,...,m− 1}.

Thanks to the convexity of W and W

, up to multiplying u

by a C

∞

cut-

oﬀ function near t = 0, we may assume, by continuity of u

and (u

)

t = 0, that the range of u

lies in W and the range of (u

)

lies in W

,as

requested. The fact that ∂

)

|t=0

≡ 0 for all p ≤ m − 1 directly follows from

the construction of the u

,and∂

)

|t=0

≡ 0 is a consequence of the equations

∂



g(x, 0) =





q=0







(∂

−q

)(x, 0) u

(x)

for q ≤ m − 1 contained in the compatibility conditions (CC

m−1

). Finally, the

fact that f

and g

are H

follow in a classical way from Proposition C.11 and

Theorem C.12. 

Once we have the ‘approximate solution’ u

, the ‘approximate’ forcing term

and the ‘approximate’ boundary date g

, we can apply Step 1) to the IBVP

with zero initial data

= f

)

= g

, (u

)

|t=0

=0.

By uniqueness of the solution u of the original IBVP, we have u = u

+ u

which therefore belongs to CH

since both u

and u

do: for the latter, observe

that u

∈ H

m+1

d−1

× R

× R) implies indeed

∈ H

k+1

([0,T]; H

m−k

d−1

× R

)) → C

([0,T]; H

d−1

× R

))

for all k ≤ m.

288 Variable-coeﬃcients initial boundary value problems

Step 3) It relies on two ingredients: the observation that u

can be approx-

imated by smoother initial data u

, still compatible with f and g up to order

m − 1; energy estimates of a new kind for the IBVP with zero forcing term f

and zero boundary data g. We omit the details; see [136] for their proof in the

more complicated framework of shock-waves stability. 

PART III

NON-LINEAR PROBLEMS

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THE CAUCHY PROBLEM FOR QUASILINEAR SYSTEMS

We turn to ‘realistic’ systems, which are non-linear and most often read

∂f

(u)

∂t



j=1

∂f

(u)

∂x

= c(u) , (10.0.1)

where f

and c are vector ﬁelds in R

. The time ﬂux f

will be assumed

throughout this chapter to be a diﬀeomorphism. Observing that each row in

the left-hand side of (10.0.1) is the divergence with respect to (t, x) of a vector

ﬁeld (f

(u),f

(u), ···,f

(u)) in R

d+1

, we say that (10.0.1) is in divergence form.

In general, (10.0.1) is called a system of balance laws. When the source term c is

not present (that is, is replaced by 0), (10.0.1) is called a system of conservation

laws. As in the previous chapters, we shorten the notations and write (10.0.1) as

∂

(u)+



j=1

∂

(u)=c(u).

Among the well-known examples of systems of conservation laws, one inevitably

encounters the Euler equations, governing the motion of a compressible inviscid

and non-heat-conducting ﬂuid (see [80, 195], and also Chapter 13). These read











∂

ρ + ∇·(ρ u)=0,

∂

(ρu)+∇·(ρ u ⊗ u)+∇p =0,

∂



|u|

+ e



+ ∇·





|u|

+ e



+ p



=0,

(10.0.2)

where ρ>0 denotes the density of the ﬂuid, u its velocity, e>0 its internal

energy per unit mass and (ρ, e) → p(ρ, e) > 0 is a given pressure law. The

operator ∇ is acting in space only. The importance of Euler equations in both

the physical applications and the development of the mathematical theory of

conservation laws has prompted us to devote a whole part of the book (part IV)

to these equations.

292 The Cauchy problem for quasilinear systems

10.1 Smooth solutions

As a ﬁrst approach to (10.0.1), we may look for smooth solutions. Smooth

solutions of (10.0.1) are equivalently solutions of the quasilinear system

∂

u +



j=1

(u) ∂

u = b(u) , (10.1.3)

where the n × n matrices A

and the source term b smoothly depend on u ∈ R

They are related to the ﬂuxes f

and the original source term c by

(u)=df

(u)

−1

(u) ,b(u)=df

(u)

−1

c(u).

Remark 10.1 It can be that a change of variables u → u = ϕ(u) leads to a

more convenient quasilinear system than (10.1.3), that is, a system

∂

u +



j=1



(u) ∂

u =



b(u),

where the matrices



are sparser than the A

. As an example, one may compare

the quasilinear form of the Euler equations (10.0.2) in variables (ρ, u,e)toits

counterpart in variables (p, u,s), where s is the physical entropy. Indeed, recalling

that s is deﬁned by the diﬀerential relation

T ds =de −

dρ,

(where the integrating factor T is the temperature), we easily see that (10.0.2)

is equivalent for smooth solutions to the rather simple-looking system











∂

p + ρc

∇·u + u ·∇p =0,

∂

u +(u ·∇)u + ρ

−1

∇p =0,

∂

s + u ·∇s =0,

(10.1.4)

where c

:= ∂p/∂ρ when p is regarded as a function of ρ and s. The function c

is called the sound speed.

10.1.1 Local well-posedness

For general quasilinear systems of the form (10.1.3), the Cauchy problem can

hardly be dealt with. However, there is a very important class of systems for

which (local) H

-well-posedness is known to hold true – for s large enough. This

class is a natural generalization of Friedrichs-symmetrizable linear systems.

Deﬁnition 10.1 Let U be an open subset of R

. The quasilinear system

(10.1.3) is called Friedrichs-symmetrizable in U if there exists a C

∞

mapping

Smooth solutions 293

S : U → Sym

such that S(u) is positive-deﬁnite, and the matrices S(u)A

(u)

are symmetric for all u ∈ U .

Example The system (10.1.4) admits the diagonal symmetrizer

diag (1/(ρc

),ρ,···,ρ,1)

in the domain {ρ>0 ,c

> 0}, that is, away from vacuum and where the sound

speed is well-deﬁned (which corresponds to hyperbolicity).

In more generality, systems admitting a strictly convex entropy are known

to be Friedrichs-symmetrizable – see [46, 184] for an introduction to the notion

of mathematical entropy and an extensive analysis of systems endowed with an

entropy. For this reason, since mathematical entropies are often related to phys-

ical entropy (or energy), most ‘physical’ systems are Friedrichs-symmetrizable.

In the case of Euler equations, various symmetrizations are discussed in Section

13.2.3. Also, see [71, 167], concerning the symmetrization of more complicated

systems, or systems endowed with non-convex entropies [21,46, 188].

The main result regarding the Cauchy problem for Friedrichs-symmetrizable

systems is the following. It was proved by several authors independently [67,94].

Theorem 10.1 Let U be an open subset of R

containing 0. We assume that

and b are C

∞

functions of u ∈ U , all vanishing at 0, and that (10.1.3) is

Friedrichs-symmetrizable in U . We consider the Cauchy problem associated with

(10.1.3) and initial data g ∈ H

) with s>

+1 taking values in U .There

exists T>0 such that (10.1.3) has a unique classical solution u ∈ C

× [0,T])

achieving the initial data u(0) = g. Furthermore, u belongs to C ([0,T]; H

) ∩

([0,T]; H

s−1

Observe that this result does not apply straight to the Euler equations, which

in general are not symmetrizable up to ρ = 0. But a slightly modiﬁed version of

Theorem 10.1, allowing solutions that do not vanish at inﬁnity, does apply: it

suﬃces to replace 0 by some u

,takeg ∈ u

+ H

and obtain u(t) ∈ u

+ H

See Section 13.3 for more details on the Euler equations.

Proof The proof is based on the iteration scheme

S(u

) ∂

k+1



j=1

S(u

) A

) ∂

k+1

= S(u

) b(u

). (10.1.5)

Since we are looking for smooth solutions of (10.1.3), we are only concerned

with smooth solutions of (10.1.5). By Theorem 2.12 in Chapter 2 we know that

(10.1.5) admits a C

∞

solution u

k+1

provided that u

and the initial data are

∞

. This is why we are going to smooth back the initial data.

We introduce a molliﬁer ρ

, being deﬁned as usual by

(x)=ε

−d





294 The Cauchy problem for quasilinear systems

with ε

> 0 tending to 0 (in a way to be speciﬁed later) and ρ ∈ D(R

; R

)

being supported in the unit ball,



ρ = 1. We shall use the approximate initial

data g

:= g ∗ ρ

, which is a C

∞

function that tends to g in H

when ε

goes to

0. Furthermore, we have a uniform estimate

g

− g

≤ Cε

g

(10.1.6)

for ε

small enough, say ε

≤ ε. From now on, this upper bound will be assumed

to hold true.

We initialize the iteration scheme by taking u

independent of t as u

(t):=g

Then we consider the induction process in which u

k+1

is deﬁned by (10.1.5)

and u

k+1

(0) = g

k+1

. An additional diﬃculty is due to the restricted domain of

deﬁnition of S, A

and b. The process must be able to control the L

∞

norm of

the iterates u

so that they do not approach ∂U too closely. This will be done

by using the following observation. Since g

tends to g in H

when ε

goes to 0,

by the classical Sobolev embedding

) → L

∞

) (10.1.7)

for s>d/2, we can ﬁnd ε

, δ>0 and a relatively compact open subset V of U

such that any smooth u satisfying the estimate

u − g



≤ δ (10.1.8)

only achieves values belonging to V . Furthermore, another useful fact will be

that for u satisfying (10.1.8), ∇

u is bounded (this follows from the Sobolev

embedding (10.1.7) applied to s − 1, which is still greater than d/2 by assump-

tion).

The relatively compact subset V will serve as a reference for energy estimates.

We take β>0sothat

βI

≤ S(u) ≤ β

−1

(10.1.9)

for all u ∈

V .

Finally, other essential ingredients in the proof are the following Moser

estimates [144].

Proposition 10.1

i) If u and v both belong to L

∞

∩ H

with s>0 then their product also

belongs to H

and there exists C>0 depending only on s such that

uv

≤ C ( u

∞

v

+ v

∞

u

) . (10.1.10)

ii) If u belongs to H

∩ L

∞

with s>0 and b is a C

∞

function vanishing at

0 then the composed function b(u) also belongs to H

, and there exists

a continuous function C :[0, +∞) → [0, +∞) (depending on s and b as

parameters) such that

b(u) 

≤ C (u

∞

) u

. (10.1.11)