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

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THE MIXED PROBLEM FOR QUASILINEAR SYSTEMS

Physical problems are not usually posed in the whole space: in ﬂuid mechanics,

for instance, a spatial domain typically has an entrance, an exit and walls;

however, this kind of mixed-type and non-smooth boundary yields unsolved

yet mathematical problems. The purpose of this chapter is to show how to

deal with more regular non-linear initial boundary value problems (IBVP). The

name IBVP refers explicitly to the initial data (at time zero) and the boundary

data (on the boundary of the spatial domain). IBVPs are equivalently called

mixed problems (regardless of the nature of boundary data) just because of the

‘mixing’ between initial data and boundary data: we will use either one of those

names.

The general mixed problem for a system of balance laws reads











∂

(u)+



j=1

∂

(u)=c(u) , Ω × (0,T),

b(u)=b

,∂Ω × (0,T)

|t=0

= u

, Ω,

where Ω is a connected open subset of R

, the ﬂuxes f

(u), the source term

c(u) and the boundary term b(u) are expressed through supposedly smooth non-

linear vector-valued mappings f

, c and b, respectively, and b and u

are smooth

functions encoding, respectively, the boundary data and the initial data. We

shall also assume that the level sets of the nonlinear mapping b are smooth

submanifolds of the phase space R

We will consider only smooth solutions – recall that even the Cauchy problem

is still most open for weak solutions in several space dimensions – so we may use

the chain rule in the derivatives of ﬂuxes and rewrite the PDEs in quasilinear

form. More generally, we consider a quasilinear system of PDEs, not necessarily

coming from equations in divergence form, and its associated initial boundary

value problem











∂

u +



j=1

(u) ∂

u = h(u) , Ω × (0,T),

b(u)=b

,∂Ω × (0,T),

|t=0

= u

, Ω.

(11.0.1)

316 The mixed problem for quasilinear systems

Here above, the A

are C

∞

mappings on an open subset U of R

, with values

in M

(R), and the mappings h : U → R

and b : U → R

(with p a ﬁxed

integer), are also C

∞

.InfactA

, h and b may all depend on (x, t) as well but

we avoid this reﬁnement for the sake of simplicity. The boundary data b

: ∂Ω ×

[0,T] → R

and the initial data u

:Ω→ R

will be assumed to lie in Sobolev

spaces: the main purpose of this chapter is indeed to solve the quasilinear IBVP

(11.0.1) in Sobolev spaces of suﬃciently high index (and for T small enough).

The resolution of (11.0.1) is a most open problem for general domains Ω,

but under suitable assumptions on the geometry of Ω and on the boundary

conditions, it is possible to prove well-posedness results: in the simpler one

dimensional case – i.e. with Ω a segment of the real line – the (hardly obtainable)

book by Li Ta Tsien and Yu Wen Ci [116] deals with IBVPs and other related

problems; in higher dimensions our main references are the (unpublished) PhD

thesis of Mokrane [140] and the lecture notes of M´etivier [136] (see also the

early work of Rauch and Massey [165], and more recent papers dealing with

characteristic problems by Gu`es [76], the Japanese school [149–151,192,193] and

Secchi and coworkers [43, 44, 176–179]).

11.1 Main results

11.1.1 Structural and stability assumptions

We enter now into detailed assumptions under which (11.0.1) is known to have a

unique smooth solution (for T small enough, the non-linearities precluding global

solutions in general).

A basic requirement is that the ‘linearized’ versions of the problem, including

those with variable coeﬃcients, fall into the framework of the results known for

linear initial boundary value problems (LIBVP). Here come the restrictions on

the domain Ω: Chapter 9 has shown we may hope to deal with either a half-space

(up to a change of variables) or a smooth bounded Ω. We shall implicitly assume

either one of these situations.

The LIBVP considered will be of the form











∂

u +



j=1

(v) ∂

u = f, Ω × (0,T),

db(v) · u = g, ∂Ω × (0,T)

|t=0

=0, Ω,

(11.1.2)

with v a given function of possibly limited regularity, and f , g arbitrary data (in

spaces to be speciﬁed afterwards).

Our starting point is the classical, even though not universal, constant

hyperbolicity assumption:

Main results 317

(CH) the matrices

A(w, ξ):=



j=1

(w)

are diagonalizable with real eigenvalues of constant multiplicities on U ×

d−1

This condition is known to be violated, for instance, by the (challenging)

system of ideal magnetohydrodynamics [17]. Nevertheless, (CH) is satisﬁed by

many physically relevant systems, and in particular by the Euler equations of gas

dynamics, the basic application considered in this book. For systems with variable

multiplicities, we refer to the recent work by M´etivier and Zumbrun [135], which

goes far beyond the scope of this book.

We now import from Chapter 9 some assumptions on Ω and the matrix-valued

function B := db.

(NC) for all w ∈ U and all normal vector ν to ∂Ω, the matrix A(w, ν)is

non-singular,

(N) the boundary matrix B is of constant, maximal rank and

=kerB(w) ⊕ E

(A(w, ν)) for all (w, ν) ∈ U ×S

d−1

with ν an outward normal vector to ∂ΩandE

(A(w, ν)) the stable

subspace of the (hyperbolic) matrix A(w, ν),

(T) the vector bundle kerB is trivializable, that is, kerB(w) admits a basis

depending smoothly on w ∈ U .

The latter is not very demanding: thanks to a classical diﬀerential topology

result (see [85], p. 97), it is satisﬁed as soon as B is smooth (and of constant

rank) and U is contractible (which is the case if it is a ball, for instance).

On the other hand, assuming that the whole boundary is non-characteristic

in such a strong sense as in (NC) is quite restrictive, especially when ∂Ωisa

connected bounded manifold. Indeed, observe that for all ξ ∈ S

d−1

,detA(w, ξ)=

(−1)

det A(w, −ξ) by homogeneity: assume then that ∂Ω is a smooth connected

manifold and that there are two points x

and x

in ∂Ω where the normal vectors

are opposite to each other; if, moreover, the dimension n of the phase space is

odd, the mean-value theorem trivially implies that det A(w, .) vanishes on the

connected set of normal vectors to ∂Ω along the path from x

to x

.Insome

cases, the vanishing of det A(w, ν) can even occur whatever the parity of n: in full

gas dynamics, for instance, if ∂Ω is a sphere, the set of its normal vectors is S

d−1

which intersects any hyperplane u

⊥

; this means the eigenvalue λ

= u · ξ (with

u the velocity of the ﬂuid) does vanish at some points ξ ∈ S

d−1

, irrespective of

the parity of d (or equivalently the parity of n = d + 2); and even in isentropic

gas dynamics, both eigenvalues u · ξ ± c |ξ| vanish at some ±ξ ∈ S

d−1

when the

318 The mixed problem for quasilinear systems

ﬂow is supersonic (|u|≥c). These facts urge us to weaken (NC) by taking into

account the boundary data. This is done in the following.

(NC

) For all (x, t) ∈ ∂Ω × [0,T], for all w ∈ U such that b(w)=b(x, t),

the matrix A(w, ν(x)) is non-singular (where ν(x) denotes the outward

unit normal to ∂Ω at point x).

This is obviously more likely to be satisﬁed than (NC) (as we see in the example

of full gas dynamics, for which (NC

) is true provided that boundary data impose

a non-singular and non-sonic velocity ﬁeld normal to the boundary) and suﬃcient

for our purpose. Even (NC

) is not necessary though, but mixed problems with a

(partly) characteristic boundary (for which (NC

) is false) are much trickier; see

[43,44,76,127,149–151,176–179,192,193]. The assumption (N) is to be weakened

accordingly:

) the boundary matrix B(w) is of constant, maximal rank for all (x, t) ∈

∂Ω × [0,T] and all w ∈ U such that b(w)=b

(x, t), and

=kerB(w) ⊕ E

(A(w, ν(x))).

In geometrical terms, (N

) means the level sets

(x, t):={w ∈ U ; b(w)=b(x, t)}

are submanifolds of R

of the same dimension for all (x, t) ∈ ∂Ω × [0,T], and

that for all w ∈ M

(x, t) the tangent space T

(x, t) is transverse to the stable

subspace of the characteristic matrix A(w, ν(x)).

Finally, we will of course need the uniform Kreiss–Lopatinski˘ı condition, a

draft of which is the following.

(UKL) for all (w, x,ξ, τ) ∈ U × T

∗

∂Ω × C with Re τ>0 , there exists

C>0sothat

V ≤C B(w) V  for all V ∈ E

−

(w, x, ξ, τ),

where E

−

(w, x, ξ, τ) is the stable subspace of

A(w, x, ξ, τ):=A(w, ν(x))

−1

( τI

+ iA(w, ξ)),

and ν(x) denotes the outward unit normal to ∂Ωatpointx; and the

same is true for Re τ = 0 once the subspace E

−

has been extended by

continuity.

Again, (UKL) is to be replaced by a weaker version (UKL

), obtained by asking

the estimate only for those w that are in M

(x, t)forsomet ∈ [0,T].

11.1.2 Conditions on the data

The resolution of (11.0.1) is possible in Sobolev spaces under two ‘technical’

conditions: 1) that 0 is a solution of the special IBVP with zero initial data and

Main results 319

boundary data b

(·,t= 0), which amounts to asking

h(0) = 0 and b(0) = b(x, 0) for all x ∈ ∂Ω

(recall that the vanishing of the source term at 0 was also asked for the Cauchy

problem in Theorem 10.1); 2) that the boundary data and the initial data satisfy

some compatibility conditions.

Of course, 1) assumes that zero belongs to the domain U . It could be modiﬁed

into

h(w)=0 and b(w)=b

(x, 0) for all x ∈ ∂Ω

for some ﬁxed state w ∈ U . This would yield in ﬁne solutions in aﬃne spaces w +

instead of H

.Wesetw = 0 just to simplify the presentation. The point 2)

is undoubtedly crucial: we are bound to look for smooth solutions (because of

non-linearities), and we know that the existence of smooth solutions even in the

linear case does require compatibility conditions (see Section 9.2.3).

Deﬁnitely unpleasant to write down explicitly, compatibility conditions are

nevertheless very natural. Indeed, assume that u is a smooth – in particular

continuous up to the boundary – solution of (11.0.1), then necessarily

b(u

(x)) = b(x, 0) for all x ∈ ∂Ω.

Now, if u is C

up to the boundary,

∂

b(x, 0) = db(u

(x)) · ∂

u(x, 0) = db(u

(x)) ·



h(u

(x)) −



j=1

(x))∂

(x)



for all x ∈ ∂Ω. More generally, u being C

up to the boundary implies

∂

b(x, 0) = C

(x), Du

(x), ..., D

(x)) for all x ∈ ∂Ω (11.1.3)

for some complicated nonlinear function C

, which can be computed by induction

from C

(u)=b(u) through the formula

p+1

(u, Du, ..., D

p+1

u)=



k=0

(u, Du, ..., D

u) · D



h(u) −



j=1

(u) ∂



where d

denotes the diﬀerential of C

with respect to its (k + 1)th argument

(belonging to R

!), and D

denotes the k-th order diﬀerentiation with respect

to x ∈ R

11.1.3 Local solutions of the mixed problem

A rough statement of the main result in the theory of quasilinear mixed problems

is,

existence and uniqueness of smooth solutions for smooth enough and compatible

initial data and boundary data.

320 The mixed problem for quasilinear systems

In the accurate statement below, the regularity of the solution is a little weaker

than the regularity of the data.

Theorem 11.1 We assume U is convex, 0 ∈ U , h(0) = 0, (CH), (T),and

we take m an integer greater than (d +1)/2+1.

For al l b

∈ H

m+1/2

(∂Ω × [0,T]) such that b(·, 0) ≡ b(0) and all u

∈

m+1/2

(

Ω) with values in U satisfying the compatibility conditions (11.1.3) for

all p ∈{0,...,m− 1},aswellas(NC

), (N

),and(UKL

), there exists T>0

so that the problem (11.0.1) admits a unique solution u ∈ H

(

Ω × [0,T]) having

atraceon∂Ω that belongs to H

(∂Ω × [0,T]).

This result was announced by Rauch and Massey [165], and actually proved

by Mokrane [140]; see Section 11.2.2 for a rather detailed proof.

11.1.4 Well-posedness of the mixed problem

In the case of Friedrichs symmetrizability (and under other simplifying but less

crucial assumptions) it is possible to improve Theorem 11.1 and obtain solutions

with the same regularity as the data. This has been done by M´etivier [136], in

fact in the more complicated context of shock-waves stability, but we can state

a simpliﬁed version for ‘ordinary’ mixed problems.

Theorem 11.2 We assume Ω is the half-space {x =(y, x

) ∈ R

d−1

× R

}, the

domain U is convex and contains 0, we also assume (CH) and the existence

of a Friedrichs symmetrizer S

on U (i.e. S

: U → SPD

being C

∞

and such

that S

(w)A

(w) is symmetric for all j ∈{1,...,d}).

Additionally, we assume h ≡ 0 and b

≡ 0.

If m is an integer greater than d/2+1, for all u

:(y, z) → u

(y, z) in

d−1

× R

) with values in U , satisfying the compatibility conditions

(11.1.3) for all p ∈{0,...,m− 1},aswellas(NC

), (N

),and(UKL

then there exists T>0 so that the problem (11.0.1) admits a unique solution

u ∈ C ([0,T]; H

d−1

× R

)) such that ∂

u ∈ C ([0,T]; H

m−p

d−1

× R

)) for

all p ∈{1,...,m}. Furthermore, if the maximal time of existence of the solution

u is a ﬁnite T

∗

then either u(x, t) leaves every compact subset of U or

lim

tT

∗

∇

u(t)

∞

d−1

×R

)

=+∞.

In particular, this theorem contains a blow-up criterion analogous to what is

known for the Cauchy problem (see Theorem 10.3). Of course it does not tell us

in advance if (and when) blow-up will take place. In one space dimension a few

quantitative results are known, which provide either global solutions (for small

enough data and boundary damping [115, 116]), or an estimate of the blow-up

time (see, for instance, [14, 97]).

Proofs 321

11.2 Proofs

For both Theorem 11.1 and Theorem 11.2 the sketch of the proof is most classical,

and proceeds in two steps: 1) linearize and 2) use an iterative scheme. Details

are more cumbersome. We will give only the proof of Theorem 11.1. Regarding

Theorem 11.2, the reader is referred to Section 4 in M´etivier’s lectures notes [136].

11.2.1 Technical material

Topology

Let s be a real number greater than d/2 and consider a ﬁxed u ∈ H

) taking

values in U 0. Then the closure of u(R

) is a compact subset K of U ,and

there exist ρ>0andV ⊂⊂ U (which means

V is a compact subset of U )such

that K ⊂ V and for all v ∈ C (R

v − u

∞

)

≤ ρ implies v(x) ∈ V for all x ∈ R

Hence, denoting by ν

s,d

the norm of the Sobolev embedding

) → L

∞

v − u

)

≤ ρ/ν

s,d

implies v(x) ∈ V for all x ∈ R

ThesamepropertyistruewhenR

is replaced by

Ω. (Recall that H

(

Ω) is

made of functions having an extension in H

).) From now on, we ﬁx ρ

> 0

such that

v − u



∞

(Ω)

≤ ρ

implies v(x) ∈ V

⊂⊂ U for all x ∈ Ω,

where u

is the initial data in the IBVP (11.0.1), supposed to belong at least to

∈ H

(

Ω) for s>d/2. (This assumption will be reinforced later.)

Calculus

The short way of writing compatibility conditions in (11.1.3) is convenient but

it conceals some technical details needed for the proof of Theorem 11.1.

Here is a more explicit (though ugly) way, using repeatedly Fa´a di Bruno’s

formula for the nth derivative of a composition

(f ◦ u)

(n)



m=1



+···+i

,...,i

f ◦ u) · (u

)

,...,u

)

(The actual value of coeﬃcients c

,...,i

is all but important here.) The time

derivatives of a C

function u such that

∂

u = −





∂

u +



322 The mixed problem for quasilinear systems

should satisfy the induction formula

∂

u = −

i−1



=0



i − 1







j=1

(∂





) ∂

∂

i−1−

u + ∂

i−1



for all i ∈{1,...,p}.When



= A

◦ u and



h = h ◦ u,wemayuseFa´adi

Bruno’s formula to expand the derivatives ∂





and ∂

i−1



h. Using also Fa´adi

Bruno’s formula to expand ∂

(b ◦ u)forp ≥ 1, we eventually ﬁnd the non-linear

mapping C

introduced in Section 11.1.2 above alternatively reads

(u, Du,...,D

u)=



m=1



+···+i

,...,i

b(u) · (u

,...,u

), (11.2.4)

where the functions u

: x → u

(x) are such that











= u, u

= h ◦ u −



j=1

◦ u) ∂

i−1



k=1





+···+

=i−1



,...,

h ◦ u) · (u



,...,u



)

−



j=1

◦ u) ∂

i−1

−

i−1



=1



i − 1







j=1





k=1





+···+

=



,...,

◦ u) · (u



,...,u



) ∂

i−1−

for all i ∈{2,...,p}.

(11.2.5)

Lemma 11.1 If s>d/2 and u ∈ H

(

Ω), then for p the largest integer less than

s, there exists a ﬁnite sequence (u

,...,u

) satisfying (11.2.5) and such that

for all i ∈{0,...,p}, u

∈ H

s−i

(

Ω). Moreover, using this sequence in (11.2.4)

we get a well-deﬁned function x → C

(u(x), Du(x),...,D

u(x)) that belongs to

s−p

(

Ω).

Proof By assumption, u

= u belongs to H

(

Ω) = H

s−0

(

Ω). Let us see what

happens with u

. By assumption (our ‘technical’ condition 1)), h vanishes at 0, so

that Theorem C.12 implies h ◦ u belongs to H

. Furthermore, we may decompose

the other terms into

◦ u) ∂

u =(A

◦ u − A

(0)) ∂

u + A

(0) ∂

where the latter obviously belongs to H

s−1

and the former is a product of

one term in H

, thanks to Theorem C.12 again, and one term in H

s−1

:since

Proofs 323

s +(s − 1) − (s − 1) >d/2 by assumption, Theorem C.10 ensures the product is

in H

s−1

Let us now proceed by induction. We take i ≥ 2 and assume that u

∈

s−k

(

Ω) for all k ∈{0,...,i− 1}. Then the ﬁrst sum in u

is found to be in

s−i

exactly as in the case i = 1. In the second term we ﬁnd only products in

· H

s−

···H

s−

(using the same trick as for A

(0) to cope with the non-zero

h(0)), with

s +(s − 

)+···+(s − 

)=s + ks − (i − 1) >s− (i − 1) + d/2

and min(s, s − 

,...,s− 

)=s − (i − 1) , so that those products belong to

s−(i−1)

by Theorem C.10. Finally, the quadruple sum in u

involves products

in H

· H

s−

···H

s−

· H

s−i+

, with

s +(s − 

)+···+(s − 

)+(s − i + )=s + ks −  + s − i + >2s − i + d/2

and min(s, s − 

,...,s− 

,s− i + )=s − i +1, so that those products

belong to H

s−i

(since s − i<min(2s − i, s − i + 1)).

To ﬁnd that C

(u(x), Du(x),...,D

u(x)) belongs to H

s−p

we use exactly the

same argument as for the double sum in u

. 

‘Approximate solution’

The next technical step towards the proof of Theorem 11.1 consists in reducing

the problem to an IBVP with zero initial data. This is done thanks to an

appropriate lifting of the initial data u

Lemma 11.2 Under the assumptions of Theorem 11.1, there exists T

> 0 and

∈ H

m+1

(

Ω × R) vanishing for |t|≥2T

so that

)

|t=0

= u

, u

(x, t) − u

(x)≤

for all (x, t) ∈

Ω × [−T

and f

:= −L

+ h ◦ u

and g

:= −(b ◦ u

)

|∂Ω×R

+ b are such that

∂

≡ 0, ∂

≡ 0 at t =0 for all p ∈{0,...,m− 1}.

Furthermore, f

belongs to H

(

Ω × R), g

belongs to H

(∂Ω × R), and both

vanish for |t|≥2T

Proof By Lemma 11.1 we can construct u

∈ H

m+1/2−i

(

Ω) satisfying (11.2.5)

for all i ∈{1,...,m− 1} with u = u

. Then, by trace lifting (see, for instance,

[1], pp. 216–217), we ﬁnd u

∈ H

m+1

(

Ω × R) such that u



m+1

(

Ω×R)



u



m+1/2

(

Ω)

and

∂

)

|t=0

= u

for all i ∈{0,...,m− 1}.

By construction, u

belongs to C (I; H

(

Ω)) for all compact intervals I of R.

This implies, in particular, the existence of T

so that

u

(t) − u

(0)

(

Ω)

≤

2ν

m,d

324 The mixed problem for quasilinear systems

for all t ∈ [−T

], hence

u

(t) − u

(0)

∞

(Ω)

≤

which means

u

(x, t) − u

(x)≤

for all (x, t) ∈

Ω × [−T

This ensures, in particular, that u

(x, t) stays in U for all (x, t) ∈

Ω × [−T

Furthermore, multiplying u

by a suitable C

∞

cut-oﬀ function in time, we may

assume without loss of generality that u

vanishes for |t|≥2T

(or any number

greater than T

) and, thanks to the convexity of U and the fact that 0 belongs

to U ,thatu

stays in U for all t ∈ R. This precaution allows us to speak about

◦ u

, h ◦ u

and b ◦ u

and therefore to deﬁne f

:= −L

+ h ◦ u

and

:= −(b ◦ u

)

|∂Ω×R

+ b.Thatf

and g

are in H

follows in a classical way

from Proposition C.11 and Theorem C.12. The vanishing of ∂

at t = 0 for

p ≤ m − 1 follows from the construction – (11.2.5) – of the u

= ∂

)

|t=0

.The

vanishing of ∂

at t = 0 for p ≤ m − 1 is a consequence of the compatibility

conditions in (11.1.3) and the deﬁnition of the non-linear functions C

in (11.2.4).

Finally, that f

and g

vanish for |t|≥2T

follows from our ‘technical’ condition

1) and the fact that u

does so. 

Once the ‘approximate solution’ u

is available, the resolution of the IBVP

(11.0.1) is equivalent to the resolution of the IBVP with zero initial data











+ v)=h(u

+ v) , Ω × (0,T) ,

b(u

+ v)=b ,∂Ω × (0,T) ,

|t=0

=0, Ω .

(11.2.6)

Iterative scheme

The resolution of (11.2.6) will be done using the natural iterative scheme











k+1

= −L

+ h(u

+ v

) , Ω × (−∞,T] ,

B(u

+ v

) v

k+1

= B(u

+ v

) v

− b(u

+ v

)+b ,∂Ω × (−∞,T] ,

k+1

|t<0

=0, Ω .

(11.2.7)

In what follows we shall denote by I

the time interval (−∞,T].

Let us introduce the following notations, extending those of Lemma 11.1 to

a non-zero perturbation v of u

= −L

+ h ◦ (u

+ v) ,g

= B ◦ (u

+ v) · v − b ◦ (u

+ v)+b.