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

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Smooth solutions 295

iii) If s>1 and α is a d-uple of length |α|≤s, there exists C>0 such that

for all u and a in H

with ∇u and ∇a in L

∞

 [ ∂

,a∇] u 

≤ C ( ∇a

∞

u

+ ∇u

∞

a

) . (10.1.12)

The estimates in (10.1.10) and (10.1.11) are classical (see [207], pp. 10–11,

or [7], pp. 98–103). For completeness, their proof is given in Appendix C (see

Proposition C.11 for i) and Theorem C.12 for ii)). The last one, (10.1.12), is

actually a consequence of i) (see Proposition C.13 in Appendix C).

Note that the L

∞

assumption in i) (respectively, in iii)) automatically follows

from the Sobolev embedding (10.1.7) if s>d/2 (respectively, if s>d/2+1).

We can now give a detailed proof of Theorem 10.1. There are two main steps,

in terms of the so-called ‘high norm’ and ‘low norm’: 1) controlling the high norm

u

k+1

− g



C ([0,T ];H

)

and, 2) showing a contraction property for the sequence

)

k∈N

in C ([0,T]; L

). Both steps appear to work for T small enough. In the

ﬁrst one, we shall assume s is an integer in order to stay at a more elementary

level. The result is true for any real number s greater than the critical regularity

index d/2 + 1 though. (Also see Theorem 10.2 below.)

High-norm boundedness For simplicity, we assume that s is an integer. We

proceed by induction. Initially, we have u

(t)=g

and thus

u

− g



C ([0,T ];H

)

=0≤ δ

for any T and δ. We assume that for all  ≤ k, u



is deﬁned inductively by

(10.1.5) and u



(0) = g



and satisﬁes the estimate

u



− g



C ([0,T ];H

)

≤ δ. (10.1.13)

We are going to show the same estimate (10.1.13) for  = k + 1, provided that

T is suitably chosen.

We introduce the notations v

k+1

:= u

k+1

− g

, w

k+1

:= ∂

k+1

where α is

any d-uple of length |α|≤s. By deﬁnition, v

k+1

must solve the Cauchy problem











∂

k+1



j=1

) ∂

k+1

= b(u

) −



j=1

) ∂

k+1

(0) = g

k+1

− g

(10.1.14)

and, diﬀerentiating, w

k+1

must solve











∂

k+1



j=1

) ∂

k+1

= f

k+1

(0) = ∂

( g

k+1

− g

) ,

(10.1.15)

296 The Cauchy problem for quasilinear systems

where the right-hand side f

k+1

is deﬁned by

k+1

:= ∂

b(u

) −



j=1

∂

( A

) ∂

) −



j=1

[ ∂

) ∂

] v

k+1

From (10.1.13) and the remark above, we know there is a constant C, depending

only on

ε and δ so that

u





C ([0,T ];H

)

≤ C, u





∞

≤ C, ∇





∞

≤ C (10.1.16)

for all  ≤ k. By Proposition 10.1 (i) and ii)), this shows that the ﬁrst terms in

k+1

,namely∂

b(u

)and



j=1

∂

( A

) ∂

) are bounded in C ([0,T]; L

)(by

a constant depending only on

ε and δ). Furthermore, (10.1.16), Proposition 10.1

(iii)) and the Sobolev inequality

∇

k+1

(t)

∞

≤ C



v

k+1

(t)

show that





j=1

[ ∂

) ∂

] v

k+1



C ([0,T ];L

)

≤ C



v

k+1



C ([0,T ];H

)

where C



is another constant depending only on ε and δ. For simplicity, we omit

the primes. Thus we have an estimate

f

k+1



C ([0,T ];L

)

≤ C (1 + v

k+1



C ([0,T ];H

)

) . (10.1.17)

A similar estimate also holds for S(u

) f

k+1

since u

is bounded in L

∞

.Now,

using the estimates in (10.1.16) for  = k and  = k − 1 and recalling from (10.1.5)

for k − 1that

∂

= −



j=1

k−1

) ∂

+ b(u

k−1

we see that ∂

is also bounded in L

∞

. Together with the L

∞

bound for ∇

this implies

∂

S(u



∂

( S(u

))

∞

≤ C

(10.1.18)

for some constant C

depending only on ε and δ. Therefore, we can apply

the estimate (2.1.4) from Chapter 2 (Proposition 2.2) to the Cauchy problem

(10.1.15) and λ such that β(λ − 1) ≥ C

. We ﬁnd that

w

k+1

(t)

≤ e

λt

g

k+1

− g





λ (t−τ )

f

k+1

(τ)

dτ.

Smooth solutions 297

Using (10.1.17) and summing on m all the inequalities for w

k+1

, we get another

constant C

(depending only on s)sothat

sup

t∈[0,T ]

v

k+1

(t)

≤ C

λT

( g

k+1

− g



+2T (1 + sup

t∈[0,T ]

v

k+1

(t)

)).

For 4C

λT

T ≤ β

, we can absorb the H

norm of v

k+1

into the left-hand side.

This yields the estimate

sup

t∈[0,T ]

v

k+1

(t)

≤ 2 C

λT



g

k+1

− g



+2T



Since g

tends to g in H

when ε

goes to 0, we can ensure that

sup

t∈[0,T ]

v

k+1

(t)

≤ δ

by choosing ε

small enough, up to diminishing T again.

Low-norm contraction Subtracting (10.1.5) for k − 1totheonefork,we

see that W

k+1

:= u

k+1

− u

, solves the Cauchy problem











S(u

) ∂

k+1



j=1

S(u

) A

) ∂

k+1

= F

k+1

(0) = g

k+1

− g

(10.1.19)

where

k+1

= S(u

) b(u

) − S(u

k−1

) b(u

k−1

) − ( S(u

) − S(u

k−1

))∂

−



j=1

( S(u

) A

) − S(u

k−1

) A

k−1

))∂

In view of (10.1.16) for  = k and  = k − 1 and the fact that ∂

is also bounded

in L

∞

, the mean-value theorem implies that

F

k+1



C ([0,T ];L

)

≤ C W



C ([0,T ];L

)

. (10.1.20)

Applying the estimate (2.1.4) from Chapter 2 to the Cauchy problem (10.1.19),

we have for β(λ − 1) ≥ C

(the bound C

being as in (10.1.18))

sup

t∈[0,T ]

W

k+1

(t)

≤ e

λT

g

k+1

− g



+ C

T e

λT

sup

t∈[0,T ]

W

(t)

Provided that T is small enough, more precisely if

T e

λT

≤ β

we thus have the uniform estimate

sup

t∈[0,T ]

u

k+1

(t) − u

(t)

≤

sup

t∈[0,T ]

u

(t) − u

k−1

(t)

λT

g

k+1

− g



298 The Cauchy problem for quasilinear systems

Hence (u

)

k∈N

will be a Cauchy sequence in C ([0,T]; L

) provided that the series



g

k+1

− g



is convergent. The estimate in (10.1.6) shows that this holds true for ε

−k

ε,

for instance. With this choice, the sequence (u

)

k∈N

has a limit u in C ([0,T]; L

It remains to prove additional regularity on u. (The method will generalize the

proof of the continuity with values in H

of the solution in Theorem 2.9.)

Regularity and uniqueness Since (u

(t))

k∈N

is bounded in H

(see

(10.1.16)) and convergent in L

(for t ≤ T ), the limit u(t)mustbeinH

weak compactness of bounded balls in H

and uniqueness of the limit in the

sense of distributions. Furthermore, by L

–H

interpolation, u

is found to

converge in C ([0,T]; H



) for all s



∈ (0,s). In particular, we can choose s



greater

than 1 + d/2 (like s). Standard Sobolev embeddings then imply the convergence

holds in C ([0,T]; C

)). Thus we can pass to the limit in the iteration scheme

(10.1.5), which shows that ∂

tends to ∂

u in C ([0,T]; C

)) and the limit u

is a C

solution of (10.1.3). The initial condition is trivially satisﬁed, by passing

to the limit in the initial condition for u

. It is not diﬃcult to show that the C

solution constructed by the iteration scheme (10.1.5) is the only one satisfying

(10.1.8) in the time interval [0,T]. As a matter of fact, the L

norm of the

diﬀerence between two solutions u and v can be estimated similarly as in the low-

norm calculation on u

k+1

− u

. (We can even show the uniqueness of classical

solutions in the wider class of entropy solutions, see [46] and Section 10.2 below.)

We already know that u belongs to C ([0,T]; H



) for all s



<s. In fact, we can

show that u belongs to C ([0,T]; H

) (which automatically implies that u belongs

to C

([0,T]; H

s−1

) by the equation in (10.1.3)). The proof is not obvious though.

As a ﬁrst step, we can check that u

(t) converges uniformly on [0,T]inH

the space H

equipped with the weak topology. We just take any φ in H

−s

choose ψ in the dense subspace H

−s



(for s



<s) close enough to φ,andmakea

standard splitting. Using the estimates in (10.1.16), we get

sup

t∈[0,T ]

|φ, (u

−u)(t)

−s

|≤Cφ − ψ

−s

+sup

t∈[0,T ]

|ψ, (u

−u)(t)

−s



The ﬁrst term in the right-hand side can be made arbitrarily small, and the

second term is known to tend to 0 since u

converges to u in C ([0,T]; H



Hence the left-hand side also tends to 0.

Secondly, up to translating or/and reversing time, it is suﬃcient to prove the

right continuity in H

of the limit u at t = 0. Furthermore, by a straightforward

ε/3 argument, we have that u(t) − g tends to 0, as t tends to 0+, in H



for all



<s, and so by the same splitting as before, u(t) tends to g in H

. In particular,

we have

lim inf

t0

u(t)

≥g

Smooth solutions 299

Hence, to prove the strong convergence, it just remains to show that

g

≥ lim sup

t0

u(t)

It appears that this inequality is easier to show using the equivalent norm on

, deﬁned by

v

s,g



|α|≤s

S(g) ∂

v, ∂

v .

More generally, because of (10.1.9), the norm ·

s,u

associated with any func-

tion u satisfying the estimate in (10.1.8) may serve as an equivalent norm. In

particular, we see that u(t) satisﬁes (10.1.8) by taking the liminf as  tends to

+∞ in (10.1.13). And in fact, an additional useful observation is that

lim sup

t0

u(t)

s,u(t)

= lim sup

t0

u(t)

s,g

This follows from the pointwise continuity of u at t = 0 and its boundedness in

. Therefore, we are led to showing

g

s,g

≥ lim sup

t0

u(t)

s,u(t)

. (10.1.21)

The third and last step is based on an energy estimate very similar to that

previously computed in the high norm. For all d-uple m of length |m|≤s, u

k+1

∂

k+1

solves the Cauchy problem











∂

k+1



j=1

) ∂

k+1

= h

k+1

(0) = ∂

k+1

where the right-hand side h

k+1

is deﬁned by

k+1

:= ∂

b(u

) −



j=1

[ ∂

) ∂

] u

k+1

Similarly as in (10.1.17), we have

h

k+1



C ([0,T ];L

)

≤ C (1 + u

k+1



C ([0,T ];H

)

Now we compute that

S(u

) u

k+1

 = ( ∂

S(u



∂

(S(u

)) ) u

k+1



+2Re S(u

) u

k+1

.

300 The Cauchy problem for quasilinear systems

By (10.1.18) and the uniform boundedness of u

k+1

in H

, we ﬁnd after integration

that

S(u

(t)) u

k+1

(t) ,u

k+1

(t) ≤S(g

) g

k+1

 + C



Summing on m we get

u

k+1

(t)

s,u

(t)

≤g

k+1



s,g

+ C



where the right-hand side tends to g

s,g

+ C



t as k → +∞ (because of the

construction of g

through molliﬁcation of g). As to the left-hand side, it is such

that

u(t)

s,u(t)

≤ lim sup

k→+∞

u

k+1

(t)

s,u

(t)

because of the weak convergence of u

k+1

(t)inH

and the uniform pointwise

convergence of u

(t). Therefore, we have

u(t)

s,u(t)

≤g

s,g

+ C



which yields (10.1.21). 

In fact, the local well-posedness result shown in Theorem 10.1 is also valid,

under the same restriction of the regularity index, for systems admitting a sym-

bolic symmetrizer in the following natural sense (see Deﬁnition 2.4 in Chapter 2).

Deﬁnition 10.2 A symbolic symmetrizer associated with the quasilinear system

(10.1.3) is a C

∞

mapping

S : R

× (R

\{0}) → M

(C),

homogeneous degree 0 in ξ such that

S(u, ξ)=S(u, ξ)

∗

> 0 and S(u, ξ) A(u, ξ)=A(u, ξ)

∗

S(u, ξ)

for all (u, ξ) ∈ R

× (R

\{0}), where A(u, ξ):=



j=1

(u).

Theorem 10.2 We assume that A

and b are C

∞

functions of u ∈ R

,with

b(0) = 0, and that (10.1.3) admits a symbolic symmetrizer. For all initial data

g ∈ H

) with s>

+1, there exists T>0 such that (10.1.3) has a unique

solution u ∈ C ([0,T]; H

) such that u(0) = g.

This theorem is shown for higher s by Taylor in [205]. M´etivier [132] proves

it under the optimal condition s>

+ 1. In fact, he also allows the matrices

and the reaction term b to depend on (x, t) inside a compact set, and b to

not necessarily vanish at 0. Moreover, the same precautions as in Theorem 10.1

would allow those matrices A

and the reaction term b to be deﬁned only on an

open subset U of R

. We omit these reﬁnements for simplicity.

Smooth solutions 301

Proof Unsurprisingly, the proof relies on the linear result in Theorem 2.7 and

an iterative scheme. After initialization by u

= g, the scheme merely reads

k+1

= b(u

) ,u

k+1

(0) = g,

where

= ∂



(u(x, t)) ∂

This deﬁnes inductively u

∈ C ([0,T]; H

) ∩ C

([0,T]; H

s−1

). Furthermore, at

least for small T, we can control inductively the norms of u

governing the

constants K

, C

and γ

in the energy estimate

u

k+1

(t)

≤ e



u(0)

+ C

t L

k+1



∞

(0,T ;H

)



In other words, we can make K

, C

and γ

independent of k, provided that

t ≤ T is small enough. Recall that K

a priori depends on u



∞

([0,T ];W

1,∞

)

and

, γ

a priori depend on

sup



u



∞

([0,T ];H

)

, ∂



∞

([0,T ];H

s−1

)



We take M

> g

1,∞

, M

> g

and M

> 0 (to be enlarged later)

and look for conditions under which

u



∞

([0,T ];W

1,∞

)

≤ M

, u



∞

([0,T ];H

)

≤ M

∂



∞

([0,T ];H

s−1

)

≤ M

independently of k.

Assume that this is the case for u

. By Theorem C.12, we know there exists

= c

)sothatb(u

)

≤ c

). Therefore, we have the estimate

u

k+1

(t)

≤ e

γ(max(M

))T



K(M

) g

+ TC(max(M

)) c

)



Hence, the ﬁrst condition to keep u

k+1

(t)

less than M

γ(max(M

))T



K(M

) g

+ TC(max(M

)) c

)



≤ M

Assume that this is the case. Then, also by Theorem C.12 plus Proposition C.11

and the Sobolev embedding H

s−1

→ L

∞

, there exists c

= c

)sothat





) ∂

k+1



s−1

≤ c

Consequently, ∂

k+1

= b(u

) −



) ∂

k+1

satisﬁes the estimate

∂

k+1



∞

([0,T ];H

s−1

)

≤ c

)+c

Finally, we need an estimate of

u

k+1



∞

([0,T ];W

1,∞

)

≤u

k+1

− u

k+1

(0)

∞

([0,T ];W

1,∞

)

+ g

∞

([0,T ];W

1,∞

)

302 The Cauchy problem for quasilinear systems

Since

u

k+1

− u

k+1

(0)

∞

([0,T ];W

1,∞

)

≤ C



u

k+1

− u

k+1

(0)

∞

([0,T ];H



)

for s



>d/2 + 1, we can bound this term by L

–H

interpolation. For d/2+1<



<s,weget

u

k+1

− u

k+1

(0)

∞

([0,T ];W

1,∞

)

≤ C



1−s



∂

k+1



1−s



∞

([0,T ];L

)

u

k+1

− u

k+1

(0)



∞

([0,T ];H

)

≤ 2



1−s



1−s



≤



s,s



max(M

) T

1−s



In conclusion, the following successive choices appear to work. We take M

g

1,∞

, M

> max(1,K(M

)

1/2

) g

and M

≥ c

)+c

)andthen

choose T small enough so that

g

1,∞



s,s



max(M

) T

1−s



≤ M

γ(max(M

))T



K(M

) g

+ TC(max(M

)) c

)



≤ M

Once we have these strong bounds on the sequence (u

) we can show its con-

vergence in C ([0,T],H

s−1

). Indeed, we have by construction (u

k+1

− u

)(0) =

0and

k+1

− u

)=b(u

) − b(u

k−1

) − (L

− L

k−1

) u

The right-hand side above can be estimated using Corollary C.3 (Appendix C).

Hence, applying the a priori estimate of Theorem 2.4 with m = s − 1, we obtain

u

k+1

− u



C ([0,T ];H

s−1

)

≤ CTe

γT

u

− u

k−1



C ([0,T ];H

s−1

)

where C and γ are independent of k – they can be evaluated in terms of M

. So the sequence (u

) is convergent in C ([0,T],H

s−1

)ifT is so small that

CTe

γT

< 1.

The rest of the proof is the same as for Theorem 10.1. 

10.1.2 Continuation of solutions

Once we have local existence results like Theorems 10.1 and 10.2, we may wonder

whether it is possible to continue, and for how long, local-in-time solutions.

The continuation principle stated hereafter gives a simple answer to the ﬁrst

part of the question. Suppose T is the maximal time of existence of a solution

u ∈ C ([0,T); H

)). There are roughly two alternatives. Either T is inﬁnite or

u(t)

1,∞

)

is unbounded as t approaches T . In the latter case, a possibility

is that u(x, t) escapes any compact set of R

. In fact, if the matrices A

and the

reaction term b are well-deﬁned only on an open subset U of R

, the correct

statement is that, if T is ﬁnite, either u(x, t) escapes any compact subset of

U or ∇

u(t)

∞

)

is unbounded. The ﬁrst case is analogous to the ﬁnite

Smooth solutions 303

time blow-up in ordinary diﬀerential equations. The second one corresponds

to the formation of shocks, a phenomenon that is classically explained on the

one-dimensional example of Burgers’ equation, corresponding to A

(u)=u.

For more details on blow-up (in one space dimension), we recommend, for

instance, the book by Whitham [219], p. 42–46. Also see the more recent book by

Alinhac [6].

We give below a continuation result in the most general framework of systems

admitting a symbolic symmetrizer. We shall use again para-diﬀerential calculus

in the proof. Of course, a more elementary proof can be given for Friedrichs-

symmetrizable systems (see, for instance, Majda [126], p. 46–48).

Theorem 10.3 Let U be an open subset of R

containing the closed ball

B(0; ω). We assume that A

and b are C

∞

functions of u ∈ U , with b(0) = 0,

and that (10.1.3) has a symbolic symmetrizer in U .Ifu ∈ C ([0,T); H

) with

0 <T <+∞ and s>

+1 is a solution of (10.1.3) such that

u

∞

([0,T ];W

1,∞

))

≤ ω

then u is continuable to a solution in C ([0,T



]; H

) with T



>T.

Proof The main ingredient will be a uniform estimate of u(t)

for all

t<T. In fact, this estimate is almost contained in the proof of Theorem 2.4 in

Chapter 2 applied to v = u. For clarity, we give below the crucial points. We

shall use the same notations as in the proof of that theorem. We ﬁrst introduce

the para-diﬀerential operator

= ∂



(u)

∂

For all t<T, P

u(t) − L

u(t) belongs to H

because of Proposition C.9 and

Theorem C.12, and we have a uniform estimate

P

u − L

u 

≤ K u

where K = K(ω). This is the ﬁrst point. Then we proceed as in the proof

of Theorem 2.4. The new fact here is that ∂

u

∞

×[0,T ])

is bounded by a

constant C = C(ω). This is simply due to the equation

∂

u = −



(u)∂

u + b(u).

So, the estimate obtained at the end is

u(t)

≤

(ω)

β(ω)



γt

u(0)



γ (t−τ )

P

u(τ)

dτ



304 The Cauchy problem for quasilinear systems

valid for all γ ≥ 1+ 2β(ω)

−1

(ω)+C(ω)). Now, using that P

u = P

u −

u + L

u, we get the estimate

u(t)

≤



γt

u(0)



γ (t−τ )

( K + K

) u(τ)

dτ



where K

= K

(ω) comes from Theorem C.12 applied to F = b.Soinshort,

there is a constant C

depending only on ω and T such that

u(t)

≤ C



u(0)



u(τ)

dτ



By Gronwall’s Lemma, this implies

u(t)

≤ C

u(0)

Thus u belongs to L

∞

([0,T]; H

). Furthermore, u belongs to C

([0,T); H

s−1

)–

because of the equation ∂

u = −



(u) ∂

u + b(u)andu ∈ C ([0,T); H

)–

and thus to C

([0,T); L

∞

) – by Sobolev embedding – and we have a uniform

bound

∂

u

∞

×[0,T ])

≤ C.

Therefore, u is continuable to a C ([0,T]; L

∞

) function. Hence, by weak compact-

ness of bounded sets in Hilbert spaces and uniqueness of the limit in the sense

of distributions, we have

u(t) u(T )inH

when t  T.

Consequently, we have u ∈ L

∞

([0,T]; H

) ∩ C ([0,T]; H

To show that u actually belongs to C ([0,T]; H

), we can now make use of

the uniqueness part of Theorem 2.7 in Chapter 2 – applied to v = u and f =

b(u). Indeed, f belongs to L

∞

([0,T]; H

) ∩ C ([0,T]; L

∞

) – like u – and thus to

∞

([0,T]; H

) ∩ C ([0,T]; H

). So the theorem does apply, and the weak solution

u ∈ L

([0,T]; H

)of∂

u +



(u) ∂

u = f is a strong solution and belongs

in fact to C ([0,T]; H

Once we know this, we can apply the local existence result in Theorem 10.2

to u(T ) as initial data, and this completes the proof. 

10.2 Weak solutions

As suggested by our previous remarks and everyday experience of shocks in

gas dynamics, for instance, blow-up in ﬁnite time does occur for generic initial

data. This urges us to consider weak solutions, i.e. solutions in the sense of

distributions that are not even continuous. For general quasilinear systems, the

meaning of discontinuous solutions is unclear. However, for systems in divergence

form (10.0.1), there is no ambiguity, and the associated Cauchy problems admit

natural weak formulations (see, for instance, [46], p. 50–51 or [184], p. 86–87).