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

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Proofs 325

These are well-deﬁned provided that u

+ v only achieves values in U :bythe

estimate of u

in Lemma 11.1, this will be the case as long as v

∞

≤ ρ

/2.

If v belongs to H

(

Ω × I

) and its trace on ∂Ω belongs to H

(∂Ω × I

)then

Proposition C.11 and Theorem C.12 imply that f

also belongs to H

(

Ω × I

)

and g

belongs to H

(∂Ω × I

). In fact, we have the following, more precise

result, which will be useful to make the scheme in (11.2.7) work.

Lemma 11.3 For al l T ∈ (0,T

], for all v ∈ H

(

Ω × I

), of norm less than

/(2ν

m,d+1

), having a trace in H

(∂Ω × I

) and such that v

|t<0

≡ 0, we have

∂

)

|t=0

=0 and ∂

)

|t=0

=0 for all p ∈{0, ..., m − 1}.

Furthermore, for all M ∈ (0,ρ

/(2ν

m,d+1

)) there exist C

= C

(M) and C

(M) so that for all T ∈ (0,T

v

(

Ω×I

)

≤ M

implies

f



(

Ω×I

)

≤ C

(M) and g



(∂Ω×I

)

≤ TC

(M)+ε(T ),

where ε(T ) is independent of M andgoestozeroasT goes to zero.

Proof By deﬁnition of f

, the derivative ∂

)att = 0 reduces to the sum of

∂

)

|t=0

, which is known to be zero by Lemma 11.2, and terms with derivatives

of v up to order p in factor, which are zero at t = 0 by assumption. Therefore,

∂

)

|t=0

= 0. Similarly, ∂

)

|t=0

= 0 follows from the fact that ∂

)

|t=0

Since m>(d +1)/2, the estimate of f

in H

is a straightforward conse-

quence of Theorem C.12. The estimate of g

is trickier. By a second-order Taylor

expansion of b (recalling that B =db)wehave

(x, t)=



(θ − 1) d

b(u

(x, t)+θv(x, t)) · (v(x, t),v(x, t)) dθ

+ b

(x, t) − b(u

(x, t))

for all (x, t) ∈ ∂Ω × [0,T

] . Let us deﬁne

ε(T ):=b

− b ◦ u



(∂Ω×I

)

 b − b ◦ u



C (I

m−1

(∂Ω)

This does go to zero with T since (by construction of u

) b(x, 0) = b(u

(x, 0))

for all x ∈ ∂Ω. Regarding the other term in g

we have, thanks to Proposition

C.11 and Theorem C.12,





(θ − 1) d

b(u

+ θv) · (v, v)dθ



(∂Ω×I

)

≤ C



v

∞

(∂Ω×I

)

326 The mixed problem for quasilinear systems

for some new constant C



depending only on M (the H

norm of v). Further-

more, since m − 1 > (d +1)/2 by assumption, we have

v

∞

(∂Ω×I

)

≤ C



m−1,d

v

m−1

(∂Ω×I

)

Now, since all time derivatives of v up to order m − 1 vanish at time t =0,we

have by the Cauchy–Schwarz inequality

∂

∂

v

(∂Ω×I

)

≤ T ∂

p+1

∂

v

(∂Ω×I

)

for all p ∈{0,...,m− 1} and all d-uple α of length less than or equal to m −

1 − p. This shows that

v

m−1

(∂Ω×I

)

≤ T v

(∂Ω×I

)

hence the result aimed at, with C

= C



m−1,d

M. 

11.2.2 Proof of Theorem 11.1

Construction of the sequence (v

)

We ﬁx M ∈ (0,ρ

/(2ν

m,d+1

)). We set v

= 0, and construct v

by induction.

Assume that v

has been constructed in such a way that for some T ∈ (0,T











∈ H

(

Ω × I

) ,v

|∂Ω

∈ H

(∂Ω × I

) ,

with v



(

Ω×I

)

≤ M,

and v

|t<0

≡ 0 .

(11.2.8)

Then by Theorem 9.21, the IBVP in (11.2.7) admits a unique solution v

k+1

∈

(

Ω × I

) having a trace in H

(∂Ω × I

) and satisfying the estimate

v

k+1



(

Ω×I

)

+ v

k+1



(∂Ω×I

)

≤C



T f



(

Ω×I

)

+ g



(∂Ω)



where C depends only on the H

norm of u

+ v, i.e. depends only on M

(+u



(

Ω×R)

, which is ﬁxed anyway). This implies in, particular,

v

k+1



(

Ω×I

)

≤

√



T f



(

Ω×I

)

√

T g



(∂Ω)



or using Lemma 11.3,

v

k+1



(

Ω×I

)

≤

√



(M)+T

3/2

(M)+T

1/2

ε(T )



which we can assume to be less than or equal to M , up to diminishing T .This

enables us to construct a whole sequence (v

)

k∈N

satisfying (11.2.8).

Convergence of the sequence (v

)

From the uniform H

bound in (11.2.8) we already know there is a subsequence

of (v

) converging weakly in H

. The next point consists in proving the (whole)

Proofs 327

sequence converges in L

. More precisely, we are going to prove that both (v

)

and the sequence of traces on ∂ΩconvergeinL

In this respect, we look at the diﬀerence w

:= v

k+1

− v

, which satisﬁes











=(L

k−1

− L

) u

+ h(u

) − h(u

k−1

) , Ω × (0,T) ,

B(u

) w

= B(u

k−1

) w

k−1

+ b(u

k−1

) − b(u

) ,∂Ω × (0,T) ,

k+1

|t=0

=0, Ω ,

where we have denoted for simplicity u

:= u

+ v

. Hence, by the L

estimate

(9.2.51) in Theorem 9.18, there exists c

> 0 depending only on u



1,∞

(

Ω×I

)

so that

w



(

Ω×I

)

+ w

|∂Ω



(∂Ω×I

)

≤ c

T (L

k−1

− L

) u

+ h(u

) − h(u

k−1

)

(

Ω×I

)

+ c

B(u

k−1

) w

k−1

+ b(u

k−1

) − b(u

)

(∂Ω×I

)

Observing that

u



1,∞

(

Ω×I

)

 u



(

Ω×I

)

since m>(d +1)/2 + 1, we see the constant c

depends in fact only on M.

Merely by the mean value theorem we obtain c



> 0, depending only on the

maximum of u



∞

(

Ω×I

)

and u

k−1



∞

(

Ω×I

)

such that

(L

k−1

− L

) u

+ h(u

) − h(u

k−1

)

(

Ω×I

)

≤ c



(1 + u



1,∞

(

Ω×I

)

) w

k−1



(

Ω×I

)

And by a second-order Taylor expansion of b we ﬁnd that

B(u

k−1

) w

k−1

+ b(u

k−1

) − b(u

)

(∂Ω×I

)

≤ c



w

k−1



∞

(∂Ω×I

)

w

k−1



(∂Ω×I

)

where c



depends only on the maximum of u



∞

(∂Ω×I

)

and u

k−1



∞

(∂Ω×I

)

Now since m>d/2,

w

k−1



∞

(∂Ω×I

)

 w

k−1



(∂Ω×I

)

≤ 2

√

(M)+TC

(M)+ε(T ))

thanks to Lemma 11.3 and the H

estimate on v

|∂Ω

and v

k−1

|∂Ω

. So, ﬁnally, we

have

w



(

Ω×I

)

+ w

|∂Ω



(∂Ω×I

)

≤ c

T w

k−1



(

Ω×I

)

+ ε

(T ) w

k−1

|∂Ω



(∂Ω×I

)

328 The mixed problem for quasilinear systems

for c

depending only on M and ε

(T ) going to zero as T goes to zero.

Consequently, up to diminishing T so that all four numbers

ε

(T ) , c

T, ε

(T ) T, c

are less than or equal to 1/4, we have

w



(

Ω×I

)

≤2

−k

w



(

Ω×I

)

and w

|∂Ω



(∂Ω×I

)

≤2

−k

w



(∂Ω×I

)

This implies both (v

)and(v

|∂Ω

) are Cauchy sequences in L

. Let us call v and

their respective limits.

Conclusion

As already said, the limit v of (v

) is necessarily in H

(

Ω × I

). Similarly,

because of the uniform H

bound (

√

C (

√

(M)+T

(M)+ε(T

)))for

the traces, the limit v

of (v

|∂Ω

) is necessarily in H

(∂Ω × I

). Moreover, by

–H

interpolation, (v

) converges strongly to v in H

(

Ω × I

)and(v

|∂Ω

)

converges strongly to v

in H

(∂Ω × I

) for all s ∈ [0,m). Therefore v = v

|∂Ω

and v solves the IBVP (11.2.6), so that u = u

+ v solves the original IBVP

(11.0.1). 2

PERSISTENCE OF MULTIDIMENSIONAL SHOCKS

Shock waves are of special importance in such diverse applications as aero- and

gas dynamics, materials sciences, space sciences, geosciences, life sciences and

medicine. Research in this ﬁeld is a century old but still very active, as the

existence of a research journal precisely entitled Shock Waves attests.

We know very well from everyday experience what shock wave means (espe-

cially in gas dynamics). However, it is a mathematical issue to prove the existence

and/or the stability of (arbitrarily curved) shocks for general hyperbolic systems,

and in particular for the Euler equations.

The formal, linearized stability of shock waves was addressed in the 1940s

by several physicists and engineers. Then, the mathematical analysis of the fully

non-linear problem waited for the independent works of Majda [124–126] and

Blokhin [18] in the 1980s, and was more recently revisited by M´etivier and co-

workers [56, 133, 136, 140].

Following Freist¨uhler [58,59] in his work on non-classical shocks, we use here

the term persistence (in particular in the title of this chapter) as a shortcut for

existence-and-stability. Both notions are indeed closely related, and we can view

the stability problem as a preliminary step to the existence problem: assume

a special shock-wave solution is known (e.g. a planar shock propagating with

constant speed, which is not diﬃcult to ﬁnd); one may ask whether a small

initial pertubation (of the shock front and of the states on either side) will

destroy its structure, or lead to a solution (local in time) still made of smooth

regions separated by a (modiﬁed) shock front; when the reference shock falls

into the latter case for a suﬃciently large set of initial perturbations, it may

be called ‘structurally stable’, and thus serve as a model to construct, in other

words to show the existence of, a non-planar shock. Alternatively, one may put

the problem slightly diﬀerently: consider a Cauchy problem where the initial

data consist of two smooth regions separated by a given hypersurface; under

what conditions (on the initial data) does this Cauchy problem admit a solution

made of smooth regions separated by a (moving) shock front? The answer is

twofold, as the initial data must satisfy compatibility and stability conditions.

The necessity of compatibility conditions is easy to understand: even in one space

dimension, two arbitrary, uniform states are not connected by a single shock wave

in general. (The corresponding Cauchy problem is called a Riemann problem, and

its solution involves, in general, as many waves as there are characteristic ﬁelds.

See, for instance, [24, 46, 88, 184].) Those compatibility conditions come from

330 Persistence of multidimensional shocks

the Rankine–Hugoniot jump conditions across fronts of discontinuities. Stability

conditions are of a diﬀerent nature: as shown by Majda, the stability of a planar

shock wave (propagating with constant speed) is roughly speaking equivalent

to the well-posedness of a (non-standard) constant-coeﬃcients Initial Boundary

Value Problem, which is itself encoded by the so-called (generalized) Lopatinski˘ı

condition. It also turns out that a (arbitrarily curved) shock front is ‘structurally

stable’ provided that all local jump discontinuities along the interface correspond

to stable planar shock waves.

The purpose of this chapter is to explain how all this works. Before going into

technical details, we can describe roughly the methodology. The general problem

is a hyperbolic Cauchy problem with initial data discontinuous across a given

hypersurface, and solutions are sought in a class of functions that are smooth

on either side of a moving, unknown hypersurface. Thus various diﬃculties are

involved:

several space dimensions,

non-linearity,

free boundary.

The latter can be overcome in a standard way by ﬁxing the free boundary

through a change of variables (even though there is some arbitrariness in the

choice of this change of variables). If, for instance, the unknown boundary

stays close to a hyperplane, the free boundary problem (FBP) is easily changed

into a mixed problem, or Initial Boundary Value Problem (IBVP) in a half-

space. For clarity, we shall present most of the analysis in that framework, and

come only at the end to shock fronts that are (smooth) compact manifolds (as

in Majda’s memoir [124]). The main novelty compared to Chapter 11 is that

we have to deal with non-standard IBVP, in which the boundary conditions

(coming from the Rankine–Hugoniot jump conditions across the front) are of

diﬀerential type in the (unknown) front location. In fact, the front location can

be eliminated and we get pseudo-diﬀerential boundary conditions in the main

dependent variables, which can be treated almost as standard ones thanks to

symbolic calculus. So the main point is the understanding of the elimination

step.

The non-linearity of equations of course also plays an important role. In

particular, it is in turn responsible for the smallness of the time of existence – as

already seen for the Cauchy problem (Chapter 10) and for the regular IBVP

(Chapter 11). However, the most important part of the job is in fact linear,the

main problem being to deal with linear equations with coeﬃcients of limited

regularity: once the well-posedness of linearized problems about approximate

solutions is proved, the solution of the non-linear problem is obtained (unsur-

prisingly) through a suitable iterative scheme – ‘a straightforward adaptation of

the standard proof of short-time existence for smooth solutions of the Cauchy

problem’, Majda stated [124].

From FBP to IBVP 331

As regards the several space dimensions, they can be tackled as in standard

IBVP, by using a Fourier transform in the direction of the boundary, as far as the

linearized, frozen coeﬃcients problem is concerned: the resulting normal modes

analysis is rather similar to what is done for standard IBVP; see Section 12.2

below.

12.1 From FBP to IBVP

12.1.1 The non-linear problem

Consider a system of conservation laws

∂

(u)+



j=1

∂

(u)=0, (12.1.1)

and the associated Rankine–Hugoniot condition

(u)] +



j=1

(u)]=0. (12.1.2)

A very general problem is the following.

(FBP) Find a codimension one surface Σ in R

× [0,T], splitting (R

[0,T])\Σ in two connected component Ω

−

and Ω

, and ﬁnd u such

that u

|Ω

∈ C

(Ω

) satisfy (12.1.1) in Ω

and (12.1.2) – plus some

admissibility condition, to be speciﬁed later – across Σ, with N

, N

..., N

being the components in the directions t, x

,..., x

of a vector

N orthogonal to Σ, and the brackets [·] stand for jumps:

(u)](x, t) = lim

ε0

( f

(u((x, t)+εN(x, t))) −f

(u((x, t) − εN (x, t)))),

(x, t) ∈ Σ .

We readily see that a necessary condition for having a solution to (FBP) is that

the vector



j=1

|t=0

)],

where n

,...,n

are the components of a normal vector to the initial shock front

|t=0

, is parallel to [f

|t=0

)]. This is the ﬁrst, natural compatibility condition.

There are many (almost) trivial solutions to (FBP), which correspond to

planar shock waves, with Σ a ﬁxed hyperplane – corresponding to the propagation

at constant speed of a hyperplane in the physical space – and u

|Ω

≡ u

independent of t and x. The derivation of planar shock waves is mostly algebraic.

Indeed, choose a direction of propagation, for instance x

, and consider the so-

called Hugoniot set passing through a reference state w ∈ U (the domain in R

332 Persistence of multidimensional shocks

where the ﬂux functions f

are well-deﬁned and smooth):

H (w):={u ∈ U ; f

(u) − f

(w)  f

(u) − f

(w) }.

(This set is studied in detail in Chapter 13 for the Euler equations of ‘real’

gas dynamics.) Assuming, for instance, the strict hyperbolicity of (12.1.1),

we can easily show that H (w) is locally the union of n curves. Take u

on any of these curves and u

−

= w. By deﬁnition of H (w) there exists

σ ∈ R such that f

) − f

−

)=σ ( f

) − f

−

) ). Then take Σ :=

{(x

,...,x

,t); x

= σt} and you get a planar ‘shock’ wave propagating at

speed σ in the direction x

. At this stage, the ‘shock’ might be a contact

discontinuity or any other kind of discontinuous wave. (We postpone, on purpose,

the discussion of the admissibility of discontinuities.)

It is much more diﬃcult in general to ﬁnd solutions to (FBP) with non-

planar Σ. The aim of this chapter is to show that such solutions do exist,

for T small enough, provided we choose initial data satisfying: 1) compatibility

conditions and 2) stability conditions. The compatibility conditions imply, in

particular, that initially, at each point of the front, the states on either side of

the front are connected by a planar shock wave in the direction normal to the

front. The stability conditions require additionally that this shock be uniformly

stable (in the sense of the uniform Kreiss–Lopatinski˘ı condition) with respect to

multidimensional perturbations. We shall make this more precise below.

12.1.2 Fixing the boundary

For clarity, in what follows we seek solutions close to a planar reference shock.

The case of arbitrary (compact) fronts of discontinuities (actually dealt with by

Majda in [124]) is postponed to the end of this chapter.

Provided that the system (12.1.1) is invariant under rotation – again this is

the case for the Euler equations of gas dynamics, for instance – we may choose

co-ordinates, without loss of generality such that the direction of propagation of

the reference shock is the last co-ordinate x

. Then this reference shock can be

represented by a mapping

:(R

× R

)\Σ → R

(x, t) → u

if x

≷ σt,

with Σ

:= {(x

,...,x

,t); x

= σt},andf

) − f

−

)=σ ( f

) −

−

) ). We look for a perturbed shock

u :(R

× R

)\Σ → R

(x, t) → u(x, t)=u

(x, t)ifx

≷ χ(x

,...,x

d−1

,t) ,

with Σ := {(x

,...,x

,t); x

= χ(x

,...,x

d−1

,t)} the perturbed shock front

(supposedly close to Σ

), the unknowns u

and χ being such that both the interior

equations (12.1.1) and the jump conditions (12.1.2) are satisﬁed.

From FBP to IBVP 333

It will turn out that u exists at least on a small ﬁnite time interval

[0,T],

if the reference shock is uniformly stable (in a sense to be speciﬁed later). For

the proof, and even for the precise statement of this result, we need some more

material.

The set of equations for u

and χ is











∂



j=1

∂

)=0 forx

≷ χ(x

,...,x

d−1

,t) ,

(u)] ∂

χ +

d−1



j=1

(u)] ∂

χ − [f

(u)]=0 atx

= χ(x

,...,x

d−1

,t) .

We introduce our usual shortcut y := (x

,...,x

d−1

) for the independent vari-

ables along the boundary. As long as the function χ stays close – in the class of

functions say – to the reference function (y, t) → σt, both mappings

:(y, x

,t) → ( y, ±(x

− χ(y, t)) ,t)

are diﬀeomorphisms from Ω

= {(y,x

,t); x

≷ χ(y,t)} to the half-space

Ω:={(y, z, t); z>0},

and both Φ

−

and Φ

map Σ to the hyperplane {z =0}.Fromnowon,

we replace both unknowns u

by u

◦ Φ

−1

, which we still denote by u

for

simplicity, and consider the IBVP on Ω obtained by this (unknown) change of

variables for the double-size unknown u =(u

−

Notations Denoting (slightly diﬀerently from Chapter 10)

(w)=df

(w) for all j ∈{0,...,d} and w ∈ U ,

we introduce, for χ :(y, t) → χ(y, t) ∈ R at least diﬀerentiable once,

(w, dχ):=A

(w) −

d−1



j=1

(∂

χ) A

(w) − (∂

χ) A

(w).

This is only a shortcut for

A(w, −∂

χ, −∂

χ,...,−∂

d−1

χ, 1) where A(w, ξ):=



j=0

(w)

denotes the (generalized) characteristic matrix. For convenience, in what follows

we denote indiﬀerently ∂

or ∂

the derivative with respect to t in the (y, z,t)-

variables and for all j ∈{1,...,d− 1}, ∂

now stands for the derivation with

respect to y

in the (y, z, t)-variables, as long as no confusion can occur.

As said before, the short-time existence is due to the (high) non-linearity of the problem.

334 Persistence of multidimensional shocks

In these new variables, the interior equations read

d−1



j=0

) ∂

± A

, dχ) ∂

=0, for z>0 , (12.1.3)

and the boundary conditions are

d−1



j=0

( f

) − f

−

))∂

χ =(f

) − f

−

)) at z =0. (12.1.4)

Observe that the BVP in (12.1.3) and (12.1.4) for the unknown (u

,χ)

involves the unknown funtion dχ both in the interior equations (12.1.3) and

in the boundary conditions (12.1.4).

Remark 12.1 There has been some arbitrariness in the way we have changed

the domains {(y,x

,t); x

≷ χ(y,t) } into a half-space; for a discussion of

alternatives, see [56]. One speciﬁc problem is that the resulting interior equations

in (12.1.3) depend on (y, t) – through the derivatives of χ – for all values of x

.In

other words, the ﬂattening of the boundary inﬂuences the far-ﬁeld behaviour of

equations. As pointed out by M´etivier [136], an alternative consists in localizing

the change of variables around the boundary. (A similar trick was used by Majda

for compact boundaries, see Section 12.4.2.) More precisely, if ϕ ∈ D(R)isa

positive cut-oﬀ function equal to 1 in say [0, 1], one may choose a positive κ,

depending only on ϕ





∞

and χ

∞

d−1

×[0,T ])

so that

:(y, z, t) → ( y, x

= ±κz + ϕ(z) χ(y,t) ,t)

are diﬀeomorphisms from

{(y, z, t); z>0 ,t∈ (−T,T) }

{(y, x

,t); x

≷ χ(y,t) ,t∈ (−T,T) },

and Ψ

both map the hyperplane {z =0} to the front {(y, x

,t); x

χ(y, t) }. This choice does not alter the boundary conditions in (12.1.4), and

the corresponding interior equations (which involve only the derivatives of Ψ

)

are (almost) the same (up to trivial rescaling and symmetry z → x

= ±κx)as

the original ones for z large enough. So this overcomes the problem of the far-

ﬁeld behaviour. The drawback is that interior equations look more complicated:

for this reason we shall perform most of the analysis with the diﬀeomorphisms

−1

insteadofΨ

12.1.3 Linearized problems

Once the non-linear problem is set on a ﬁxed domain Ω, it can reasonably be

linearized – the other way round would not have been possible. As we have