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

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Resolution of non-linear IBVP 375

Our main purpose is to study the persistence of the front of discontinuity

under (short) time evolution, as in Majda’s second memoir [124]. In other words,

we want to ﬁnd a solution to (FBP) with u

|t=0

= u

and Σ

|t=0

=Σ

. This will

indeed be possible under some further (higher-order) compatibility conditions

and the uniform stability of initial discontinuities. The meaning of the latter

should be clear to the reader from the previous sections. However, the statement

of higher-order compatibility conditions requires some more work, and is (as

usual) quite technical.

Fixing the boundary

A preliminary step is (as in Section 12.1.2 ) to make a change of variables to

ﬁx the unknown front. More precisely, as in Remark 12.1, we can make (locally

in time) a change of variables that does not inﬂuence the far-ﬁeld equations

and sends all front locations Σ(t) to the ﬁxed (initial) surface Σ

.Thiscanbe

done explicitly by considering tubular neighbourhoods of Σ

, X

⊂ X

say, such that for all x ∈ X

there exists a unique (y,θ) ∈ Σ

× R such that

x = y + θn(y), and a cut-oﬀ function ϕ that is equal to one on X

and zero

outside X

.SinceΣ(t) is expected to remain in X

for t small enough, it may

be described as

Σ(t)={x = y + χ(y,t) n(y); y ∈ Σ

for some unknown function (‘the front location’) χ.NowforT small enough, we

can deﬁne the mapping

Φ: R

× (−T,T) → R

× (−T,T)

(x, t) → ( x := x − ρ(x) χ(Y (x),t) n(Y (x)) ,



t := t) ,

where for x ∈ X

, Y (x) ∈ Σ

is uniquely determined by the requirement that

x − Y (x) be parallel to n(Y (x)), and Y (x) is extended arbitrarily for x outside

:sinceρ is identically zero near the boundary of X

, Φ inherits in the whole

space the regularity of Y in X

. By construction Φ maps every subset of the

form Σ(t) ×{t} onto Σ

×{t}, up to diminishing T so that Σ(t) lies in fact in

for all t ∈ (−T,T). Furthermore, Φ is a global diﬀeomorphism provided that

sup{|χ(y,t)| + d

χ(y, t); y ∈ Σ

,t∈ (−T,T)}

is small enough, which is indeed what we expect for T small enough.

Modiﬁed bulk equations

The inverse mapping Φ

−1

is of the form (x,



t ) → (x = ψ(x,



t ),t =



t ). (We shall

give more details on ψ later.)

Therefore, it is a calculus exercise to check that the quasilinear system

(u) ∂

u +



j=1

(u) ∂

u =0

376 Persistence of multidimensional shocks

is satisﬁed by u(x, t)=u(x,



t ) outside Σ(t) if and only if, outside Σ

(u) ∂



u +



j=1



(x,



t, u) ∂



u =0,

with



(x,



t, u)=A

(u) −



k=1

j,k

(x,



t ) A

(u) − c

j,0

(x,



t ) A

(u),

where, for j, k ∈{1,...,d},

j,k

(x,



t )=∂

(ρ(x)χ(Y (x),



t )n

(Y (x)))

|x=ψ(x,t )

j,0

(x,



t )=(ρ(x) ∂

χ(Y (x),



t )n

(Y (x)))

|x=ψ(x,t )

What is hidden in these formulae is that ψ, as Φ and thus also Φ

−1

depends on

the unknown front χ. In fact, we can make this dependence more explicit.

Assume indeed that the tubular neighbourhoods X

, i =0, 1, 2 are given by

= {x = y + θn(y); y ∈ Σ

,θ∈ (−ε

,ε

)},

for 0 <ε

<ε

= ε

+ ε

, and that

sup{|χ(y,t)|; y ∈ Σ

,t∈ (−T,T)}≤ε

With the mapping Y introduced above, and Θ deﬁned on X

Θ(x)=(x − Y (x)) · n(Y (x)),

(in such a way that x = Y (x)+Θ(x) n(Y (x))), we may deﬁne the cut-oﬀ ρ by

ρ(x)=ϕ(Θ(x)), where ϕ is a C

∞

cut-oﬀ function R → [0, 1] being equal to one

on (−ε

,ε

) and equal to zero outside (−ε

,ε

). For x outside X

, x = x,and

for x ∈ X

, x belongs to X

and by construction Y (x)=Y (x). Therefore, to

ﬁnd a representation for the mapping ψ,itjustremainstolookatΘ(x)interms

of x. By deﬁnition, for x ∈ X

x = Y (x)+(Θ(x) − ϕ(Θ(x)) χ(Y (x),t))n(Y (x)),

hence (using the fact that t =



t and Y (x)=Y (x))

Θ(x)=Θ(x) − ϕ(Θ(x)) χ(Y (x),



t )).

Now, if ε

< 1and|z| <ε

, the function θ → θ − ϕ(θ) z is an increasing dif-

feomorphism from R to R. Therefore, by the implicit function theorem and

a connectedness argument, there exists a C

∞

mapping g : R × (−ε

,ε

) → R

such that, for all (



θ, θ, z) ∈ R × R × (−ε

,ε

θ − ϕ(θ) z =



θ ⇔ θ = g(



θ, z).

Resolution of non-linear IBVP 377

Consequently, for x ∈ X

Θ(x)=g(Θ(x),χ(Y (x),



t ))

and therefore

ψ(x,



t )=Y (x)+g(Θ(x),χ(Y (x),



t )) n(Y (x)).

Denoting

χ : R

× (−T,T) → R

(x,



t ) → χ(x,



t ):=χ(Y (x),



t ) ,

the discussion above shows that the coeﬃcients c

j,k

are in fact of the form

j,k

= c

j,k

(x,



t, χ(x,



t ), dχ(x,



t ))

for some (complicated) non-linear mappings

c

j,k

: R

× (−T,T) × R × (R

d+1

)



→ R

independent of the unknown front χ. So ﬁnally, if we assume additionally (as

usual) that A

(u) is everywhere non-singular, the bulk equations in the new

variables read,

∂



u +



j=1

(x,



t, u, χ, dχ) ∂



u =0, (x,



t ) ∈ (R

\Σ

) × (−T,T) ,

(12.4.50)

with

(x,



t, u, χ, dχ):=A

(u)

−1



(u) −



k=0

c

j,k

(x,



t, χ(x,



t), dχ(x,



t)) A

(u)



Before dropping the tildas deﬁnitively, we have to write also the jump

conditions in the new variables.

Jump equations

For simplicity, let us rewrite the Rankine–Hugoniot condition (12.1.2) as

(u)] + N

[f(u)] = 0,

where N

stands for the row vector of components (N

,...,N

)andf(u) stands

for the d × n matrix whose column vectors are f

(u),...,f

(u). By assumption,

this condition is satisﬁed by u = u

at any point y ∈ Σ

, with N

= −σ(y)and

N = n(y). The N

are of course to be modiﬁed when the discontinuity surface

moves along

Σ(t)={x = y + χ(y,t) n(y); y ∈ Σ

378 Persistence of multidimensional shocks

To be more precise, let us denote by W the Weingarten map on Σ

, characterized

W (y

) · h =

(n(y(s)))

|s=0

for s → y(s)anycurveonΣ

such that y(0) = y

and y



(0) = h,andby

∇

χ(y, t) the tangential gradient (along Σ

) of the mapping y → χ(y, t): by

deﬁnition, for all y ∈ Σ

, ∇

χ(y, t) is in the tangent space T

, viewed as a

subspace of R

, such that the linear form d

χ(y, t)onT

is given by

χ(y, t) · h = ∇

χ(y, t),h

for all h ∈ T

,where,  stands for the standard inner product in R

(which

gives the ﬁrst fundamental form on T

). Then it is a standard diﬀerential

geometry exercise to show (using the symmetry of the Weingarten map with

respect to the ﬁrst fundamental form) that the normal vector to Σ = ∪

Σ(t) × t

at point (y + χ(y,t) n(y),t) is parallel to



n(y) − (1 + χ(y,t) W (y))

−1

∇

χ(y, t)

−∂

χ(y, t)



Therefore, the Rankine-Hugoniot condition (12.1.2) equivalently reads

−∂

χ(y, t)[f

(u)](y, t)+



n(y) −(1 + χ(y,t) W (y))

−1

∇

χ(y, t)



[f(u)](y, t)=0

for all (y, t) ∈ Σ

× (−T,T). Noting that by construction, χ(x,



t )=χ(x,



t )for

x ∈ Σ

, we can rewrite this jump condition in the new co-ordinates as



n(x) − (1 + χ(x,



t ) W (x))

−1

∇

χ(x,



t )



[f(u)](x,



t )

= ∂



χ(x,



t )[f

(u)](x,



t ) , (x,



t ) ∈ Σ

× (−T,T) .

(12.4.51)

Our original problem is clearly equivalent to ﬁnding (u, χ) solving (12.4.50)–

(12.4.51) with the initial data u(x, 0) = u

(x), χ(x, 0) = 0.

From now on, we drop all tildas.

Higher-order compatibility conditions

Our next task is to ﬁnd higher-order compatibility conditions between the initial

maps u

and σ. We recall that ‘zeroth-order’ compatibility conditions are

−σ ( f

) − f

−

)) + n(x)

( f(u

) − f

−

) ) = 0 on Σ

(12.4.52)

As usual, higher-order compatibility conditions will be obtained by diﬀerentiating

successively the bulk equations (12.4.50) and the jump conditions (12.4.51),

and by equating the results on Σ

×{0}. For convenience we rewrite all these

equations without the tilda, omitting a number of obvious independent variables

Resolution of non-linear IBVP 379

but introducing the distinction between u

−

the restriction of u to the inside D

−

of Σ

and u

the restriction of u to the outside D

of Σ











∂



j=1

(x, t, u

,χ,dχ) ∂

=0,x∈ D



n − (1 + χW)

−1

∇



[f(u)] = (∂

χ)[f

(u)] on Σ

(12.4.53)

We shall not write down explicitly all compatibility conditions, which are

quite terrible (see [124], pp. 24–25 for some details). We concentrate on the ﬁrst-

order ones. Assume that (u

,χ) is a solution of (12.4.53) with u

of class C

× [0,T)andχ of class C

on R

× [0,T), with u

|t=0

= u

and ∂

|Σ

×{0}

= σ

satisfying the zeroth-order compatibility condition in (12.4.52). In addition, we

assume that for all y ∈ Σ

, the discontinuity between u

−

(y)andu

(y) is a 1D-

stable Lax shock (of speed σ(y)) in the direction n(y). We are going to exhibit

(n − 1) independent equations on u

:= ∂

|t=0

and χ

:= ∂

|t=0

valid on

×{0}.

On the one hand, in view of the deﬁnition of A

, the ﬁrst equation in (12.4.53)

imposes that

(x)=−



j=1



(x))

−1

(x)) − σ(x) n

(x)



∂

(x) (12.4.54)

for x ∈ Σ

. On the other hand, diﬀerentiating once with respect to t the second

equation in (12.4.53) and evaluating at t =0,weget

−(∂

χ(x, 0))



(x)) − f

(x))





A(u

(x),n(x)) − σ(x)A

(x))



∂

(x, 0)

−



A(u

−

(x),n(x)) − σ(x)A

−

(x))



∂

−

(x, 0)

=(∇

(x, 0))



f(u

(x)) − f (u

(x))

(12.4.55)

for x ∈ Σ

, where we have used our usual notation A(u, n)=



j=1

(u).

Now, the 1D-stability of the Lax shock (u

−

(x),u

(x),σ(x)) can be used in the

following way: introducing P

(x) the eigenprojector onto the unstable subspace

(x)ofA(u

(x),n(x)) − σ(x)A

(x)) and P

−

(x) the eigenprojector onto

380 Persistence of multidimensional shocks

the stable subspace E

−

(x)ofA(u

−

(x),n(x)) − σ(x)A

−

(x)), the mapping

L(x): R × E

(x) × E

−

(x) → R

(β,v

−

) →−β



(x)) − f

(x))





A(u

(x),n(x)) − σ(x)A

(x))



−



A(u

−

(x),n(x)) − σ(x)A

−

(x))



−

is an isomorphism. Therefore, if we denote by B(x), V

(x), V

−

(x) the compo-

nents of the inverse mapping L(x)

−1

, (12.4.55) equivalently reads











∂

χ(x, 0) = B(x)(q(x)) ,

(x)(u

(x)) = V

(x)(q(x)) ,

−

(x)(u

−

(x)) = V

−

(x)(q(x)) ,

(12.4.56)

where

q(x):= (∇

(x, 0))



f(u

(x)) − f (u

(x))

−



A(u

(x),n(x)) − σ(x)A

(x))



(I − P

(x))(u

(x))



A(u

−

(x),n(x)) − σ(x)A

−

(x))



(I − P

−

(x))(u

−

(x)) .

The ﬁrst equation in (12.4.56) is a ‘bonus’ here (it would serve to deﬁne χ

we were to derive second-order compatibility conditions). The others make the

announced n − p + p − 1=n − 1 ﬁrst-order compatibility conditions.

More generally, the compatibility conditions to order s (as in (CC

))consist

the deﬁnition (by induction) of a sequence (u

,χ

), p ∈{1, ···,s}, depend-

ing only on u

, σ and their derivatives,

for each p ∈{1, ···,s},(n − 1) equations on u

and ∇

in terms of u



and χ



,  ∈{1, ···,p},onΣ

Main persistence result

The following is a slight adaptation of Majda’s theorem ([124], p. 8), where the

‘block-structure condition’ has been replaced by the more explicit constant hyper-

bolicity assumption: indeed, as shown by M´etivier [134], constant hyperbolicity

and Friedrichs symmetrizability together imply the block structure condition of

Majda. The Sobolev index m is supposed to be a large enough integer: Majda

assumed it to be at least equal to 2[d/2] + 7; the work of M´etivier suggests that

m>(d +1)/2 + 1 is suﬃcient. Finally, the initial data is assumed for simplicity

Resolution of non-linear IBVP 381

to vanish at inﬁnity, but any data asymptotically constant at inﬁnity would also

work (as stated by Majda in [124]).

Theorem 12.11 (Majda) Structural assumptions are the following.

(CH) There exist open subsets of R

,sayU

−

and U

0 such that for all

w ∈ U

the matrix A

(w) is non-singular, and the operator A

(w) ∂



j=1

(w) ∂

is constantly hyperbolic in the t-direction;

(S) The operator A

(w) ∂



j=1

(w) ∂

is Friedrichs symmetrizable in U

We assume that Σ

is C

∞

compact manifold in R

and denote by D

−

its inside

and D

its outside. We consider u

∈ H

m+1

), taking values in U

,and

σ ∈ H

m+1

(Σ

) satisfying

all compatibility conditions up to order m − 1, including (12.4.52),

for all x ∈ Σ

, the planar discontinuity between u

−

(x) and u

(x), of speed

σ(x) in the direction n(x), is a uniformly stable Lax shock in the sense of

the Kreiss–Lopatinski˘ı condition.

Then there exists T>0, C>0, and a solution (u

−

,χ) of (12.4.53) such that

= u

and χ =0,∂

|Σ

= σ at t =0,

and u

∈ H

× [0,T]), χ ∈ H

m+1

× [0,T]).

The proof is most technical, but the ideas are basically the same as for

Theorem 12.9:

derivation of an ‘approximate’ solution;

convergence of an iterative scheme, thanks to the analysis of linear IBVP

with zero initial data.

Here, the linear IBVP (as we can guess from the form of (12.4.50)) involve

coeﬃcients depending on (x, t)inaC

∞

manner as well as on (u

(x, t),χ

(x, t)),

(an element of the iterative scheme) of limited regularity. For simplicity in this

book, we have not dealt with IBVP (nor even Cauchy problems) with coeﬃcients

of the form A(x, t, v(x, t)) say, but only either of the form A(x, t)orofthe

form A(v(x, t)). However, coeﬃcients of the form A(x, t, v(x, t)) do not conceal

any additional, fundamental diﬃculty. Another diﬀerence between Theorem 12.9

and Theorem 12.11 is that in the latter we have to deal with IBVP in a (ﬁxed)

compact domain instead of a half-space: as seen in Chapter 9 for standard IBVP,

this is mainly a matter of co-ordinate charts.

Anyway, we shall not produce the complete proof of Theorem 12.11 here.

The reader may refer to [124] (which relies on [125] for the analysis of linearized

problems). In [124] (p. 26–28), Majda also shows, interestingly, how to construct

a large class of compatible initial data. This relies on an important observation

382 Persistence of multidimensional shocks

on the structure of compatibility conditions: as we can see in (12.4.54),

)

|Σ

= −



j=1



)

−1

,n) − σ



∂u

∂n

+ tangential derivatives,

and a similar (though more intricate) fact is true for all the u

, p ≥ 2 (which we

did not deﬁne here, see [124], p. 24).

Remark 12.6 A similar observation on the structure of (CC

) is used by

M´etivier in [136] (Proposition 4.2.4) to prove that the existence result in Theorem

12.9 is valid with slightly less regular initial data, namely with u

− u ∈ H

instead of H

m+1/2

PART IV

APPLICATIONS TO GAS DYNAMICS

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