Proceedings of the International Congress of Mathematicians
August 16-24, 1983, Warszawa
ANDREW MAJDA*
Systems of Conservation Laws in Several Space
Variables
We describe some recent progress in the short-time existence of discontinuous solutions for the Oauchy problem for an m x m system of hyperbolic
•conservation laws in w-space variables/
du . v i d
dt
u)
>0
+È-k**
-°>
*
'
i
jL-i
ox*
—
(1)
u(x, 0) = u°(x),
"where ' x = (a?u ..., xJ e Rn, u = t(u1, ..., um) is an m-vector, and the
Fj(u) are smooth nonlinear mappings of Rm to Rm with A5(u) = dFôjdu
the corresponding mxm Jacobian matrices for 1 < j < n. The prototypical
«example of a system of conservation laws is' given by the compressible
Euler equations of fluid dynamics, (where m = n+2)
do
•JL+div(m) = 0 ,
orni
,. ImrnA
dp
_ i + d l v ( _ l ) + _ | = „,
,_!
„
m
expressing conservation of mass, momentum, and total energy. In (2),
Q is the density with 1/g = r the specific volume, t> = '(f^,..., i?n) is the
fluid velocity with gt? = m the momentum vector, p is the scalar pressure,
* Partially supported by N.8.F. Grant
5^483964-20530.
[1217]
^MCS-81-02360 and A.R.O. Grant
1218
Section 11: A. Majda
and E = %(m-m,)lQ +Qe(r,p) is the total energy with e the internal
energy, a given function of (r,p) defined through thermodynamic considerations. Other hyperbolic conservation laws often occur in classical
physics and engineering — in particular, in describing magneto-fluid
dynamics, combusion in certain regimes, shallow water waves, and petroleum reservoir engineering ([1], [10]).
Despite the abundance of concrete problems associated with systems
of hyperbolic conservation laws, the rigorous mathematical theory, especially in several space dimensions (n > 1), is only beginning. Here we
describe the recent theoretical progress regarding discontinuous solutions
in several space variables ([8], [9]), we contrast the new phenomena for
n > 1 with those when n = 1, and also we mention some of the interactionsof this theory with more concrete applied problems ([11], [12]). To emphasize this concrete point of view, we mostly state these results in the context
of the compressible Euler equations in (2). The reader can consult t h e
above papers for the general framework as well as the author's recentlectures ([10]) for a leisurely discussion of many of the topics mentioned
briefly below.
For linear hyperbolic equations, jump discontinuities in solutionsalways follow characteristic hypersurfaces for short enough times before
focusing occurs. The most interesting manifestation of the strong nonlinearity in (1) is t h a t unlike the linear case, jump discontinuities of (1)
do wot typically follow characteristic surfaces. To motivate this, we comment
that a piecewise smooth weak solution of (1) which is smooth except for
a jump across the space-time hypersurface S(t) with respective sides G+r
GL and space-time normal (nt, n19..., nn) necessarily satisfies the quasilinear equations
du.
v~i
du,
M
^f+i> ±>äf= o
(3
>
in the respective smooth regions G± for u as well as the nonlinear boundary
conditions across the hypersurface, 8, given by
1Ä
ZA
l
where [ ] denotes the jump across 8, i.e., [u]\s = (% — tt-)U- * n general
the highly nonlinear boundary conditions in (4) force the weak solution
to jump across noncharacteristic surfaces, 8(t)9 called shock fronts (see [1]
Systems ol Conservation Laws in Several Space Variables
1219
for plane wave solutions of (2) and (4) for the elementary theory of genuinely nonlinear plane waves where 8(t) is always noncharacteristic).
Hext, we describe initial data for (1) where intuitively one would
expect that a shock front solution of (1) with the qualitative structure
described in (3), (4) would be generated by this initial data for sufficiently
ßhort times. We take discontinuous initial data so that there is a smooth
initial hypersurface, M, parametrized by a, with two sides ß + , ß _ so
that
u°(x) =
U°JL(X)
for
a? in Q, ,
u°__(x) for
x in ß__,
where u°± are smooth functions. Given the qualitative structure in (4)
anticipated for all small times t > 0, it is natural that we also require
for this initial data that there is a scalar function, a(a), so that
-cf(a)[nQ'} + ^nj[Fj(u0ma)=:0
(6)
for all as M where ri *= (n19..., nn) is the normal to M. Initial data
satisfying (5) and (6) are called shoclc front initial data. What conditions
are needed in several space variables to guarantee the existence and structural stability of solutions satisfying (3) and (4) with the initial data from
(5), (6)? We state the least technical version of the main theorem in [9]
specialized to the compressible Euler equations in (2). Before doing this,
we introduce three important physical parameters associated with the
shock front initial data for (2).
The normal Mach numbers,
\v, •% — a\
M±(a)=±-±1(a)
with 0 the speed of sound,
The compression ratio, a (a) = \-—\ (a).
The Gruneisen coefficient, r_, measuring the equation of state,
r_
=[Q-ep(r_,p_))-l>0.
We have
THEOEEM. Assume that the shoclc front initial data for the compressible
Euler equations belong to the Sobolev space, HS(Q±), s > [w/2] + 7 and
satisfy (6) as well as the related compatibility conditions up to order s—1
1220
Section 11: A. Majda
(see [9]). Asume that the normal Mach numbers satisfy
M2+ (a)>l>
Ml (a)
for all a e M
(A)
and for n^2 that the Gruneisen coefficient, compression ratio, and normal
Mach numbers also satisfy
.
*
(ft(a)-l)Ml(a)<lir(a)+l
(B>
for all ae M. Then for sufficiently short times, there is a G2 hypersurface
8(t) and G1 functions u± defined on the respective sides of this hypersurface
satisfying (3), (4), and defining a shock front solution of (2) with the given
initial data. Furthermore, any compressive shock front initial data (i.e.
[K,(a) > 1) for idealpolytropic gases where e = px(r —-I)-1, r > 1 automatically
satisfies (A), (B) and therefore has a shock-front solution.
We remark here that for n = 1, under assumption (A), sharper results.
are known and a complete theory of the perturbed Eiemann problem hasbeen developed in [6], [7]. Also, with the above rigorous theorem, some
of the formal calculations in [13] can be justified. The condition in (A)
is very natural and corresponds to Lax's geometric entropy conditionsin the general case ([14]). The additional condition in (B) for n > 2 mighti
seem at the moment to be a technical restriction; however, the evidence,,
both physical and mathematical, is overwhelming that when (B) is violated for n > 2, more complex inherently multi-D wave patterns occur
rather than shock fronts ([11], [12]). I n [5], [15], interesting general
geometric entropy conditions always implying (A) are developed. A natural
question arises: Does every jump discontinuity for (2) satisfying these
general entropy conditions automatically satisfy the inequalities in (B)?
The answer is no and the corresponding examples are constructed in [8] y
[10].
The shock front problem described in (3), (4) can be viewed as a highly
nonlinear free boundary value problem for a quasi-linear hyperbolic
system since 8(t) is non-characteristic and must be determined as part
of the solution of the problem. There are three main steps in the proof
of the above theorem.
(I) Map to a fixed domain.
(II) Linearization of perturbed shock fronts.
(Ill) Construction of the shock front solution via a classical iteration
scheme.
Without giving any details, we illustrate some of the main points of
this proof in the very special case of perturbed steady planar shock fronts
Systems of Conservation Laws in Several Space Variables
1221
in two space variables. We assume the shock front initial data has the
special form
W\+v\(x,y), x>0,
* > , » ) - | .\ul+v»_(x,y),
o
x<0,
where v± eG™(R2) and 2Pi(u\) = F^ul). For small positive times, the
anticipated shock front emanating from x = 0 should have the form of
a graph, i.e., 8(t) can be described by the equation x = <p(y,t). We carry
out step (I) in this special case by mapping the unknown shock surface
x = cp onto x = 0. Thus, to construct the shock front solution, we need
to find functions u±(x,y,t), <p(y,t) in a new co-ordinate system still
denoted by (x,y,t) satisfying the interior equations,
8u
±- + ( i , ( « i ) - R i - M i M - ^ + ^ ( % ) - ~ = o
for
u±(x,y90)
= < +<,
; j j , * > 0 , (8)
?(y,0) - 0
and the nonlinear boundary conditions,
nW+cpylF^uft-m
=0 on x =0,
J>0.
(9)
Step (II) in the outline of the proof involves linearizing (8), (9) at a typical
varying perturbed state and analyzing the associated linear problem.
At the special unperturbed state u± ES u°±, cp s= 0, the linearized problem
for the unknowns (5 ± , cp) becomes the constant coefficient boundary
value problem,
aS
d
^
± +. ^A(/„o
A )ufi
^ JF < )x- i*>±
+ J . 2 ,( <
- f^- =
:t
-for
œ
> ° *>0,
ÄM+^Kl^l^iKlv^i^)5- = *
( 10 )
for x = 0, J > 0
together with appropriate initial conditions. The boundary conditions
in (10) should be regarded as an over-determined evolution equation for
the perturbed shock front boundary, q>, coupled to the boundary values
of solutions of hyperbolic equations. A general variable coefficient theory
for the problems in (10) is developed in [8]. What estimates define the
well-posedness for the mixed problem in (10)? Looking back at the full
1222
Section 11; A. Majda
nonlinear problem in (8), (9), we see that it is crucial to gain a derivative
of cp beyond the regularity of v± to avoid "loss of derivatives" in the nonlinear iteration scheme. Such shock fronts with an associated linearized
problem allowing for both this gain in regularity and also a well-posed
interior mixed problem are called uniformly stable in [8] and admit an
algebraic characterization analogous to the uniform Lopatinski condition
for standard mixed problems ([3], [14]). Beturning to the physical example
of linearized shock fronts for the compressible Euler equations, we have
the following facts:
For the compressible Euler equations
I. Shock fronts are uniformly stable for n = l iff (A) from the theorem
is satisfied.
II. Assuming (A) shock fronts are uniformly stable for n^2 iff (B)
from the theorem is satisfied.
I I I . When the inequality
PROPOSITION.
(lL-l)M2__>l+M_ir_
is satisfied, shock fronts are violently unstable for w > 2 (see [8], [11]).
TV. In the transition regime between the inequalities from (B) of the
theorem and I I I , i.e., when
-~I<
(A*-i)Äi <
i+Mjr_
causal radiating boundary wave solutions (v+9 ï_, <p)for (10) exist (see [11]).
In [11], [12], the special linearized solutions mentioned in IV are used
as the starting point of a formal asymptotic expansion which also incorporates nonlinear effects and leads to a theory which predicts the experimentally observed formation of Mach stems in reacting shock fronts —
thus, the theorem on stable shock fronts requiring condition (B) for n > 2
is sharp. As regards.the general linear problem from (10) in multi-D for
the general system (1), we have the following fact:
PROPOSITION. A necessary condition for any shock front for a system
of conservation laws in Rn to be uniformly stable is that the number of equations9
m, satisfies
m^n.
In particular, in contrast to the case of a single space variable, shock
fronts for the scalar conservation law in two space variables,
du
d
,
d
Systems of Conservation Laws in Several Space Variables
1223
are less stable than those for polytropic gases in 2-D (see the extended discussion in [10]).
The final main step in the proof of the theorem is the convergence of
a nonlinear iteration scheme based on the linearized problems analyzed
in part (B). Here the strategy follows that used in the oauchy problem
( [2], [10]) but the technical details are more complex due to both the strong
nonlinearity in the boundary conditions and also to the use of square
integrable weighted norms in space-time as opposed to maximum norms
in time where the linearized problem is well-posed. It is desirable to
find a simpler and sharper proof of the convergence than the one given
in [9].1
The results described here are the only rigorous ones known to the
author regarding discontinuous solutions of Multi-D conservation laws.
Obviously, this is a field in its mathematical infancy and a large number
of very interesting open problems remain. The author hopes that this
lecture stimulates the interest of other mathematicians in this important
subject.
Bibliography
[1] Courant R. and Friedrichs K. 0., Supersonic Flow and Shoclc Waves, Wiley Interscience, New York, 1949.
[2] Kato T., The Cauchy problem for quasi-linear symmetric systems, Arch. Bat.
Mech. Anal. 58 (1975), pp. 181-205.
[3] Kriess H. 0., Initial boundary value problems for hyperbolic systems, Oomm.
Pure Appi. Math. 23 (1970), pp. 277-298.
[4] Lax P . D . , Hyperbolic Systems of Conservation Laws and the Mathematical Theory
of Shook Waves, S.I.A.M. Regional Conf. Ser. Appi. Math.#ll
(1973), Philadelphia.
[5] Liu T. P., The Riemann problem for general systems of conservation laws,
J. Diff. Equations 18 (1975), pp. 218-234.
[6] Li Da-qian and Yu Wen-ci, Some existence theorems for quasi-linear hyperbolic
systems of partial differential equations in two independent variables, I I , Scientia
Sinica 13 (1964), pp. 551-564.
[7] Li Da-qian and Yu Wen-ci, The local solvability of boundary value problems
for quasilinear hyperbolic systems, Sdenta Sinica 23 (1980), p p . 1357-1367.
[8] Majda A., The stability of multi-dimensional shock fronts, Memoirs
A.M.S.
# 2 7 5 , January 1983.
[9] Majda A., The existence of multi-dimensional shock fronts, Memoirs
A.MS.
# 2 8 1 May 1983.
1
A quite different but much simpler proof for the shock-front theorem for
2nd order wave equations has been given in [16],
25 — Proceedings..., t, II
1224
Section 11: A. Majda
[10] Majda A., 0.1. M. E. Lectures on Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (to appear in Springer-Verlag Applied
Math. Science Series).
[11] Majda A. and Rosales R., A theory for spontaneous Mach stem formation in
reacting shock fronts, I : the basic perturbation analysis, (to appear in S.I.A.M.
J. Appi. Math. 43 (1983), p p . 1310-1334.
[12] Majda A. and Rosales R., A theory for spontaneous Mach stem formation in
reacting shock fronts, I I : the evidence for breakdown, (to appear in Studies in
Appi. Math in 1983).
[13] Maslov V. P . , Propagation of shock waves in an isentropic non viscous gas,
J. Sov. Math. 13 (1980), p p . 119-163.
[14] Sakamoto R., Mixed problems for hyperbolic equations, I, I I , J. Math. Kyoto
Univ. 10 (1970), p p . 349-373 and p p . 403-417.
[15] Wendorff B., The Riemann problem for materials with nonconvex equations of
state I I . General flow, J. Math. Anal. Appi. 38 (1972), p p . 649-658.
Added in proof:
[16] Majda A. and Thomann E., Multidimensional
wave equations (to appear),
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF CALIFORNIA
BERKELEY, OA 94720 USA
shock fronts for second order