Global solutions to the compressible Euler equations with
Transcript Of Global solutions to the compressible Euler equations with
Commun. Math Phys 180, 153193 (1996)
Communications ΪΠ
Mathematical Physics
© SpringerVerlag 1996
Global Solutions to the Compressible Euler Equations with Geometrical Structure
GuiQiang Chen1, James Glimm2 1 Department of Mathematics, Northwestern University, Evanston, Illinois 60208, USA Email: [email protected] nwu edu 2 Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, New York 11794, USA. Email: [email protected] sunysb edu
Received: 17 October 1995/Accepted: 30 January 1996
Abstract: We prove the existence of global solutions to the Euler equations of compressible isentropic gas dynamics with geometrical structure, including transonic nozzle flow and spherically symmetric flow. Due to the presence of the geometrical source terms, the existence results themselves are new, especially as they pertain to radial flow in an unbounded region, \x\ ^ 1, and to transonic nozzle flow. Arbitrary data with L°° bounds are allowed in these results. A shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions. The convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework.
Contents
1. Introduction
153
2. Nonlinear Waves and Riemann Solutions
158
3. SteadyState Solutions
162
4. Approximate Solutions
176
5. L°° Estimates
181
6. H~x Compactness Estimates
182
7. Convergence and Consistency
186
8. Transonic Nozzle Flow: Proof of Theorem A
189
9. Spherically Symmetric Flow: Proof of Theorem B
189
References
1. Introduction
We develop new mathematical existence theory and numerical schemes for global discontinuous solutions to the Euler equations of compressible isentropic gas
154
GQ Chen, J Glimm
dynamics with large initial data and with geometrical structure. The compressible Euler equations are of the following conservation form:
= 0,
(L1)
where p, m, and p are the density, the momentum, and the pressure of the gas, respectively. On the nonvacuum state, u = m/p is the velocity. For polytropic gas, p(p) = py/y, where the adiabatic exponent γ is restricted to the interval (1, 5/3], as is usual for gases.
Our results are new for the problem of spherically symmetric flow in an unbounded domain (\x\ ^ 1) and for transonic nozzle flow: existence of global discontinuous solutions for general initial data in L°°. The central difficulty in the unbounded domain is the reflection of waves from infinity and their strengthening as they move radially inward. The central difficulty for the transonic nozzle flow lies in the associated steadystate equations, which change type from elliptic to hyperbolic at the sonic point; such steady state solutions are fundamental building blocks in our approach and in early work on nozzle flow.
The Cauchy problem models transonic nozzle flow through a variablearea duct (cf. [EM, GL, GM, L1L3, CF, Wh]). In [LI] the existence of global solutions of this problem was obtained by first incorporating the steadystate building blocks with the random choice method [Gl], provided that the initial data have small total variation and are bounded away from both sonic and vacuum states. A generalized random choice method was introduced to compute transient gas flows in a Laval nozzle in [GL, GM]. A mathematical analysis of the qualitative behavior of nonlinear waves for nozzle flow was given in [L3]. In this paper we introduce a Godunov shock capturing scheme to obtain L°° estimates and compensated compactness of the corresponding approximate solutions. Our method incorporates natural building blocks from Riemann solutions and steadystate solutions. Such estimates lead to the convergence of the approximate solutions and to an existence theory of global weak entropy solutions for measurable initial data inZ°°.
The initialboundary problem is motivated by many important physical problems such as flow in a jet engine inlet manifold and stellar dynamics including supernovae formation. A global weak entropy solution with spherical symmetry was constructed in [MU] for the isothermal case γ = 1 and the local existence of such a weak solution for the general case 1 < y ^ 5/3 was also discussed in [MT]. A theorem has also been established for the general case in [C3] to ensure the existence of L°° spherically symmetric weak solutions in the large for a class of L°° Cauchy data of arbitrarily large amplitude, which model outgoing blast waves and largetime asymptotic solutions.
In Sects. 24 we develop a first order Godunov shock capturing scheme, with piecewise constant building blocks replaced by piecewise steady ones. The main point is to use the steadystate solutions, which incorporate geometrical source terms, to modify the wave strengths in the Riemann solutions. This construction yields better approximate solutions, and permits uniform L°° bounds. There are two technical difficulties which we overcome to achieve this goal, both due to transonic phenomena. One is that no smooth steadystate solution exists in each cell in general. This problem is easily solved by introducing a standing shock at the center
Global Solutions to Compressible Euler Equations with Geometrical Structure
155
of the cell, as discussed in Sect. 2. The other is that the constructed steadystate solution in each cell must satisfy the following requirements:
(a) The oscillation of the steadystate solution around the Godunov value must be of the same order as the cell length to obtain the L°° estimate for the convergence arguments;
(b) The difference between the average of the steadystate solution over each cell and the Godunov value must be higher than first order in the cell length to ensure the consistency of the corresponding approximate solutions with the Euler equations.
These requirements are satisfied by smooth steadystate solutions bounded away from the sonic state in the cell. The general case must include the transonic steadystate solutions. The sonic difficulty is overcome, as in experimental physics, by introducing an additional standing shock with continuous mass and by adjusting its left state and right state in the density and its location to control the growth of the density. These requirements also enable us to make H~ι compactness estimates for corresponding entropy dissipation measures to deduce the strong convergence of the approximate solutions with the aid of the compactness framework (see [Cl, C2]).
We rewrite (1.1) with geometrical structure as
( pt + mx= a(x)m ,
or in a compact form:
vt + f(υ)x = a(x)g(v),
(1.3)
where m is the momentum of the gas, a(x) is a C2 function in the region of x = x under consideration, v = ( p , m ) τ , f(v) = (m,m2/p + p(p))Ύ, and g(v) = (m,m2/p)τ. The function a(x) can be represented by
a(χ) = A'(x)/A(x) with A(x) = eS* a{y)dy .
(1.4)
The function A(x) represents the crosssectional area at x in a variablearea duct for transonic nozzle flow, A(x) — 2πx for cylindrically symmetric flow, and A(x) = 4πx2 for spherically symmetric flow. For cylindrical and spherical flow, we impose reflecting boundary conditions at x = 1 to exclude the singularity at the origin, but, as a principal new result of this paper, we are able to handle successfully the difficulties at infinity.
We consider the Cauchy problem:
v\t=o = vo(x),
(1.5)
and the initialboundary value problem:
{ϊ,vf
with the initial data VQ(X) € L°°. A pair of mappings (η,q) : R2 —> R2 is called an entropyentropy flux pair [Lai]
if it satisfies an identity
(1.7)
156
GQChen, J Glimm
Furthermore, if, for any fixed ^ G (00,00),f/ vanishes onthe vacuum p = 0, then η is called a weak entropy. For example, the mechanical energyenergy flux pair
\m2
1
f\m2 Py~x
is a strictly convex weak entropyentropy flux pair. Onecanprove that, for 0 ^
\Vη\ ^ const,
(1.9)
and
\V2η(r,r)\ g const.V2^*(r,r) ,
(1.10)
for any weak entropy η, where r is any vector and the constant is independent of r.
Definition 1.1.A pair of measurablefunctions v(x,t) = (p(x,t),m(x,t)) is called a
global weak entropy solution of the Cauchy problem (1.3) and (1.5) //,for any test function φ G CQ(Ω) with Ω C R 2 + Ξ R X R + ,
t + f(v)φx + a{x)g{v)φ)dxdt + / ^ ( x , 0 ) ώ = 0 ;
supp 0( , 0)
(1.11)
shock wave with left state v, right state t;+, α/?J speed σ,
^  ) ) " (q(v+)  q(v)) ^ 0 ,
(1.12)
for any convex weak entropyentropy flux pair (η,q) It is called a global weak entropy solution of the initialboundary problem (1.3) and(1.6) provided that
1 i+e
 fm(x9t)dx>0,
ε 1
mL£(R+),
αn>0;
(1.13)
and, for any convex weak entropy pair (rj,q) and any Cι test function φ with suppφ C (l,oo) x R+, both (1.11) and (1.12)
For the initialboundary problem for the compressible Euler equations (1.1) with
(m xfμi = 0 ,
\x\ ^ 1 ,
we introduce the following conventional notion of weak entropy solution.
Definition 1.2. A measurable vector function (p(x, t), m(x, t)) is called a global weak entropy solution of the initialboundaryproblem (1.1)and (1.14) provided that
(1) The vector function (p(x9t),m(x,t)) satisfies the Euler equations (1.1) in
the sense of distributions with respect to the test function space {φ GCO°({JΓ >
1} x R+)0(jΓ,r) = ^(j?,ί)} (2)
1 ^+ε  fm(x9t)
£
x dx^0,
X
as s 10, m Z 1 ^ 1 x R + ) ;
(1.15)
Global Solutions to Compressible Euler Equations with Geometrical Structure
157
(3) Along any shock wave propagating in the direction v e RN, v = 1, with left and right states (p±,m±) and speed s = s(p,p+,m,m+;v),
m\2
p+ J
\ λp
p_
(1.16)
where e = p,' . w ί/ze internal energy.
In these definitions, the entropy conditions (1.12) and (1.16) are equivalent to the corresponding Lax entropy conditions along the shock waves (cf. [Lai, D2, Sm]).
Our main results of this paper are included in the following theorems. For the Cauchy problem (1.3)—(1.5), which models the transonic nozzle flow, we have
Theorem A. Assume that A(x) is a C2 function bounded away from zero for all x GR and the initial data satisfy
0 S po(x) ^ Co,
mo(x)
(1.17)
for some Q > 0. Then there exists a global weak entropy solution (p(x9t)9rn(x9t)) of the Cauchy problem (1.3)—(1.5) in the sense of Definition 1.1 satisfying
p(x,t)
(1.18)
for some C(T) ^ Co in the region R x [0, T] for any fixed T e (0,oo).
For the initialboundary problem (1.1) and (1.14), which models the spherically symmetric flow, we have
Theorem B. Assume that the initial data are of the form (1.14) with (po(x),mo(x)) G L°°({x ^ 1 } ) satisfying (1.17). Then there exists a global weak entropy solution (p(x,t),m(x,t)) of the initialboundary problem (1.1) and (1.14) in the sense of Definition 1.2, which takes the form
(p,m)(x, t) = ί p{x90,rn(x, t)
\
x
with (p(x,t)9m(x,t)) GL°°({x ^ 1} x R+) satisfying (1.18).
Note that it is sufficient to show that v(x9t) = (p(x9t)9rn(x,t)) is a global weak entropy solution of the initialboundary problem (1.3) and (1.6), or the Cauchy problem (1.3) and (1.5), in the sense of Definition 1.1. To achieve these results, we also apply a compensated compactness framework (7.1)—(7.2) (Sect. 7) in [Cl, C2] (also see [DC 12, Di]): uniform boundedness (7.1) of the approximate solutions (ph(x,t),mh(x,t)) and H~ι compactness (7.2) of the corresponding entropy dissipation measures imply the strong convergence of the approximate solutions (ph(x,t),mh(x,t)) to the global weak entropy solution (p(x, t)9 m(x, t)) G
158
G.Q Chen, J. Glimm
L°° of the initialboundary problem (1.3) and (1.6) and the Cauchy problem (1.3) and (1.5), respectively, almost everywhere with the same property (7.1). The importance of this framework is that it takes the vacuum into account in correct physical variables (p,m) near the vacuum, rather than (p,w) that is physically incorrect on the vacuum. This framework was proved in [Di] for the case 7 = 1 + 2^+\>m = ^ integers, and in [Cl, C2, DC1] for the general case of gases 1 < γ ^ 5/3. Further discussions on this framework for other cases can be found in [LP].
In Sect. 2 we construct two solutions which will serve as building blocks for our construction: Riemann solutions for the homogeneous system of gas dynamics and (exact and approximate) steadystate solutions for the inhomogeneous system (1.2). We discuss their basic properties in Sects. 2 and 3.
Section 4 is devoted to the construction of the shock capturing scheme and the corresponding approximate solution of the problems (1.5) and (1.6) for (1.3). Some basic properties of the approximate solutions are discussed. It is proved in Sect. 5 and Sect. 6 that the approximate solutions satisfy the compensated compactness framework (7.1)(7.2) (see [Cl, C2]). The existence theory is established in Sect. 7.
Then the existence theory is applied to the transonic nozzle flow in Sect. 8 (Theorem A) and the spherically symmetric flow in Sect. 9 (Theorem B).
By the methods developed here, we have also proved existence for the initial boundary value problem that models the cylindrically symmetric flow in the unbounded domain £ ^ 1 (see [CG]). The ideas developed here have been also applied to solving the compressible EulerPoisson equations with geometrical structure that model semiconductor devices and biological channel proteins (see [CW]).
2. Nonlinear Waves and Riemann Solutions
In this section we first review some nonlinear waves in gas dynamics and construct Riemann solutions for the homogeneous system of gas dynamics. Then we discuss their basic properties for use in subsequent developments.
2.1. Shock Waves and Rarefaction Waves for 1D Gas Dynamics. Consider the Riemann problem for the onedimensional system of isentropic gas dynamics:
u + (i + P(P))X = o,
ί with
(p_,m_), JC < xo j
(p+,/w+), x > xo ,
where xo E (—oo, oo), p± ^ 0 and m± are constants satisfying  —  < oo.
The eigenvalues of the system are
(2.2)
λ\ = P
c = c(M — 1),
Λ,2 = \ c = c(M + 1), P
Global Solutions to Compressible Euler Equations with Geometrical Structure
159
where the sound speed c = pθ, the Mach number M = ^ , and θ — ^γm Corresponding Riemann invariants are
Any discontinuity in the weak solutions to (2.1) must satisfy Hugoniot condition:
σ(v ~ v0) = f(v)  f(v0),
the Rankine
where σ is the propagation speed of the discontinuity, and v0 — (po,wo) and v — (p,m) are the corresponding left state and right state. This means that
_ m—niQ __ mo i / _ρ_ _
p—po
Po
V PQ
(2.4)
A discontinuity is called a shock if it satisfies the entropy condition (see [La]):
σ(η(v)  η(v0))  (g(v)  q(υ0)) έ 0 ,
(2.5)
for any convex entropy pair (η,q). There are two distinct types of rarefaction waves and shock waves denoted by
1Rw or 2Rw and 1shock or 2shock, respectively, in the isentropic gases. If a state (βo,mo) or (po,wo) is given, the possible states (p,m) or (p,u) that can be connected to (po,mo) ° n the right by a Rw or shock are
Rx(0) :mmo R2(0) :rnmo
=
—(p Po

p0) 
%θ v
 pθ0X
= — (p  Po) + % θ  Po),
Po
0
P < Po , P > Po ,
,:mmo mo == ^—(P(pppoo))  Jg P(P)  ^(p
Po
V Po P  Po
 p0), p > p0 > 0 ,
£2(0) : m — m0 = — ( p — p 0 ) + \ —
Po
V Po
respectively. Along the curve 7?i(0),
_ mo " Po
and along the curve
Po _ ^ + 1 nθ
β
β P'
P  Po
d2m 'dp1
 PoX P < Po ,
mo Pπ
dp
Po 0
> 0.
R2(0)
This shows that the curve R\(0) is concave and the curve R2Φ) is convex in the p — m plane.
160 Along the curve Si(0),
G.Q Chen, J. Glimm
P~Pθ
_ mo _ / P PiP.
V PO i
_d_ ί fn—rriQ \
dp\PPθ)\Sλm~I5Ί(O)
and along the curve
Pp
p—Pp
< 0 n *> n ^>
2yj£/('p^((pp)(pp)(p0p))((ppo)p)0()ppo) = ' μ P0
ΪYl — ϊϊlQ
PPO S2(0)
— mo I / p p(p)p(po) PO V PO PPO
p(p)/?O0)
This shows that the curve 5Ί(0) is concave and the curve ^ ( 0 ) is convex with respect to (po,mo) in the p — m plane.
One wave is of particular interest, namely, the standing shock with the speed cr = O. For this case, the RankineHugoniot conditions are
m = mo9
— + p=n^ + p(p0).
P
Po
Equations (2.6) can be rewritten as
«'(β+I),
M2 0
=
^
i
j = l + ( θ + l ) ( ί  l ) +O ( k  l  2 ) ,
where t = p/p0. From (2.7), one has
(2.6)
(2.7)
One can easily check that MQ is a monotonically increasing function of t with M Q ( 0 ) = 0 , M Q ( 1 ) = 1, and MQ(OO) = oo; and M2 is monotonically decreasing function of t with M2(0) = oo,M2(l) = 1, and M2(oo) = 0.
The entropy condition (2.5) for the standing shock (2.6) or (2.7) is
J M < 1 < Mo, \ M < —1 < Mo,
when m0 > 0 , when mo < 0 .
2.2. Riemann Solutions. Similarly, given a state (po?^o) or (PO,UQ) for p 0 > 0, the locus of possible states (p,m) or (p,u) for p > 0 that can be connected to the state on the left by a shock wave S~ι or rarefaction wave R~ι defines what is called an inverse shock wave curve or inverse rarefaction wave curve. It has behavior similar to that of S or R.
From the behavior of these curves in the phase plane (p,m), we can construct the unique solution for the Riemann problem
V, X < Xo , V+, X > Xo ,
(2.8)
Global Solutions to Compressible Euler Equations with Geometrical Structure
161
and the Riemann initialboundary problem
t>ί=o = v+, m\x=ι = 0 .
(2.9)
For the problem (2.8), we can get a diagram of the first family of elementary wave curves for given left state V and a diagram of the second family of inverse elementary wave curves for given right state υ+ to determine a unique intersection point to obtain the unique solution. Forthe problem (2.9), we can draw a diagram of the second family of inverse elementary wave curves for given right state v+ to determine a unique intersection point with the line m = 0 to obtain the unique solution.
Theorem 2.1. There exists a unique piecewise smooth entropy solution (p(x,ί), m(x,t)) containing the vacuum state (p— 0) on the upper plane t > 0 for each problem o/(2.8) and (2.9) satisfying
(1) For the Riemann problem (2.8),
w(p(x9t)9rn(x,t)) S max(w(p_,m_),vy(p+,m+)), z(ρ(x,t),m(x,t)) ^ min(z(p_,m_),z(p+,m+)), w(p(x,t),m(x,t)) z(p(x,tχm(x,t)) ^ 0;
(2) For the Riemann problem (2.9),
' w(ρ(x,t)9m(x,t)) g max(w(p+,m+),z(p+,m+)), z(ρ(x,t),m(x,t)) ^ min(z(p+,m+),0), w(p(x, t)9 m(x, 0 )  z(p(x, t\ m(x, f)) ^ 0 . Such Riemann solutions have the following properties:
Lemma 2.1. The regions Σ = {(P>m) '• w =w^z =zo>w ~~z = 0} are invariant with respect to both of the Riemann problem (2.8) and the average of the Riemann solutions in x. More precisely, if the Riemann data lie in ]P, the corresponding Riemann solutions (p(x, t),m(x, t)) lie in J^, and their corresponding averages in x also lie in ]Γ:
\ fp(x,t)dx, 1 fmb (x,t)dx \
\ba a
bbaa a
JJ
Furthermore, for the Riemann initialboundary problem (2.9), the regions ^ = {(p,m) : w ^ wo,z ^ zo,w — z ^ O},zo ^ 0 ^ w°^"z°, are invariant with respect to both of the Riemann problem (2.9) and theaverage of the corresponding Riemann solutions in x.
The proof of Lemma 2.1 can be found in [Cl, MT].
Lemma 2.2. The rate of entropy production of a shock with left state t;_ and right state v+ for an arbitrary weak entropy η is dominated by the associated rate of entropy production for η* in the following sense:
\σ(η(v+)  η(υ))  (q(v+)  q(υ))\
where the constant C depends only on η and max(p± +  —  ) . The proof of this fact can be found in [Cl].
162 3. SteadyState Solutions
GQ Chen, J Glimm
Travelling waves and Riemann solutions of the homogeneous problems (2.8) and (2.9) have been discussed in Sect. 2. The purpose of this section is to provide important estimates on steadystate solutions of the inhomogeneous problem (1.2) determined by the following system of ordinary differential equations:
subject to the boundary condition
(P,m)\x=xo =(po,mo).
(3.2)
The nonsonic and transonic cases are distinct, as the former produces smooth solutions and the latter may contain a standing shock wave. The L°° estimates are derived based on Riemann invariant inequalities and are required for the compensated compactness framework. The Lx estimates areneeded for consistency and verification of the entropy condition.
In this section we always assume that A(x) is a C2 function satisfying A(x) ^ Co > 0 in the interval under consideration.
3.1. Smooth SteadyState Solutions for the Nonsonic Case. We first consider the nonsonic case MQ φ 1, where MQ — M{x — XQ) — ^.
The first equation can be directly integrated to obtain
A{x)m = A(xo)rπo .
(3.3)
The second equation can berewritten as
, x m2
and, using (3.3), that is,
m (  ) + P(P)X = 0 , = 0.
Therefore, one has
*+ίiΰ!id.»+jmlb.
Z
Q
S
A
0
S
In terms of p,M, and Mo, the system (3.3)(3.4) becomes
A(x)M
M
A(xo)Mo {
( l γ p0
(3.4,
(16)
Communications ΪΠ
Mathematical Physics
© SpringerVerlag 1996
Global Solutions to the Compressible Euler Equations with Geometrical Structure
GuiQiang Chen1, James Glimm2 1 Department of Mathematics, Northwestern University, Evanston, Illinois 60208, USA Email: [email protected] nwu edu 2 Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, New York 11794, USA. Email: [email protected] sunysb edu
Received: 17 October 1995/Accepted: 30 January 1996
Abstract: We prove the existence of global solutions to the Euler equations of compressible isentropic gas dynamics with geometrical structure, including transonic nozzle flow and spherically symmetric flow. Due to the presence of the geometrical source terms, the existence results themselves are new, especially as they pertain to radial flow in an unbounded region, \x\ ^ 1, and to transonic nozzle flow. Arbitrary data with L°° bounds are allowed in these results. A shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions. The convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework.
Contents
1. Introduction
153
2. Nonlinear Waves and Riemann Solutions
158
3. SteadyState Solutions
162
4. Approximate Solutions
176
5. L°° Estimates
181
6. H~x Compactness Estimates
182
7. Convergence and Consistency
186
8. Transonic Nozzle Flow: Proof of Theorem A
189
9. Spherically Symmetric Flow: Proof of Theorem B
189
References
1. Introduction
We develop new mathematical existence theory and numerical schemes for global discontinuous solutions to the Euler equations of compressible isentropic gas
154
GQ Chen, J Glimm
dynamics with large initial data and with geometrical structure. The compressible Euler equations are of the following conservation form:
= 0,
(L1)
where p, m, and p are the density, the momentum, and the pressure of the gas, respectively. On the nonvacuum state, u = m/p is the velocity. For polytropic gas, p(p) = py/y, where the adiabatic exponent γ is restricted to the interval (1, 5/3], as is usual for gases.
Our results are new for the problem of spherically symmetric flow in an unbounded domain (\x\ ^ 1) and for transonic nozzle flow: existence of global discontinuous solutions for general initial data in L°°. The central difficulty in the unbounded domain is the reflection of waves from infinity and their strengthening as they move radially inward. The central difficulty for the transonic nozzle flow lies in the associated steadystate equations, which change type from elliptic to hyperbolic at the sonic point; such steady state solutions are fundamental building blocks in our approach and in early work on nozzle flow.
The Cauchy problem models transonic nozzle flow through a variablearea duct (cf. [EM, GL, GM, L1L3, CF, Wh]). In [LI] the existence of global solutions of this problem was obtained by first incorporating the steadystate building blocks with the random choice method [Gl], provided that the initial data have small total variation and are bounded away from both sonic and vacuum states. A generalized random choice method was introduced to compute transient gas flows in a Laval nozzle in [GL, GM]. A mathematical analysis of the qualitative behavior of nonlinear waves for nozzle flow was given in [L3]. In this paper we introduce a Godunov shock capturing scheme to obtain L°° estimates and compensated compactness of the corresponding approximate solutions. Our method incorporates natural building blocks from Riemann solutions and steadystate solutions. Such estimates lead to the convergence of the approximate solutions and to an existence theory of global weak entropy solutions for measurable initial data inZ°°.
The initialboundary problem is motivated by many important physical problems such as flow in a jet engine inlet manifold and stellar dynamics including supernovae formation. A global weak entropy solution with spherical symmetry was constructed in [MU] for the isothermal case γ = 1 and the local existence of such a weak solution for the general case 1 < y ^ 5/3 was also discussed in [MT]. A theorem has also been established for the general case in [C3] to ensure the existence of L°° spherically symmetric weak solutions in the large for a class of L°° Cauchy data of arbitrarily large amplitude, which model outgoing blast waves and largetime asymptotic solutions.
In Sects. 24 we develop a first order Godunov shock capturing scheme, with piecewise constant building blocks replaced by piecewise steady ones. The main point is to use the steadystate solutions, which incorporate geometrical source terms, to modify the wave strengths in the Riemann solutions. This construction yields better approximate solutions, and permits uniform L°° bounds. There are two technical difficulties which we overcome to achieve this goal, both due to transonic phenomena. One is that no smooth steadystate solution exists in each cell in general. This problem is easily solved by introducing a standing shock at the center
Global Solutions to Compressible Euler Equations with Geometrical Structure
155
of the cell, as discussed in Sect. 2. The other is that the constructed steadystate solution in each cell must satisfy the following requirements:
(a) The oscillation of the steadystate solution around the Godunov value must be of the same order as the cell length to obtain the L°° estimate for the convergence arguments;
(b) The difference between the average of the steadystate solution over each cell and the Godunov value must be higher than first order in the cell length to ensure the consistency of the corresponding approximate solutions with the Euler equations.
These requirements are satisfied by smooth steadystate solutions bounded away from the sonic state in the cell. The general case must include the transonic steadystate solutions. The sonic difficulty is overcome, as in experimental physics, by introducing an additional standing shock with continuous mass and by adjusting its left state and right state in the density and its location to control the growth of the density. These requirements also enable us to make H~ι compactness estimates for corresponding entropy dissipation measures to deduce the strong convergence of the approximate solutions with the aid of the compactness framework (see [Cl, C2]).
We rewrite (1.1) with geometrical structure as
( pt + mx= a(x)m ,
or in a compact form:
vt + f(υ)x = a(x)g(v),
(1.3)
where m is the momentum of the gas, a(x) is a C2 function in the region of x = x under consideration, v = ( p , m ) τ , f(v) = (m,m2/p + p(p))Ύ, and g(v) = (m,m2/p)τ. The function a(x) can be represented by
a(χ) = A'(x)/A(x) with A(x) = eS* a{y)dy .
(1.4)
The function A(x) represents the crosssectional area at x in a variablearea duct for transonic nozzle flow, A(x) — 2πx for cylindrically symmetric flow, and A(x) = 4πx2 for spherically symmetric flow. For cylindrical and spherical flow, we impose reflecting boundary conditions at x = 1 to exclude the singularity at the origin, but, as a principal new result of this paper, we are able to handle successfully the difficulties at infinity.
We consider the Cauchy problem:
v\t=o = vo(x),
(1.5)
and the initialboundary value problem:
{ϊ,vf
with the initial data VQ(X) € L°°. A pair of mappings (η,q) : R2 —> R2 is called an entropyentropy flux pair [Lai]
if it satisfies an identity
(1.7)
156
GQChen, J Glimm
Furthermore, if, for any fixed ^ G (00,00),f/ vanishes onthe vacuum p = 0, then η is called a weak entropy. For example, the mechanical energyenergy flux pair
\m2
1
f\m2 Py~x
is a strictly convex weak entropyentropy flux pair. Onecanprove that, for 0 ^
\Vη\ ^ const,
(1.9)
and
\V2η(r,r)\ g const.V2^*(r,r) ,
(1.10)
for any weak entropy η, where r is any vector and the constant is independent of r.
Definition 1.1.A pair of measurablefunctions v(x,t) = (p(x,t),m(x,t)) is called a
global weak entropy solution of the Cauchy problem (1.3) and (1.5) //,for any test function φ G CQ(Ω) with Ω C R 2 + Ξ R X R + ,
t + f(v)φx + a{x)g{v)φ)dxdt + / ^ ( x , 0 ) ώ = 0 ;
supp 0( , 0)
(1.11)
shock wave with left state v, right state t;+, α/?J speed σ,
^  ) ) " (q(v+)  q(v)) ^ 0 ,
(1.12)
for any convex weak entropyentropy flux pair (η,q) It is called a global weak entropy solution of the initialboundary problem (1.3) and(1.6) provided that
1 i+e
 fm(x9t)dx>0,
ε 1
mL£(R+),
αn>0;
(1.13)
and, for any convex weak entropy pair (rj,q) and any Cι test function φ with suppφ C (l,oo) x R+, both (1.11) and (1.12)
For the initialboundary problem for the compressible Euler equations (1.1) with
(m xfμi = 0 ,
\x\ ^ 1 ,
we introduce the following conventional notion of weak entropy solution.
Definition 1.2. A measurable vector function (p(x, t), m(x, t)) is called a global weak entropy solution of the initialboundaryproblem (1.1)and (1.14) provided that
(1) The vector function (p(x9t),m(x,t)) satisfies the Euler equations (1.1) in
the sense of distributions with respect to the test function space {φ GCO°({JΓ >
1} x R+)0(jΓ,r) = ^(j?,ί)} (2)
1 ^+ε  fm(x9t)
£
x dx^0,
X
as s 10, m Z 1 ^ 1 x R + ) ;
(1.15)
Global Solutions to Compressible Euler Equations with Geometrical Structure
157
(3) Along any shock wave propagating in the direction v e RN, v = 1, with left and right states (p±,m±) and speed s = s(p,p+,m,m+;v),
m\2
p+ J
\ λp
p_
(1.16)
where e = p,' . w ί/ze internal energy.
In these definitions, the entropy conditions (1.12) and (1.16) are equivalent to the corresponding Lax entropy conditions along the shock waves (cf. [Lai, D2, Sm]).
Our main results of this paper are included in the following theorems. For the Cauchy problem (1.3)—(1.5), which models the transonic nozzle flow, we have
Theorem A. Assume that A(x) is a C2 function bounded away from zero for all x GR and the initial data satisfy
0 S po(x) ^ Co,
mo(x)
(1.17)
for some Q > 0. Then there exists a global weak entropy solution (p(x9t)9rn(x9t)) of the Cauchy problem (1.3)—(1.5) in the sense of Definition 1.1 satisfying
p(x,t)
(1.18)
for some C(T) ^ Co in the region R x [0, T] for any fixed T e (0,oo).
For the initialboundary problem (1.1) and (1.14), which models the spherically symmetric flow, we have
Theorem B. Assume that the initial data are of the form (1.14) with (po(x),mo(x)) G L°°({x ^ 1 } ) satisfying (1.17). Then there exists a global weak entropy solution (p(x,t),m(x,t)) of the initialboundary problem (1.1) and (1.14) in the sense of Definition 1.2, which takes the form
(p,m)(x, t) = ί p{x90,rn(x, t)
\
x
with (p(x,t)9m(x,t)) GL°°({x ^ 1} x R+) satisfying (1.18).
Note that it is sufficient to show that v(x9t) = (p(x9t)9rn(x,t)) is a global weak entropy solution of the initialboundary problem (1.3) and (1.6), or the Cauchy problem (1.3) and (1.5), in the sense of Definition 1.1. To achieve these results, we also apply a compensated compactness framework (7.1)—(7.2) (Sect. 7) in [Cl, C2] (also see [DC 12, Di]): uniform boundedness (7.1) of the approximate solutions (ph(x,t),mh(x,t)) and H~ι compactness (7.2) of the corresponding entropy dissipation measures imply the strong convergence of the approximate solutions (ph(x,t),mh(x,t)) to the global weak entropy solution (p(x, t)9 m(x, t)) G
158
G.Q Chen, J. Glimm
L°° of the initialboundary problem (1.3) and (1.6) and the Cauchy problem (1.3) and (1.5), respectively, almost everywhere with the same property (7.1). The importance of this framework is that it takes the vacuum into account in correct physical variables (p,m) near the vacuum, rather than (p,w) that is physically incorrect on the vacuum. This framework was proved in [Di] for the case 7 = 1 + 2^+\>m = ^ integers, and in [Cl, C2, DC1] for the general case of gases 1 < γ ^ 5/3. Further discussions on this framework for other cases can be found in [LP].
In Sect. 2 we construct two solutions which will serve as building blocks for our construction: Riemann solutions for the homogeneous system of gas dynamics and (exact and approximate) steadystate solutions for the inhomogeneous system (1.2). We discuss their basic properties in Sects. 2 and 3.
Section 4 is devoted to the construction of the shock capturing scheme and the corresponding approximate solution of the problems (1.5) and (1.6) for (1.3). Some basic properties of the approximate solutions are discussed. It is proved in Sect. 5 and Sect. 6 that the approximate solutions satisfy the compensated compactness framework (7.1)(7.2) (see [Cl, C2]). The existence theory is established in Sect. 7.
Then the existence theory is applied to the transonic nozzle flow in Sect. 8 (Theorem A) and the spherically symmetric flow in Sect. 9 (Theorem B).
By the methods developed here, we have also proved existence for the initial boundary value problem that models the cylindrically symmetric flow in the unbounded domain £ ^ 1 (see [CG]). The ideas developed here have been also applied to solving the compressible EulerPoisson equations with geometrical structure that model semiconductor devices and biological channel proteins (see [CW]).
2. Nonlinear Waves and Riemann Solutions
In this section we first review some nonlinear waves in gas dynamics and construct Riemann solutions for the homogeneous system of gas dynamics. Then we discuss their basic properties for use in subsequent developments.
2.1. Shock Waves and Rarefaction Waves for 1D Gas Dynamics. Consider the Riemann problem for the onedimensional system of isentropic gas dynamics:
u + (i + P(P))X = o,
ί with
(p_,m_), JC < xo j
(p+,/w+), x > xo ,
where xo E (—oo, oo), p± ^ 0 and m± are constants satisfying  —  < oo.
The eigenvalues of the system are
(2.2)
λ\ = P
c = c(M — 1),
Λ,2 = \ c = c(M + 1), P
Global Solutions to Compressible Euler Equations with Geometrical Structure
159
where the sound speed c = pθ, the Mach number M = ^ , and θ — ^γm Corresponding Riemann invariants are
Any discontinuity in the weak solutions to (2.1) must satisfy Hugoniot condition:
σ(v ~ v0) = f(v)  f(v0),
the Rankine
where σ is the propagation speed of the discontinuity, and v0 — (po,wo) and v — (p,m) are the corresponding left state and right state. This means that
_ m—niQ __ mo i / _ρ_ _
p—po
Po
V PQ
(2.4)
A discontinuity is called a shock if it satisfies the entropy condition (see [La]):
σ(η(v)  η(v0))  (g(v)  q(υ0)) έ 0 ,
(2.5)
for any convex entropy pair (η,q). There are two distinct types of rarefaction waves and shock waves denoted by
1Rw or 2Rw and 1shock or 2shock, respectively, in the isentropic gases. If a state (βo,mo) or (po,wo) is given, the possible states (p,m) or (p,u) that can be connected to (po,mo) ° n the right by a Rw or shock are
Rx(0) :mmo R2(0) :rnmo
=
—(p Po

p0) 
%θ v
 pθ0X
= — (p  Po) + % θ  Po),
Po
0
P < Po , P > Po ,
,:mmo mo == ^—(P(pppoo))  Jg P(P)  ^(p
Po
V Po P  Po
 p0), p > p0 > 0 ,
£2(0) : m — m0 = — ( p — p 0 ) + \ —
Po
V Po
respectively. Along the curve 7?i(0),
_ mo " Po
and along the curve
Po _ ^ + 1 nθ
β
β P'
P  Po
d2m 'dp1
 PoX P < Po ,
mo Pπ
dp
Po 0
> 0.
R2(0)
This shows that the curve R\(0) is concave and the curve R2Φ) is convex in the p — m plane.
160 Along the curve Si(0),
G.Q Chen, J. Glimm
P~Pθ
_ mo _ / P PiP.
V PO i
_d_ ί fn—rriQ \
dp\PPθ)\Sλm~I5Ί(O)
and along the curve
Pp
p—Pp
< 0 n *> n ^>
2yj£/('p^((pp)(pp)(p0p))((ppo)p)0()ppo) = ' μ P0
ΪYl — ϊϊlQ
PPO S2(0)
— mo I / p p(p)p(po) PO V PO PPO
p(p)/?O0)
This shows that the curve 5Ί(0) is concave and the curve ^ ( 0 ) is convex with respect to (po,mo) in the p — m plane.
One wave is of particular interest, namely, the standing shock with the speed cr = O. For this case, the RankineHugoniot conditions are
m = mo9
— + p=n^ + p(p0).
P
Po
Equations (2.6) can be rewritten as
«'(β+I),
M2 0
=
^
i
j = l + ( θ + l ) ( ί  l ) +O ( k  l  2 ) ,
where t = p/p0. From (2.7), one has
(2.6)
(2.7)
One can easily check that MQ is a monotonically increasing function of t with M Q ( 0 ) = 0 , M Q ( 1 ) = 1, and MQ(OO) = oo; and M2 is monotonically decreasing function of t with M2(0) = oo,M2(l) = 1, and M2(oo) = 0.
The entropy condition (2.5) for the standing shock (2.6) or (2.7) is
J M < 1 < Mo, \ M < —1 < Mo,
when m0 > 0 , when mo < 0 .
2.2. Riemann Solutions. Similarly, given a state (po?^o) or (PO,UQ) for p 0 > 0, the locus of possible states (p,m) or (p,u) for p > 0 that can be connected to the state on the left by a shock wave S~ι or rarefaction wave R~ι defines what is called an inverse shock wave curve or inverse rarefaction wave curve. It has behavior similar to that of S or R.
From the behavior of these curves in the phase plane (p,m), we can construct the unique solution for the Riemann problem
V, X < Xo , V+, X > Xo ,
(2.8)
Global Solutions to Compressible Euler Equations with Geometrical Structure
161
and the Riemann initialboundary problem
t>ί=o = v+, m\x=ι = 0 .
(2.9)
For the problem (2.8), we can get a diagram of the first family of elementary wave curves for given left state V and a diagram of the second family of inverse elementary wave curves for given right state υ+ to determine a unique intersection point to obtain the unique solution. Forthe problem (2.9), we can draw a diagram of the second family of inverse elementary wave curves for given right state v+ to determine a unique intersection point with the line m = 0 to obtain the unique solution.
Theorem 2.1. There exists a unique piecewise smooth entropy solution (p(x,ί), m(x,t)) containing the vacuum state (p— 0) on the upper plane t > 0 for each problem o/(2.8) and (2.9) satisfying
(1) For the Riemann problem (2.8),
w(p(x9t)9rn(x,t)) S max(w(p_,m_),vy(p+,m+)), z(ρ(x,t),m(x,t)) ^ min(z(p_,m_),z(p+,m+)), w(p(x,t),m(x,t)) z(p(x,tχm(x,t)) ^ 0;
(2) For the Riemann problem (2.9),
' w(ρ(x,t)9m(x,t)) g max(w(p+,m+),z(p+,m+)), z(ρ(x,t),m(x,t)) ^ min(z(p+,m+),0), w(p(x, t)9 m(x, 0 )  z(p(x, t\ m(x, f)) ^ 0 . Such Riemann solutions have the following properties:
Lemma 2.1. The regions Σ = {(P>m) '• w =w^z =zo>w ~~z = 0} are invariant with respect to both of the Riemann problem (2.8) and the average of the Riemann solutions in x. More precisely, if the Riemann data lie in ]P, the corresponding Riemann solutions (p(x, t),m(x, t)) lie in J^, and their corresponding averages in x also lie in ]Γ:
\ fp(x,t)dx, 1 fmb (x,t)dx \
\ba a
bbaa a
JJ
Furthermore, for the Riemann initialboundary problem (2.9), the regions ^ = {(p,m) : w ^ wo,z ^ zo,w — z ^ O},zo ^ 0 ^ w°^"z°, are invariant with respect to both of the Riemann problem (2.9) and theaverage of the corresponding Riemann solutions in x.
The proof of Lemma 2.1 can be found in [Cl, MT].
Lemma 2.2. The rate of entropy production of a shock with left state t;_ and right state v+ for an arbitrary weak entropy η is dominated by the associated rate of entropy production for η* in the following sense:
\σ(η(v+)  η(υ))  (q(v+)  q(υ))\
where the constant C depends only on η and max(p± +  —  ) . The proof of this fact can be found in [Cl].
162 3. SteadyState Solutions
GQ Chen, J Glimm
Travelling waves and Riemann solutions of the homogeneous problems (2.8) and (2.9) have been discussed in Sect. 2. The purpose of this section is to provide important estimates on steadystate solutions of the inhomogeneous problem (1.2) determined by the following system of ordinary differential equations:
subject to the boundary condition
(P,m)\x=xo =(po,mo).
(3.2)
The nonsonic and transonic cases are distinct, as the former produces smooth solutions and the latter may contain a standing shock wave. The L°° estimates are derived based on Riemann invariant inequalities and are required for the compensated compactness framework. The Lx estimates areneeded for consistency and verification of the entropy condition.
In this section we always assume that A(x) is a C2 function satisfying A(x) ^ Co > 0 in the interval under consideration.
3.1. Smooth SteadyState Solutions for the Nonsonic Case. We first consider the nonsonic case MQ φ 1, where MQ — M{x — XQ) — ^.
The first equation can be directly integrated to obtain
A{x)m = A(xo)rπo .
(3.3)
The second equation can berewritten as
, x m2
and, using (3.3), that is,
m (  ) + P(P)X = 0 , = 0.
Therefore, one has
*+ίiΰ!id.»+jmlb.
Z
Q
S
A
0
S
In terms of p,M, and Mo, the system (3.3)(3.4) becomes
A(x)M
M
A(xo)Mo {
( l γ p0
(3.4,
(16)