The paper is concerned with the uniqueness and existence problem for a multidimensional reacting and convection system with the vanishing viscosity method.
ON THE UNIQUENESS AND EXISTENCE PROBLEM FOR A MULTIDIMENSIONAL REACTING AND CONVECTION SYSTEM JIEQUAN LI
A The paper is concerned with the uniqueness and existence problem for a multidimensional reacting and convection system with the vanishing viscosity method. The uniqueness theorem is obtained from the stability with respect to the initial data. To solve the existence problem, the uniqueness and existence of solutions to a viscous system are first proved, and then the L"-modulus of continuity of solutions independent of the small viscosity is obtained.
1. Introduction Consider the Cauchy problem for a multidimensional reacting and convection system 1 ct(ujqz)j ci fi(u) l 0, 2 i (1.1) 3 ct zjKφ(u) z l 0 4 with the initial data (u, z) (t l 0, x) l (u , z ) (x) ? L_(Rn). ! !
(1.2)
Here and throughout this paper x l (x , … , xn) ? Rn, t 0, ct l c\ct, ci l c\cxi, and " designates the summation taken from 1 to n unless exceptional cases are pointed out. This model is the natural generalization of a one-dimensional model to display dynamic combustion in several dimensions (cf. [10, 11] and so on). The first equation of (1.1) resembles the energy conservation laws in gas dynamics, leading to shock waves and rarefaction waves, and the second equation is a reacting equation with contact discontinuities, along which the chemical reaction occurs. We interpret the dependent variable u as a lumped variable to represent density, velocity and temperature, and z as the percentage of unburnt gas. The constant q is the chemical binding energy, K denotes the reaction rate and φ(u) (0 φ 1) is an ignition type function taken as 1 1, u u` jη, φ(u) l 23 (1.3) 0, u u` , 4 where u` is the ignition temperature and can be considered as zero, η 0. We assume that fi ? C #(R), φ ? C "(R), and Lf and Lφ are their respective Lipschitz constants.
Received 19 February 1999 ; revised 6 September 1999. 2000 Mathematics Subject Classification 35K57. J. London Math. Soc. (2) 62 (2000) 473–488 # London Mathematical Society 2000.
474
Our purpose is to prove the uniqueness and existence of weak entropy solutions to (1.1) and (1.2). The uniqueness theorem is based on the framework of Kruzkov [4] and obtained from the stability relative to the changes in the initial data. The existence problem is solved with the vanishing viscosity method ; see [3–5]. In this connection, we first prove the uniqueness and existence of solutions to the Cauchy problem for a reacting–diffusion–convection system 1
ct(ujqz)j ci fi(u) l ε∆u, ε 0, 2
i
3
(1.4)
4
ct zjKφ(u) z l 0,
where ε is the viscosity, and ∆ is the Laplacian. This result lays a solid foundation for the theory of multidimensional gas dynamic combustion [7]. Then we derive the modulus of continuity in L" of the solutions independent of ε. Relating to this work, the existence and uniqueness problem for the onedimensional case of (1.1) and (1.2) was shown in [5, 11] with the difference method and the vanishing viscosity method respectively. The corresponding Riemann problem was studied in [9, 11]. The one-dimensional case of (1.4) was taken by Majda [10] to model the dynamic combustion and the limit of viscous profile related to the strong detonation was proved to be a travelling wave solution as t _. The proof of the stability of viscous strong detonation as t _ was due to [6, 8]. This paper is organized as follows. We give the definition of the weak entropy solution to (1.1) and (1.2) in Section 2. On the basis of this definition, we follow Kruzkov and Levy [4, 5] to prove the uniqueness of solutions. The result is obtained from the stability of solutions with respect to the initial data, for which a more exact estimate than that in [5] is given. This is the content of Section 3. Section 4 contributes to the uniqueness and existence problem for the viscous system (1.4) and (1.2). In Section 5, we prove the existence of a weak entropy solution to (1.1) and (1.2) by using the estimates on the L"-modulus of continuity.
2. Definition of a weak entropy solution and entropy condition on simple discontinuities As is well known, the solutions of (1.1) and (1.2) may not exist after a finite time T * 0 in the classical sense even though the initial data (1.2) is smooth enough, and some additional condition should be posed in order to characterize an admissible solution among all their solutions in the generalized sense. In this section we will derive the definition of weak entropy solutions of (1.1), which contains the entropy condition. The method is frequently used for conservation laws ; see for example [3]. For convenience of statement, we often use the notation ΠT l [0, T ]iRn.
(2.1)
For simplicity, assume that (u, z) is a smooth solution to (1.4). Then (1.4) is equivalent to the system 1
ct uj ci fi(u)kKqφ(u) z l ε∆u, 2
i
4
3
ct zjKφ(u) z l 0.
(2.2)
475
We multiply the first equation of (2.2) by a convex function Φh(u) (Φd(u) 0) to obtain ct Φ(u)j ci Fi(u)kKqφ(u) zΦh(u) l ε∆uΦh(u) i
0
1
l ε ∆Φ(u)kΦd(u) (ci u)# ε∆Φ(u), where
i
(2.3)
&
Fi(u) l fi (u) Φh(u) du.
(2.4)
Since the solutions uε of (2.2) are locally bounded and have a limit as ε will be seen in later sections, we obtain
0, which
& (Φ(u) c gj F (u) c gjKqφ(u) zΦh(u) g* dx dt 0, Π
t
i
i
T
for all 0 g(t, x) ? C _(ΠT). ! Taking
i
Φ(u) l ((uk)#jµ#)"/#
(2.5)
(2.6)
with ? R and µ 0, we conclude that Φ(u) l QukQ
and
Fi(u) l sign(uk) ( fi(u)kfi())
as µ tends to zero, where sign(x) lp1 if px 0. Therefore, the definition of weak entropy solutions to (1.1) and (1.2) can be given as follows. D 2.1. A pair of bounded measurable functions (u, z) (t, x) is a weak entropy solution to the Cauchy problem (1.1) and (1.2) in the strip ΠT for a bounded " (Rn)), T 0 if (u, z) (t, x) ? C([0, T ], Lloc (u, z) (0, x) l (u , z ) (x), ! !
and 1
2 3
4
& &
(
almost everywhere in ΠT,
*
(2.7)
sign(uk) (uk) ct gj ( fi(u)kfi()) ci gjKqφ(u) zg dx dt 0,
Π
T
Π
T
i
(2.8)
ozct gkKφ(u) zgq dx dt l 0,
for any 0 g ? C _(ΠT) and ? R. ! This definition implies the integral indentity by setting lpsup(t,x)?Π Qu(t, x)Q so T that
& (uc gj f (u) c gjKqφ(u) zg* dx dt l 0, Π
t
T
i
i
i
(2.9)
for any g ? C _(ΠT) and ? R. The second equation of (2.8) is equivalent to the ! equation
& for all w ? R.
Π
sign(z(t, x)kw) o(z(t, x)kw) ct gkKφ(u) zgq dx dt l 0 T
(2.10)
476
For a discontinuity, this definition also gives an entropy condition. Let ν l (νt, ν , … , νn) be a normal to a surface of discontinuity S with (up, zp) as the limit value " of the weak entropy solution (u, z) on the side to which pν is pointing. Then, from (2.8), there holds either νt l 0 (2.11) or (u+ku−) νtj ( fi(u+)kfi(u−)) νi l 0. (2.12) i
For the former, u is continuous and z may undergo a jump. This is a contact discontinuity. For the latter, z is continuous (that is, no reaction occurs) and u undergoes a jump for which the following entropy condition should be satisfied : (u−k) νtj ( fi(u−)kfi()) νi 0,
(2.13)
i
for all ? (u−, u+). This inequality is consistent with the entropy condition in [11]. 3. Uniqueness of the weak entropy solution to (1.1) and (1.2) and stability with respect to the initial condition Denote
(
M l RuRL_(Π ), N l max ( fi (u))# T Q Q u M
i
*" #, /
(3.1)
and
R, 0 t T l min(T, RN −")q, (3.2) ! (3.3) Sτ l ΩEot l τ ; 0 τ T q. ! Then the uniqueness of the weak entropy solution of the Cauchy problem (1.1) and (1.2) follows from the following theorem concerning the stability with respect to the initial data in the norm of the L"loc-space. Ω l o(t, x) ; QxQ
T 3.1. Let (u , z ) and (u , z ) be two weak entropy solutions of the problem " " # # (1.1) and (1.2) with the initial data (u", z") and (u#, z#) respectiely. Then, for almost all ! ! ! ! t ? [0, T ], !
&
St
oQu (t, x)ku (t, x)QjQz (t, x)kz (t, x)Qq dx C " # " #
&
S!
oQu"ku#QjQz"kz#Qq dx, ! ! ! !
(3.4)
where C l exp(K(qj1) max(1, Lφ) t). Proof. Let 0 g(t, x, τ, y) ? C _(ΠTiΠT). We set l u (τ, y), g(t, x) l g(t, x, τ, y) # ! for a fixed point (τ, y) in (2.8) and then integrate over ΠT with respect to (τ, y) to obtain
&&
Π
T×ΠT
(
sign(u (t, x)ku (τ, y)) (u (t, x)ku (τ, y)) ct g(t, x, τ, y) " # " # j ( fi(u (t, x))kfi(u (τ, y))) ci g(t, x, τ, y) " # i
*
jKqφ(u (t, x)) z (t, x) g(t, x, τ, y) dx dt dτ dy 0. (3.5) " "
477
Adding the analogue of the above inequality where u , z , t, x are interchanged with " " u , z , τ, y, we get # #
&&
Π
T×ΠT
(
sign(u (t, x)ku (τ, y)) (u (t, x)ku (τ, y)) (ct gjcτ g) " # " # j ( fi(u (t, x))kfi(u (τ, y))) (cx gjcy g) " # i i i
*
jKq(φ(u (t, x)) z (t, x)kφ(u (τ, y)) z (τ, y)) g dx dt dτ dy 0. (3.6) " " # # Here and in the remainder of this section cx l c\cxi and cy l c\cyi. i i Take tjτ xjy tkτ n x ky g(t, x, τ, y) l p , χh χh i i , 2 2 2 i= 2 "
0
1 0 1 0
1
(3.7)
where p(t, x) ? C _([ ρ, Tk2ρ]iKr− ρ), Kr l ox ? Rn ; QxQ rq, 0 χh ? C _(QxQ h ! # ! ρ, x ? R) with R χh( y) dy l 1. Then, with the same method as in [4], we get from (3.6)
&
Π
T
(
sign(u (t, x)ku (t, x)) (u (t, x)ku (t, x)) ct gj ( fi(u (t, x))kfi(u (t, x))) ci g " # " # " # i
*
jKq(φ(u (t, x)) z (t, x)kφ(u (t, x)) z (t, x)) g dx dt 0 " " " #
(3.8)
by letting h approach zero. Denote by &u the set which does not contain Lebesgue points of the function u on [0, T ]. Then it has Lebesgue measure zero ; see [4]. Set αh(σ) l
&
σ
−_
χh( y) dy, 0 δγh(t, x) l 1kαh(QxQjNtkRjγ),
0 fhγ(t, x) l [αh(tkρ)kαh(tkτ)] δγh(t, x), for some 0 notation
ρ
γ 0, (3.9)
τ min( ρ, Tkτ, (τkρ)\2, ρ, τ ? [0, T ]B(&u D&u D&µ), where the ! " # µ(t) l
&
St
Qu (t, x)ku (t, x)Q dx " #
(3.10)
is used. Then we have supp δγh(t, x) 9 Ω, and
)
)
f (u )kfi(u ) # c δγ . 0 ct δγhjN ci δγh ct δγhj i " i h u ku i i " #
(3.11)
Taking g(t, x) l fhγ(t, x), one can obtain
&
Π
T
oQu (t, x)ku (t, x)Q [ χh(tkρ)kχ(tkτ)] δγhjKqQφ(u ) z kφ(u ) z Q fhγq dx dt 0. " # " " # # (3.12)
478
Since ρ and τ are the Lebesgue points of µ(t), it follows that, as γ
0,
&! ([ χ (tkρ)kχ (tkτ)] & Qu"(t, x)ku#(t, x)Q dx* dt jKq & ([α (tkρ)kα (tkτ)] & Qφ(u ) z kφ(u ) z Q dx* dt 0. " " # # ! T!
h
h
St
T
h
Letting h
h
(3.13)
St
" (Rn)), we get 0, and noting the assumption that u , u ? C([0, T ], Lloc " # µ( ρ)kµ(τ)jKq
&& τ
ρ
St
Qφ(u ) z kφ(u ) z Q dx dt 0. " " # #
(3.14)
Also, with the same arguments as above, we obtain from (2.10) that ν( ρ)kν(τ)jK
&&
where ν(t) l
τ
ρ
&
St
St
Qφ(u ) z kφ(u ) z Q dx dt 0, " " # #
Qz (t, x)kz (t, x)Q dx. " #
(3.15)
(3.16)
Since Qφ(u ) z kφ(u ) z Q Qz kz QjLφQu ku Q, " # " " # # " # we add (3.14) and (3.15) and then let ρ approach zero to arrive at µ(τ)jν(τ) µ(0)jν(0)jK(qj1) max(1, Lφ)
&! ( µ(t)jν(t)) dt. τ
(3.17)
Hence, it follows from Gronwall’s inequality that µ(τ)jν(τ) ( µ(0)jν(0)):exp(K(qj1) max(1, Lφ) τ).
(3.18)
The proof is complete.
We point out that this theorem shows the unique weak entropy solution of (1.1) and (1.2) has the finite propagation speed property. 4. Existence and uniqueness problem for the iscous system In this section, we solve the uniqueness and existence problem for (1.4) and (1.2). First, we assume that (u , z ) (x) ? C $(Rn). Then the following maximum principle is ! ! ! satisfied. T 4.1. If (u(t, x), z(t, x)) is a classical bounded solution of (1.4) and (1.2) in ΠT for a bounded T 0, then inf ou (x)q u(t, x) sup ou (x)qjKqt, ! ! ? n
x?Rn
xR
inf oz (x)q z(t, x) sup oz (x)q. ! ! ? n ? n
xR
xR
(4.1)
479
Proof. The method is classical. We only prove the upper bound claim of u(t, x). Assume that (t, x) l u(t, x)kKqtkδ(2εtjQδxQ#\2n) (δ 0) attains its maximum value at (ta , x` ) in ΠT. Then, at (ta , x` ), ct l ct ukKqk2δε 0, ∆ l ∆ukδ$ 0 and c l c ukδ$x` \n l 0, so j
j
j
0 ct kε∆ l ct ukKqk2δεkε∆ujεδ$ lk fi (u(ta , x` )) ci ujKqφ(u(ta , x` )) z(ta , x` )kKqk2δεjεδ$ i
lk fi (u(ta , x` )) i
δ$x` j kKq(1kφ(u(ta , x` )) z(ta , x` ))k2δεjεδ$ n
0
if δ is small enough. This contradiction proves our claim.
Next we consider the existence problem. We make use of the iterative method in [3, 5] to solve it. It is evident that the viscous system (1.4) is equivalent to 1
ct ukε∆u lk ci fi(u)jKqφ(u) z, 2
i
3
(4.2)
ct zjKφ(u) z l 0. 4
Define inductively (u(m)(t, x), z(m)(t, x)) (m l 0, 1, …) by the solutions of the Cauchy problems 1
ct u(m)kε∆u(m) lk ci fi(u(m−"))jKqφ(u(m−")) z(m−"), 2 3 4
i ct z(m−")jKφ(u(m−")) z(m−") l 0
(4.3)
subject to the initial data (u(m), z(m)) (t l 0, x) l (u , z ) (x). (4.4) ! ! Here u(−") should be read as zero and z(−") as z (x). Then the solutions of (4.3) and (4.4) ! can be given, by Duhamel’s principle, as 1
&! & c f (u " (s, y)) E (tks, xky) ds dy jKq & & φ(u " (s, y)) z " (s, y) E (tks, xky) ds dy ! j& u ( y) E (t, xky) dy, ! " l z (x) exp 0kK & φ(u " (s, x)) ds1 , ! ! t
u(m) lk
Rn i t
i i
(m− )
(m− )
ε
(m− )
Rn
2 3
(4.5)
ε
Rn
t
z(m− ) 4
ε
(m− )
where Eε(t, x) is the heat kernel
0 QxQ4εt#1
satisfying
&
Rn
Eε(t, x) l (4πεt)−n/# exp k
(4.6)
Qcαx Eε(t, x)Q dx l Cα(εt)−QαQ/#, t 0,
(4.7)
α l (α , … , αn) is a multi-index, QαQ l α j…jαn, and Cα is a constant depending " " only on α and C l 1. !
480 Set
(m) l u(m+")ku(m), w(m) l z(m+")kz(m).
(4.8)
Then we have, from (4.5),
&! & c [ f (u (s, y))kf (u " (s, y))] E (tks, xky) ds dy jKq & & [φ(u (s, y)) z (s, y)kφ(u " (s, y)) z " (s, y)] !
1
(m) lk
t
t
2
(m)
i i
Rn i
(m− )
i
(m)
(m)
ε
(m− )
(m− )
(4.9)
Rn
3
:Eε(tks, xky) ds dy
9 0 &! φ(u
w(m−") l z (x) exp kK ! 4
Noting that if x , x " #
t
1
0 &! φ(u
(m)(s, x)) ds kexp kK
t
1: .
(m−")(s, x)) ds
0, Qex"kex#Q
we get
&! Qφ(u KL & Q !
Qw(m−")Q K
t
(m)(s, x))kφ(u(m−")(s, x))Q ds
t
φ
Therefore
Qx kx Q, " #
(m−")(s, x)Q ds.
Rw(m−")RL_(Π ) KLφ t
&! R t
(m−")R
_
L (Πs)
ds.
(4.10)
Integrating by parts in (4.9), we arrive at
&! &
1
t
(m)(t, x) l 2 3
[ fi(u(m)(s, y))kfi(u(m−")(s, y))] ci Eε(tks, xky) ds dy
Rn i
4
jKq
&! & t
[φ(u(m)(s, y)) z(m)(s, y)kφ(u(m−")(s, y)) z(m−")(s, y)] Rn
:Eε(tks, xky) ds dy.
This gives, through the use of (4.7) and (4.10), 1
Q(m)(t, x)Q
&! & t
Q fi(u(m)(s, y))kfi(u(m−")(s, y))Q Qci Eε(tks, xky)Q ds dy
Rn i
jKq 2 3
&! & t
Rn
oQφ(u(m)(s, y))kφ(u(m−")(s, y))Q z(m)(s, y)
jφ(u(m−")(s, y)) Qz(m)(s, y)kz(m−")(s, y)Qq:Eε(tks, xky) ds dy
4
C
R "& ! t
(m−")R
_
((tks)−"/#j1) ds,
L (Πs)
481
if 0 t T for any bounded T 0, that is, R(m)RL_(Π ) C t
&! R t
"
(m−")R
((tks)−"/#j1) ds,
_
L (Πs)
(4.11)
where C l C (K, q, Lf, Lφ, T ). Hence, for 0 t T, " " R(m)RL_(Π ) Ru RL_(Rn) CTm tm/#Γ ! t
" , 0mj2 2 1 −
(4.12)
if CT C (NTj1) Γ(1\2), where Γ(s) is the gamma function. " In fact, (4.12) is true when m l 0, and since
&! (tks) " # s t
− /
m/# ds l t(m+")/#Γ
" mj3 , Γ0 0121 Γ 0mj2 2 1 2 1 −
the estimate (4.12) follows inductively ; see [3, 5]. The consequence is that _
R(m)RL_(Π ) t m=!
_,
(4.13)
if 0 t T. This implies that the sequence ou(m)q_ has a uniform limit in any strip m=! 0 t T, and so does oz(m)q_ from (4.10). That is, m=! lim u(m)(t, x) l u(t, x),
lim z(m)(t, x) l z(t, x),
m
m
_
(4.14)
_
in 0 t T for any bounded T 0. Now our task is to prove that ou(t, x), z(t, x)q defined in (4.14) is the classical solution of (1.4) and (1.2). Obviously, owing to (4.5), u(t, x) and z(t, x) satisfy u(t, x) l
&! & t
fi(u(s, y)) ci Eε(tks, xky) ds dy
Rn i
jKq
&! & t
Rn
&
φ(u(s, y)) z(s, y) Eε(tks, xky) ds dyj
Rn
u ( y) Eε(t, xky) dy, ! (4.15)
0 &! φ(u(s, x)) ds1 .
z(t, x) l z (x) exp kK !
t
(4.16)
Differentiating (4.5) with respect to xj ( j l 1, … , n), we obtain
&! & c f (u " (s, y)) c E (tks, xky) ds dy jKq & & φ(u " (s, y)) z " (s, y) c E (tks, xky) ds dy ! j& c u ( y) E (t, xky) dy. !
cj u(m)(t, x) lk
t
Rn i t
(m− )
i i
(m− )
Rn
Rn
j
ε
j
ε
(m− )
j
ε
(4.17)
482 It follows by using (4.7) that Rcj u(m)RL_(Π ) C t
&! Rc u t
#
j
(m−")R
_
(tks)−"/# dsjC , $
(4.18)
L (Πs)
where C l C (Lf, ε) and C l C (ε, K, q, Rcj u RL_(Rn)). Therefore, we obtain in# # $ $ ! ductively from (4.18) that for all m Rcj u(m)RL_(Π ) 2C eC% T, $ t
if C is so large that %
C
&! s " #e T
#
− / −C% s ds
(4.19)
". #
(4.20)
Noting that ct Eε l ε∆Eε, we get
& &
Rn
Rn
Qcj(Eε(t, xj∆x)kEε(t, x))Q dx C(ε) t−("+θ)/#Q∆xQθ, Qcj(Eε(tj∆t, x)kEε(t, x))Q dx Cg (ε) t−("+#τ)/#Q∆tQτ,
(4.21)
for 0 θ 1, 0 τ ", where ∆x ? Rn and ∆t ? R, C(ε) and Cg (ε) are constants # depending only on ε. Therefore we conclude that cj u(m) is uniformly locally Ho$ lder continuous in ΠT with respect to x for any exponent 0 θ 1, and with respect to t for any exponent 0 τ ", respectively, since the right-hand sides of (4.21) are # integrable in t. Using (4.14) and the Arzela–Ascoli theorem, one can obtain cj u(m)(t, x)
cj u(t, x),
j l 1, … , n,
(4.22)
uniformly in every compact subset of ΠT because u(m) converges to u(t, x) uniformly. Hence it follows from (4.5) that cj z(m)(t, x)
cj z(t, x),
j l 1, … , n.
(4.23)
With almost the same method, we get uniform bounds for ∆u(m) by differentiating (4.17) with respect to xj and using the uniform boundedness of cj u(m), cj z(m) and fi (u(m)). We can further obtain the uniform local Ho$ lder continuity of ∆u(m). Therefore, as in (4.22), it holds that ∆u(m)
∆u
(4.24)
uniformly in every compact subset of ΠT. Hence (u(t, x), z(t, x)) is a classical solution of (1.4) and (1.2). Such a classical solution is unique. As a matter of fact, assume (u , z ) and (u , z ) " " # # to be two solutions of (1.4) with the same initial data (u , z ), and denote ug l u ku , ! ! " # zg l z kz ; then, with the same arguments leading to (4.10) and (4.11), we obtain " # Rug RL_(Π ) C t
&! Rug R t
L
_
(Π
((tks)−"/#j1) ds, Rzg RL_(Π ) KLφ s) t
&! Rug R t
_
L (Πs)
ds. (4.25)
This results in ug l zg 0. Thus we prove the following theorem. T 4.2. There exists a unique classical solution u(t, x) and z(t, x) to (1.4) and (1.2) in any strip ΠT.
483
5. The existence of the entropy solution to (1.1) and (1.2) We will follow [4, 5] to prove that there exists a solution of (1.1) and (1.2) in the sense of Definition 2.1 which is the limit of solutions to (1.4) and (1.2) as the viscosity ε vanishes. To this end, we need to derive an a priori estimate of the modulus of continuity in L"-space of the viscous solutions independent of ε. The following lemma is necessary. L 5.1. For τ 0, γ 0, the equation L(g) l ct gj Ai(t, x) ci gjγ∆g l 0
(5.1)
i
admits a solution g(t, x) in Πτ such that g(τ, x) l g (x) ? C _(ox ; QxQ rq) and ! ! Qg(t, x)Q Rg RL_(Rn) expoγ−"[(CγjH ) (τkt)jrkQxQ]q, (5.2) ! where Ai(t, x) are bounded differentiable with bounded deriaties, C is some constant depending only on n, and H l 1j sup
(t,x)?Πτ
0 QA Q#1" #. /
i
(5.3)
i
The one-dimensional version of this lemma can be found in [5]. Proof of Lemma 5.1. The existence follows exactly from Theorem 4.2 by the setting of τg l τkt. Now we prove the remaining parts. Take Φ l γ log Rg RL_(Rn)jrkQxQ, Ψ l 1jΦ M χ, where M represents convolution, ! 0 χ 1 is a mollifier satisfying χ ? C _(Rn), Rn χ dx 1 and χ l 0 outside the unit ! ball. Then Φ Ψ Φj2 and Q∆ΨQ C for some C depending only on n. Denote h(t, x) l expoγ−"[(CγjH ) (τkt)jΨ(x)]q ;
(5.4)
then L(hpg) l ct hj Ai(t, x) ci hjγ∆h
0
1
i − " l γ kHkCγj Ai(t, x) ci Ψj (ci Ψ)#jγ∆Ψ h 0. i i
(5.5)
Therefore, by the maximum principle, pg expoγ−"[(CγjH ) (τkt)jΨ(x)]q expoγ−"[(CγjH ) (τkt)jΦj2]q l Rg RL_(Rn) expoγ−"[(CγjH ) (τkt)jrkQxQ]q. ! This proves (5.2).
In what follows, we derive the uniform estimate of solutions to (1.4) and (1.2) for the L"-modulus of continuity, which is independent of ε. L 5.2. Denote I(g, y) l and assume that
&
Rn
Qg(xjy)kg(x)Q dx,
R]u RL_(Rn), R]z RL_(Rn) ! !
y ? Rn, _.
(5.6) (5.7)
484
Then, for any bounded T 0, it holds that I(u(t, :), y)jI(z(t, :), y) C(I(u(0, :), y)jI(z(0, :), y)),
(5.8)
where C depends only on q, K, Lφ and T. Proof. First, we claim that R]u(t, :)RL"(Rn), R]z(t, :)RL"(Rn)
(5.9)
C,
where C l C(ε, K, q, Lf, Lφ, Ru RL_(Rn), R]u RL"(Rn), R]z RL"(Rn), T ). ! ! ! In fact, from (4.5), we have
&
Rn
Q]z(m−")(t, x)Q dx
&
R
Q]z (x)Q dxjKLφ ! n
&! & t
Rn
Q]u(m−")(s, x)Q dx ds.
(5.10)
While we obtain from (4.17)
&
Rn
& )& f (u " (s, y)) c u " (s, y) ]E (tks, xky) ds dy) dx jKq & & oQφh(u " (s, y)) ]u " (s, y) z " (s, y)Q
Q]u(m)(t, x)Q dx
Π
T
Rn i
i
(m− )
i
(m− )
Π
(m− )
ε
(m− )
(m− )
n T R
jQφ(u(m−")(s, y)) ]z(m−")(s, y)Qq Eε(tks, xky) ds dy dx
& &
j
Rn Rn
QEε(t, xky)Q Q]u ( y)Q dy dx. !
Using the uniform boundedness of u(m−"), z(m−"), fi (u(m−")) and φ(u(m−")), we obtain R]u(m)(t, :)RL"(Rn) C
&! [R]u t
(m−")(s, :)R " n L (R )
jR]z(m−")(s, :)RL"(Rn)] [(tks)−"/#j1] dsjR]u RL"(Rn), ! where C l C(ε, K, q, Lf, Lφ). It follows through the use of (5.10) that R]u(m)(t, :)RL"(Rn) Cg (C, T )
&! R]u t
(5.11)
(m−")(s, :)R " n [(tks)−"/#j1] ds L (R )
jC` (R]u RL"(Rn), R]z RL"(Rn), T ). ! ! Since o]u(m)q_ converges to ]u, we conclude by Fatou’s lemma that R]u(t, :)RL"(Rn) m=! C(Cg , C` , R]u RL"(Rn), R]z RL"(Rn)), for all 0 t T. Furthermore, this together with ! ! (5.10) shows that R]z(t, :)RL"(Rn) has similar bounds for 0 t T. These prove our claim (5.9). Set (u , z ) l (u(t, xjy), z(t, xjy)). Then the differences l u ku and w l z kz " " " " satisfy 1
ct j ci(ai(t, x) ) l ε∆jKq(b(t, x) z jφ(u) w), " i
4
3
2
0 &!
w l z (xjy) exp kK !
t
1
0 &!
φ(u (s, x)) ds kz (x) exp kK " !
t
1
φ(u(s, x)) ds ,
(5.12)
485
where f (u )kfi(u) φ(u )kφ(u) " ai(t, x) l i " , b(t, x) l . (5.13) u ku uku " " It follows by the assumption on fi(u) and φ(u) that ai(t, x) (i l 1, … , n) have bounded continuous first order derivatives and b(t, x) is bounded. Multiplying the first equation of (5.12) by g(t, x), which is the solution ct gj ai(t, x) ci gjε∆g l 0
(5.14)
i
with g(τ, x) ? C _(x ; QxQ R), integrating by parts and using Lemma 5.1 and the decay ! property of g, one can get
&
Rn
&
(τ, x) g(τ, x) dx l
Rn
(0, x) g(0, x) dx
jKq
&! & τ
Rn
[b(t, x) z jφ(u) w] g(t, x) dx dt, "
(5.15)
from which it follows that
&
Rn
(
(τ, x) g(τ, x) dx Rg(τ, :)RL_(Rn) I(u(0, :), y) jKq max(1, Lφ)
&! [I(u(t, :), y)jI(z(t, :), y)] dt* . τ
(5.16)
Here we use the fact that Qg(0, x)Q Rg(τ, x)RL_(Rn) by the maximum principle for (5.1). Since L_(Rn) is the dual of L"(Rn) and C _(Rn) is dense in L_ endowed with ! ωM-topology, we obtain from (5.16) I(u(τ, :), y) I(u(0, :), y)jKq max(1, Lφ)
&! [I(u(t, :), y)jI(z(t, :), y)] dt. τ
(5.17)
Similar arguments for the second equation of (5.12) give I(z(τ, :), y) I(z(0, :), y)jK max(1, Lφ)
&! [I(u(t, :), y)jI(z(t, :), y)] dt. τ
(5.18)
The summation of (5.17) and (5.18) is I(u(τ, :) y)jI(z(τ, :), y) I(u(0, :), y)jI(z(0, :), y) jK(qj1) max(1, Lφ)
&! [I(u(t, :), y)jI(z(t, :), y)] dt. τ
(5.19)
This gives the proof of Lemma 5.2 by Gronwall’s inequality, the boundedness of R]u(t, :)RL"(Rn) and R]z(t, :)RL"(Rn), and the inequality
&
Rn
Qg(xjy)kg(x)Q dx R]gRL"(Rn)Q yQ.
(5.20)
486
To prove the existence of a weak entropy solution of (1.1) and (1.2), we need to obtain the L"-continuity of the solutions of (1.4) and (1.2) with respect to t independent of ε. L 5.3. Assume the conditions of Lemma 5.2. Then, for all 0 t
&
QxQ
R
&
Qu(tjδt, x)ku(t, x)Q dx wR(δt),
QxQ
R
tjδt T,
Qz(tjδt, x)kz(t, x)Q dx wR(δt), (5.21)
where wR(δt) is a continuous nondecreasing function, being independent of ε and satisfying wR(0) l 0. Proof. Set p(x) l uε(tjδt, x)kuε(t, x),
1
ph(x) l 23 4
sign (x), QxQ Rjh, 0, otherwise,
(5.22)
for δt 0 and the regularization of ph(x) gh(x) l
&
Rn
ph(xkhy) χ( y),
0
h ? R,
(5.23)
where 0 χ ? C _(ox ? Rn ; QxQ 1q) satisfying Rn χ( y) dy l 1. Then QghQ ! C\h and Q∆ghQ C\h# ; here C only depends on the dimension n. Since
&
Rn
&
Q p(x)Q dxk
Rn
p(x) gh(x) dx l
&&
Rn×Rn
1, Q]ghQ
( p(xkhy)kp(x)) ph(xkhy) χ( y) dx dy, (5.24)
it follows by Lemma 5.2 that
&
Rn
Q p(x)Q dx
)&
Rn
)
p(x) gh(x) dx jC (I(u(0, :), h)jI(z(0, :), h)). &
(5.25)
Using Fubini’s theorem, the smoothness of ct u and the first equation of (1.4), and integrating by parts, we calculate
)&
Rn
) )& # g (x) p(x) dx)j2 & # p(x) dx )& g (x) & c u(τ, x) dτ dx)jC h % # l )& & g (x) 9k c f (u)jKqφ(u(τ, x)) z(τ, x)jε∆u(τ, x): dx dτ)jC h % # l )& & 9 f (u(τ, x)) c g (x)jKqφ(u(τ, x)) z(τ, x) g (x) # jεu(τ, x) ∆g (x): dx dτ)jC h % gh(x) p(x) dx
h
QxQ
h
QxQ
R+ h t+δt
t
t
h
QxQ R+ h t t+δt QxQ
t
R QxQ r+ h
R+ h
t+δt
R+ h
i
i
i
i i
i
h
h
δt:C
$Q
sup oQgQjQ]gQjQ∆gQqjC h %
xQ R+#h
δt:C (1jC(n)\hjC(n)\h#)jC h. $ %
h
487
It follows by Lemma 5.2 and (5.25) that
&
Rn
QQ dx δt:C (1jC(n)\hjC(n)\h#)jC hjC (I(u(0, :), h)jI(z(0, :), h)). $ % & (5.26)
The L"-modulus of continuity of z(t, x) with respect to t is rather simple. By Fubini’s theorem, from the second equation of (1.4), we have
&
QxQ
& )& l& )&
Qz(tjδt, x)kz(t, x)Q dx l R
t+δt
QxQ
R
t t+δt
QxQ
R
t
)
ct z(s, x) dt dx
K:CR:δt, where CR is the measure of ox ; QxQ
)
kKφ(u(s, x)) z(s, x) ds dx (5.27)
Rq. The proof of this lemma is complete.
On the basis of these lemmas, we can show the following theorem. T 5.1. Let u , z ? C _(Rn) such that ! ! ! ]u , ]z ? L"(Rn) and R]iu RL_(Rn), R]iz RL_(Rn) _, i l 1, 2, 3. (5.28) ! ! ! ! Then there exists a weak entropy solution to the Cauchy problem (1.1) and (1.2). Proof. The uniqueness has been proved in Theorem 3.1. Here we only need solve the existence problem. Let uε(t, x) and zε(t, x) be the global solution of (1.4) and (1.2) in any strip ΠT for a bounded T 0. Lemma 5.2 guarantees that ouεqε and ozεqε ! ! are precompact in L"loc(Rn) for 0 t T. Lemma 5.3 shows that ouεqε and ozεqε are ! ! equi-continuous from [0, T ] to L"loc(Rn). By the Arzela–Ascoli theorem, we conclude that there exists a subsequence oεkq_ of oε ; ε 0q such that ouεkq_ and ozεkq_ k=" k=" k=" " converge uniformly to u(t, :) and z(t, :) in Lloc-norm, respectively, as k tends to " (Rn)). Therefore the pair (u(t, :), z(t, :)) is a infinity, where u(t, :), z(t, :) ? C([0, T ];Lloc weak solution and satisfies the initial data almost everywhere. With the same procedure as in Section 2, we can prove this solution is entropy. T 5.2. Let u (x), z (x) ? L_(Rn), then there exists a weak solution to (1.1) ! ! and (1.2). Proof. The finite propagation speed property of solutions in Theorem 3.1 shows that it suffices to prove the existence problem with u , z ? L_(Rn) having compact ! ! support. If u , z ? L_(Rn), Qu Q M, Qz Q 1, the regularization and a cutoff far away gives ! ! ! ! sequences ouh(x)q and ozh(x)q, with Quh(x)Q M and Qzh(x)Q 0, which converge to u (x) ! ! ! ! ! and z (x) in L"loc(Rn) respectively. Also we have I(uh, y) I(u , y) and I(zh, y) ! ! ! ! I(z , y). ! From the proof of the lemmas and theorems above, we can conclude that the solutions of (1.1) and (1.2) with the initial data (uh, zh) (x) is equi-continuous from ! ! [0, T ] to L"loc(Rn). The same proof as in Theorem 5.1 gives the conclusion of this theorem.
488
Acknowledgements. The author thanks Professor Tong Zhang for helpful discussions and Professor Matania Ben-Artzi for bringing the paper [5] to his attention. This work was supported by the China Postdoctoral Science Foundation, a Postdoctoral Fellowship from the Hebrew University of Jerusalem, and a Fellowship of IAN, Department of Mathematics, Otto-von-Guericke-Magdeburg-University. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
R. A, Sobole spaces (Academic Press, 1975). R. C and K. O. F, Supersonic flow and shock waes (John Wiley, New York, 1948). L. H$ , ‘ Non-linear hyperbolic differential equations ’, lecture notes, University Lund, 1988. S. K, ‘ First-order quasilinear equations with several space variables ’, Mat. Sb. 123 (1970) 228–255 (Russian) ; Math. USSR Sb. 10 (1970) 217–273 (English). A. L, ‘ On Majda’s model for dynamic combustion ’, Comm. Partial Differential Equations 13 (1992) 657–698. D. L. L, T. P. L and D. C. T, ‘ Stability of strong detonation travelling waves to combustion model ’, J. Math. Anal. Appl. 201 (1996) 516–531. J. Q. L, ‘ Stability of viscous strong detonation fronts in several dimensions ’, preprint, Institute of Mathematics, Academia Sinica, Beijing, 2000. T. P. L and L. A. Y, ‘ Nonlinear stability of strong detonation waves for a viscous combustion system ’, SIAM J. Math. Anal. 26 (1995) 519–528. T. P. L and T. Z, ‘ A scalar combustion model ’, Arch. Rational Mech. Anal. 114 (1991) 257–312. A. M, ‘ A qualitative model for dynamic combustion ’, SIAM J. Appl. Math. 41 (1981) 70–93. L. A. Y and Z. H. T, ‘ Riemann problem for a reacting and convection hyperbolic system ’, Approx. Theory Appl. 1 (1984) 95–122. P. Z, ‘ The two-dimensional Riemann problem for the simplest combustion model ’, PhD Thesis, Institute of Mathematics, Academia Sinica, Beijing, 1996.
Institute of Mathematics Academia Sinica Beijing 100080 China
