SELFSIMILAR SOLUTIONS OF SZND MODEL

In this section, we study the selfsimilar solutions for the SZND-model.

Indeed, let !=xt, if solution (u, p, {, q) for (SZND) depends only on !, then it satisfies the following equations:

!u$&p$=0, (3.1)

!{$+u$=0, (3.2)

!E$&(up)$=0, (3.3)

!q$=k.(T ) q, (3.4)

where

.(T)=

{

The Riemann data becomes

(u, p, {, q)(&)=(u&, p&, {&, 0), (3.6) and

(u, p, {, q)(+)=(u+, p+, {+, Q). (3.7) When !{0 and the solution is smooth, then (3.1)t(3.4) can also be expressed as

{$=& #&1

#p&!²{q$, (3.8)

p$=&!²{$, (3.9)

q$=k

!.(T ) q, (3.10)

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where #p&!²{{0. If #p&!²{=0, then we require q$=0. For any !0# R¹, we may supply (3.8)t(3.10) with the initial conditions

{(!0)={0>0, p(!0)=p0>0, q(!0)=q00. (3.11) When solution (u, p, {, q)(!) for the SZND-model has a discontinuity at

!=', it then satisfies the RankineHugoniot condition,

(RH): '[u]=[ p], (3.12)

'[{]=&[u], (3.13)

[e+q]+p_r+p_l

2 [{]=0, (3.14)

where [w]=w_r&w_l, w_r=w('+0), and w_l=w('&0).

By direct computation, we can determine (u_l, p_l, {_l) in terms of (u_r, p_r, {_r), ' and [q] as follows.

p_l=1

2[(1&+²)('²{_r+p_r)]+1

2[((1&+²) '²{_r&(1++²) p_r)²+8+²'²[q]]^12, (3.15)

{_l={_r+( p_r&p_l)'², (3.16)

u_l=u_r+1

'( p_l&p_r). (3.17)

When [q]=0, (3.15)t(3.17) can be simplified to

p_l=(1&+²) '²{_r&+²p_r, (3.18) {_l=(1++²) p_r'²++²{_r, (3.19) u_l=u_r+(1&+²) '{_r&(1++²) p_r'. (3.20) Since state (&) is burnt, we may consider the solution (u, p, {, q) is also burnt in (&, 0), i.e. q(!)=0 for any ! # (&, 0]. Then (3.4) is automati-cally satisfied, and (3.1)t(3.3) is the classic, non-combustion equation.

Therefore, for any data at !=0,

{(0)={₀>0, p(0)=p₀>0, u(0)=u₀, q(0)=0. (3.21) Then (3.1)t(3.3), (3.6) and (3.21) yield unique solutions, (see Chapter 3 in [12]). Therefore, the (SZND) Riemann problem is reduced to finding suitable initial data (u0, p0, {0, 0) at !=0 such that solutions for (3.1)t(3.4), (3.7) and (3.21) exist in (0, ). Hence, it is necessary to study

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the initial-value problem (3.8)t(3.11) when !00. We first study the existence of global smooth (continuous) solutions in (!0, ), and then study solutions with discontinuities at some '>!0. The solutions for (3.1)t(3.4) depend on k( >0). For simplicity we shall omit the depen-dence of k wherever such omission does not cause confusion.

We first have the following simple observation.

Lemma 3.1. Let q(!) be the solution of (3.10) in (!₀, !1) with q(!₀)=q00. Then:

(i) If T>T_iin (!₀, !1] and ' # (!0, !1], then

q(!)=q(')

\

+

(ii) If TT_iin (!0, !1), then

q(!)=q0 in (!0, !1). (3.23) However, if T(!₀)=T_i, then we may have a non-unique solution since . is discontinuous at T=T_i. This is clarified below.

For more efficient study of (3.8)(3.11), it is convenient to divide (R⁺)³ into different regions, then study the equations for each region separately.

Definition 3.2. On (R⁺)³, denote as

G=[({, p, !) : #p&!²{>0], I⁰=[({, p, !) : p{=T_i], H=[({, p, !) : #p&!²{<0], I⁺=[({, p, !) : p{>T_i], P=[({, p, !) : #p&!²{=0], I^&=[({, p, !) : p{<T_i], G⁺=G & I⁺, etc...

Therefore, (R⁺)³consists of 4 open regions, G⁺, G^&, H⁺, H^&, 4 surfaces, P⁺, P^&, G⁰, H⁰and one curve, P⁰.

For simplicity, denote the solution of (3.8)t(3.10) as

X(!)=({(!), p(!), !), (3.24)

with initial condition

X(!₀)=X₀#({0, p₀, !₀). (3.25) We have the following demonstration of solution existence and uniqueness for all regions except G⁰ and P⁰.

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Proposition 3.3. Assume k>2. We then have:

(I) (i) If X₀# H^&, then X(!)=X0 in (!₀, ).

(ii) If X₀# H⁰, then X(!)=X0 in (!₀, ).

(iii) If X₀# H⁺, then there exists !₁>!₀ such that X(!) # H⁺ in (!₀, !₁] and X(!₁) # H⁰.

(II) (i) If X0# P^&, then there exists a !1>!0 such that

p(!)=C₁!^(2#)(#+1) and {(!)=C₂!(&2)(#+1) (3.26) for ! # (!0, !1] and X(!1) # P⁰, for some positive constants C1 and C2.

(ii) If X₀# P⁺ and q₀>0, then there exists !1>!0 such that X(!) # H⁺ in (!₀, !₁).

(III) (i) If X0# G^&, then there exists !1>!0such that X(!)=X0in (!0, !1] and X(!1) # P^&.

(ii) If X0# G⁺, then there exists a !1>!0such that X(!) # G⁺ in (!0, !1) and X(!1) # G⁰.

Proof. (I) (i) Since X0# H^&, T(!0)<T_i and #p0&!²₀{₀<0, (3.10) and (3.8) imply q$=0 and {$=0 for !>!0. Hence, X(!)=X0 for !>!0.

(ii) The proof is similar to that for (i).

(iii) When #p&!²{{0, we then have d

d!T(!)=&(#&1) k

! .(T ) p&!²{

#p&!²{} q. (3.27) If X(!) # H⁺, then (3.27) implies T$(!)<0. Furthermore, we also have

d

d!(#p&!²{)=&2!{&(#+1) !²{$<0

in H⁺. Therefore, !1>!0 exists such that T(!1)=T_i and X(!) # H⁺ in (!0, !1).

(II) (i) If X(!) # P^&, we then have

#p&!²{=0. (3.28)

For polytropic gas we also have

p{^#=C>0. (3.29)

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From (3.28) and (3.29), (3.26) follows. Since T(!)=C₁C₂!2((#&1)(#+1))

, (3.30)

T(!1)=T_ifor some !1>!0. Hence, X(!1) # P⁰ and X(!) # P^& in (!0, !1).

(ii) Since X₀# P⁺, !₀>0. If there exists !₁>!₀ such that X(!) # P⁺ in (!₀, !₁), i.e., (3.28) holds, then (3.30) implies T(!)>T_i. q₀>0 now implies q$>0 in (!₀, !₁), a contradiction, therefore the result holds.

(III) (i) Since X0# G^&, .(T )=0, thus X(!)=X0 in (!0, !1], q(!)>0. Since k>2, it can be verified by using (3.31) that X(!) can not stay at G⁺ forever. Thus, there exists !1>!0 such that X(!) # G⁺ in (!0, !1), and either X(!1) # G⁰, or X(!1) # P⁺. The latter case can be ruled out according to Theorem 3.1 [8]. The proof is complete.

Now it remains to study the problem of X0# P⁰or G⁰. X(!) # G⁺ is equivalent to T(!)>T_i which is guaranteed by (3.34). The proof is complete.

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Similarly, we have a result for X0# G⁰. Proposition 3.5. If X0# G⁰, then either

(i) X(!)=X₀in (!₀, !₁) with X(!₁) # P⁰, or

(ii) X(!) # G⁺ in (!0, !1) for some !1>!0, and (3.32)t(3.34) hold.

Proof. If (i) does not hold, then by an argument similar to that in Proposition 3.2(III)(ii) we can prove (ii), the details are omitted. The proof is complete.

The case of !0=0 is the most interesting to us, since we have to solve the Riemann problem in (0, ). When !0=0, we always assume (3.21), and then X0# G⁺_ G⁰. If X(!) # G⁺, for !>0, then (3.10) implies

q(!)=q~0!^k (3.35)

for !t0⁺, where q~00 is a parameter. In view of (3.35), we always have the freedom to chose q~0# [0, ) and it may be possible to find an appropriate q~0to fit the boundary conditions at some point '( ) when it is needed. This is stated more precisely below. The following proposition is very important in studying temperature of X(!) at G⁺, which was essen-tially proven in [8].

Proposition 3.6. If !0>0, X(!0) # G⁺ and

T$(!0)=0, (3.36)

then there exists !₁>!₀ such that X(!) # G⁺

T$(!)<0 in (!₀, !₁) and T(!₁)=T_i, (3.37) i.e., X(!₁) # G⁰.

Proof. By Theorem 3.2 in [8], there is no !>!0 such that X(!) # G⁺ and T$(!)=0. Therefore, T$(!)<0 as far as X(!) # G⁺. Now, according to Proposition 3.2 (III)(ii), there exists !1>!0 such that T(!1)=T_i. The proof is complete.

Remark 3.7. If !₀=0 and X₀# G⁺, then T$(0)=0, which does not contradict Proposition 3.5. Furthermore, according to Proposition 3.5, the temperature T(!) either strictly decreases or has exactly one maximum in (0, !₁) with X(!₁) # G⁰, where !₁ is the first ! such that T(!)=T_i. Summarizing the above results, we have the following diagram for con-structing the solutions starting at !00, see Fig. 6.

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Fig. 6. Routes in different regions.

Combining the results of Propositions 3.3, 3.4 and 3.5, we have the following global existence of solutions for (3.8)t(3.10).

Theorem 3.8. Given {(0)>0, p(0)>0 and q(0)=0, then there is a smooth (continuous) solution X(!) for (3.8)t(3.10) in (0, ). Furthermore, there is ! such that q(!)Q in (0, !) and q(!)=Q if !<.

Due to the non-uniqueness of solutions when X0# P⁰_ G⁰, we shall pay much more attention to the following simplest solutions which were con-sidered in [8].

Definition 3.9. A solution X(!) for (3.8)t(3.10) is called simple (i) if X(0) # G⁰_ G⁺, then X(!) can not jump to G⁺ in (0, !), (ii) if X(0) # G^&, then X(!) jumps exactly once in (0, !).

Otherwise, X(!) is called a non-simple solution.

Therefore, we have four types of simple solutions when q(!)Q; see Fig. 7. From (3.10), because q(!) never decreases, X(!) is a physical solu-tion of (SZND), and it is necessary that q(!)Q. It is possible to get a non-simple solution while qQ.

For type (i) solutions, in [8] Tan and Zhang considered q('0)=Q for some '₀>0. (3.8)t(3.10) are then equivalent to

{$=&k(#&1)

#p&!²{

\

+

p$=&!²{$, (3.39)

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Fig. 7. Typical diagrams for simple solutions.

where

q(!)=Q

\

+

Based on (3.38)t(3.40), Tan and Zhang were able to prove their results, the details are presented in the next section.