On time analyticity of weak solutions of the compressible Navier–Stokes equations

Physica D 229 (2007) 97–103 www.elsevier.com/locate/physd

On time analyticity of weak solutions of the compressible Navier–Stokes equations Eugene Tsyganov Max-Planck Institute for Mathematics in the Sciences, Inselstrasse 22, Germany Received 17 June 2005; received in revised form 22 January 2007; accepted 13 February 2007 Available online 18 April 2007 Communicated by R. Temam

Abstract We show that weak solutions of the compressible Navier–Stokes equations are analytic as L 2 -valued functions of time when pressure p is real analytic. As a consequence, there exist infinitely many effective viscous flux functions of higher orders. c 2007 Elsevier B.V. All rights reserved.

Keywords: Compressible Navier–Stokes equations; Weak solutions; Time analyticity

1. Introduction We study global weak solutions of the Navier–Stokes equations vt − u x = 0, u t + p(v)x = ε

(1.1) u  x

v

(1.2) x

conjecture that the results of this paper will also hold in the case of nonmonotonic pressure, though different techniques might be necessary. The pressure p is closely related to a stored-energy function W (v) by means of the equation W 0 (v) = − p(v) + c,

with Cauchy data v(·, 0) = v0 ∈ L 2 ([0, 1]),

0 < M −1 ≤ v0 ≤ M;

u(·, 0) = u 0 ∈ L 2 ([0, 1]),

(1.3) (1.4)

and the boundary conditions u(0, t) = u(1, t) = 0.

(1.5)

Here v is the specific volume, u is the velocity, ε is the viscosity constant which is positive. Time t is assumed to be nonnegative and x ∈ [0, 1]. The pressure p(v) is to satisfy the following assumptions: p 0 < 0,

lim p(v) = +∞,

v→+0

lim p(v) ≤ 0.

v→+∞

(1.6)

Remark. The assumption p 0 (v) < 0 is used only to simplify the derivation of global weak solution of (1.1)–(1.5). We E-mail addresses: [email protected], [email protected]. c 2007 Elsevier B.V. All rights reserved. 0167-2789/$ - see front matter doi:10.1016/j.physd.2007.02.015

with c to be a positive constant such that W ≥ 0 for all positive values of v. We assume that p decreases fast enough at v = +0 to guarantee limv→+0 W (v) = +∞. The other limit limv→+∞ W (v) = +∞ follows immediately from (1.6). We say that (v, u) is a weak solution of (1.1)–(1.6) provided that infx∈[0,1] v(·, t) > 0 and v(·, t) ∈ C([0, ∞); L 2 ([0, 1])); u(·, t) ∈ C([0, ∞); L 2 ([0, 1])) ∩ C((0, ∞); H 1 ([0, 1])) and for every test function ϕ ∈ C 1 ((−∞, ∞) × [0, 1]), ψ ∈ C01 ((−∞, ∞) × [0, 1]), and for all times t2 > t1 ≥ 0, t2 Z Z Z 1 t2 1 vϕdx − [vϕt − uϕx ] dx dt = 0, t1 0 0 t1 t2 Z Z Z 1 t2 1 uψdx − [uψt + p(v)ψx ] dx dt 0 t1 0 t1

98

E. Tsyganov / Physica D 229 (2007) 97–103 t2

Z

1

Z

=− 0

t1

ε

ux ψx dx dt. v

Definition. We say that a weak solution (v, u) of (1.1)–(1.5) is time analytic if there exist an open neighborhood D of (0, ∞) in C and analytic functions vc , u c : D → L2 ([0, 1]) (L2 denotes the space of complex-valued L 2 functions) such that (vc (·, t), u c (·, t)) ≡ (v(·, t), u(·, t)) for t ∈ [0, ∞). We point out that if such vc , u c exist, then they are unique. We are concerned with weak solutions that possess additional regularity properties: 0 < M −1 ≤ v(x, t) ≤ M; (1.7) Z tZ 1 Z 1 (1 ∧ s)u˙ 2 (x, s)dx ds ≤ C; (1 ∧ t) u 2x (x, t)dx + 0

0

0

(1.8) (1 ∧ t)2

1

Z

u˙ 2 (x, t)dx +

0

Z tZ 0

0

1

(1 ∧ s)2 u˙ 2x (x, s)dx ds ≤ C, (1.9)

where (1 ∧ t) = min{1, t} and u˙ = ∂u ∂t . We formulate our main result concerning time analyticity of weak solutions in the following theorem. Theorem 1.1. Let (v, u) be a weak solution of (1.1)–(1.6) with regularity properties (1.7)–(1.9). If p is real analytic, then (v, u) is time analytic. In addition, u c x , p(vc ), uvccx , (ε uvccx − p(vc ))x are also time analytic and the following equalities hold:   uc x vct = u c x , u ct = ε − p(vc ) . (1.10) vc x Moreover, the following sequence uc x − p(vc ), F0 = ε vc uc x F0x x F1 = ε − ε 2 u c x − p 0 (vc )u c x , vc vc   n d vct Fn = n ε − p(vc ) dt vc with Fm−1 x x substituted for for vct , satisfies dn vc = Fn−2 x x , dt n

dm+1 v , dt m+1 c

n ≥ 2, t > 0.

(1.11)

The paper is motivated by the result of Masuda [1] concerning analyticity of solutions of the incompressible Navier–Stokes equations. As a consequence, the solutions possess backward uniqueness property. Despite the fact that the proof is quite simple (see Foias and Temam [2]), backward uniqueness finds a number of important applications, some of which are given in Constantin, Foias, Kukavica, Majda [3]. In the paper, we show that weak solutions of the equations of one-dimensional compressible gas dynamics are time analytic when pressure p is real analytic. The interest of the proof is that it relies on the same techniques as that for the existence. An immediate application of our results is that if the flow of compressible gas reaches a state of rest at some positive time, then it is identically at rest. A sequence of equalities (1.12) is of particular importance. As shown in Hoff [7], the effective viscous flux F = ε uvx − p(v) possesses better regularity than its components. The equalities in (1.12) prove that there are cancelations of discontinuities among higher order derivatives of v and u. In addition, the time analyticity approach allows us to derive a whole sequence of equalities (1.12), which is not feasible by the standard energy estimates. The paper is structured as follows: in Section 2 we use a semidiscrete difference scheme to generate approximate solutions. Then we prove that these approximate solutions are holomorphic in an open subset D of C containing (0, ∞) and show that D is independent of the number of mesh points in the scheme. In Section 3, we take the limit as the discretization parameter tends to zero. The proof of Theorem 1.2 concludes the paper. 2. Difference approximations In this section we construct holomorphic solutions to the semidiscrete difference scheme that approximates (1.1)–(1.5). Let N be a positive integer number and h be an increment in x given by h = N1 . We set xn = nh for n = 0, 1, 2, . . . , N and x j = j h, j = 21 , 32 , . . . , N − 21 . Define the difference operator δ by δwl =

m = n, . . . , 1, and u c x

(1.12)

Remark. As a consequence of (1.12), we have that Fn ∈ H 2,∞ ([0, 1]). We will use Theorem 1.1 to prove backward uniqueness of weak solutions. Theorem 1.2. Let (v1 , u 1 ) and (v2 , u 2 ) be two weak solutions of (1.1)–(1.6) satisfying the regularity estimates (1.7)–(1.9) and pressure p be real analytic. Suppose that there is t0 ≥ 0 such that (v1 (·, t0 ), u 1 (·, t0 )) = (v2 (·, t0 ), u 2 (·, t0 )). Then (v1 (·, t), u 1 (·, t)) = (v2 (·, t), u 2 (·, t)) for any t ≥ 0.

wl+1/2 − w j−1/2 . h

We then use the following semidiscrete scheme to generate holomorphic functions v j (t) and u n (t): 1 3 1 j = , ,..., N − , 2 2 2   δu u˙ n + δpn = εδ , n = 1, 2, . . . , N − 1, v n

v˙ j − δu j = 0,

u 0 (t) = u N (t) = 0,

u n (0) = u ∗n ,

v j (0) = v ∗j .

(2.1) (2.2) (2.3)

Here p is assumed to be real analytic. It is also implied that v j and u n depend on h, but we will drop the superscript h throughout the section. Initial values v ∗j , u ∗n are assumed to satisfy 0 < M0−1 ≤ v ∗j ≤ M0 ;

(2.4)

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E. Tsyganov / Physica D 229 (2007) 97–103

X

u ∗n 2 h +

n

X

W (v ∗j )h ≤ CC0 ,

(2.5)

j

where C0 =

1 2

1

Z 0

1

Z

u 20 (x)dx +

W (v0 (x))dx.

0

For technical reasons we will also assume that X δvn∗ 2 h ≤ K .

(2.6)

n

Throughout the rest of the section, (v j , u n ) will stand for the holomorphic functions defined in D. The open set D given in Lemma 2.2 depends on the parameter N . So our next objective is to find D which is independent of N . Lemma 2.3. There exists a positive constant θ0 < π/2 and functions τ, C: (0, ∞) → (0, ∞) independent of K such that for any N ∈ N, solutions v j , u n of (2.1)–(2.3) constructed in Lemma 2.2 are holomorphic in

In the end we will let K → ∞. First we solve (2.1)–(2.3) for t ∈ R+ . The existence is given in the following Lemma.

D = {t = seiθ + a : θ ∈ (−θ0 , θ0 ), 0 ≤ s < τ (b), a ≥ b > 0}

Lemma 2.1. There exist positive constants M and C independent of K , and functions v j , u n satisfying (2.1)–(2.3) for t ∈ R+ with the following properties:

and the following estimates hold X X |δu j (seiθ + a)|2 h + |u˙ n (seiθ + a)|2 h

v j , u n : R → R; +

(2.7)

0 < M ≤ v j ≤ M; (2.8) Z tX X X u n (t)2 h + W (v j (t))h + δu 2j (s)h ds ≤ CC0 ;

(2.13)

0

j

+

X

δu 2j (t)h +

(1 ∧ t)2

u˙ 2n (t)h +

n

X

δvn2 h ≤ C K .

(1 ∧ s)

X

0

j

X

1

Z

0

u˙ 2n (s)h ds ≤ CC0 ; (2.10)

n 1

Z

(1 ∧ s)2

0

X

X

|δ u˙ n (ςeiθ + a)|2 h dς ≤ C(b);

(2.14)

h

|δvn (seiθ + a)|2 h ≤ C(b)K ,

(2.15)

n

j

(2.9) (1 ∧ t)

s

Z

−1

n

n

j

X

δ u˙ 2j (s)h ds ≤ CC0 ;

j

where seiθ + a ∈ D, a ≥ b > 0, and u˙ n (seiθ + a) stands for ∂ iθ ∂s u n (se + a). In addition, there is a positive constant Mc independent of K such that Mc−1 ≤ Re(v j (t)),

|v j (t)| ≤ Mc

(2.16)

(2.11)

for any t ∈ D and j = 12 , 32 , . . . , N − 12 .

(2.12)

Proof. First, define an open set Ω :

n

The derivation of the energy estimates in the case of Navier–Stokes equations of multidimensional compressible spherically symmetric gas flow can be found in Hoff [6]. Remark. The estimates (2.8)–(2.12) also hold in the case of nonmonotonic pressure. The proof in this case can rely on the ideas presented in Ducomet and Zlotnik [5] and in Tsyganov [4]. In the next lemma, we show that v j , u n can be continued into an open subset of C holomorphically.

Ω = {z ∈ C : distC (z, [M −1 , M]) < λ}, where λ is a positive constant which is considered to be small so that Ω is in the domain of p, and M to be the same as in (2.8). Fix a > 0, θ ∈ (−π, π ). The key to proving the lemma is the derivation of the estimates for ! X iθ 2 A(s) = sup |δu j (ςe + a)| h (2.17) 0≤ς ≤s

j

and !

Lemma 2.2. There is an open neighborhood D of (0, ∞) in C such that v j , u n can be continued holomorphically from R+ into D.

B(s) = sup

Proof. Since p is real analytic, it can be extended into an open neighborhood of (0, ∞) in C holomorphically. Thus, for any t0 ∈ R+ and initial data (v j (t0 ), u n (t0 )), we can solve (2.1), (2.2) in an open ball B(Rt0 , t0 ) ⊂ C to obtain local solution (vc j , u cn ). When restricted to B(Rt0 , t0 ) ∩ R+ , (vc j , u cn ) solve (2.1)–(2.3) with t ∈ [0, ∞). Therefore, (vc j , u cn ) coincide with (v j , u n ) on R+ due to the uniqueness of the solutions. We can now conclude that vc j , u cn are holomorphic in an open set D in C containing (0, ∞). 

0

X

0≤ς≤s s

Z +

|u˙ n (ςe + a)| h iθ

2

n

X

|δ u˙ n (ςeiθ + a)|2 h dς,

(2.18)

n

when v j ∈ Ω . From (2.2) we have that   δu iθ iθ u˙ n + e δpn = εe δ . v n

(2.19)

We multiply (2.19) by the conjugate u˙¯ n , sum over n, integrate with respect to s. Then we add the equality to its conjugate to

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E. Tsyganov / Physica D 229 (2007) 97–103

arrive at Z sX Z 2 |u˙ n |2 h dς − 0

s

Xh

0

n s

Z = −ε

" X

0

j

0

Therefore, Z X |δu j (seiθ + a)|2 h + j

0

h dς.

0

n

Z 0

X

X

|δ u˙ j |2 h dς + C

s

Z

X

0

j

|δu j |3 h dς,

j

(2.21)

j

and, therefore, !1/2 |δu j | h + C

j

X

2

|δu j | h

j

!1/2 X

2

|u˙ n | h

.

n

(2.22) We combine (2.20) with (2.22) to arrive at Z sX A(s) + |u˙ n |2 h dς ≤ C A(0) + C B(s)1/2 + C + (C

−1

n

+ C(θ ))

s

Z 0

X j

δu j vj

n



δ u˙¯ j + e−iθ



s

δ u¯ j v¯ j





δ u˙ j h dς. s

|δ u˙ j |2 h dς

X

s

Z

2

|δu j | h dς + C 0

j

X

|δu j |4 h dς.

j

This together with (2.22) leads to Z sX B(s) ≤ C B(0) + C |δu j |2 h dς 0 s

Z +C 0

j

!3 X

2

|δu j | h

dς

(2.24)

j

when v j ∈ Ω . Then the sum of (2.23) and (2.24) gives us Z s A(s) + B(s) ≤ C A(0) + C B(0) + C A3 (ς ) dς, (2.25) 0

n

j

2

s 0

n

Also, from (2.19) we obtain that |δ F j |2 ≤ |u˙ n |2 . These two inequalities give us !1/2 !1/2 X X X 2 2 2 2 |u˙ n | h |F j | ≤ C |δu j | h + C |δu j | h

0

e

j

B(s) ≤ C B(0) + C

j

|δu j | ≤ C

iθ



Z

|u˙ n |2 h dς ≤ C A(0) + C

to estimate the last term on the right side. Firstly, X X |F j |2 h ≤ C |δu j |2 h + C.

X

X

The term on the right side is equal to  iθ  Z sX e e−iθ |δ u˙ j |2 h dς −ε + vj v¯ j 0 j # Z s X" 2 2 iθ (δu j ) ˙ −iθ (δ u¯ j ) +ε e δ u¯ j + e δ u˙ j h dς. v 2j v¯ 2j 0 j

δu j − pj vj

2

s

= −ε

when v j is in Ω . Here C denotes a generic constant which depends on Ω only, C(θ ) → 0 as θ → 0. We will exploit the effective viscous flux F

j

(2.23)

We now differentiate (2.19) with respect to s, multiply by u˙¯ n h, sum over n, integrate with respect to s, and add it to its conjugate. The resulting equality is Z s Xh i X 2 |u˙ n (seiθ + a)|2 h + eiθ δ p˙ u˙¯ n + e−iθ δ p˙¯ u˙ n hς

#

(2.20)

Fj = ε

dς.

|δu j | h

j

Thus s 0

+ (C −1 + C(θ ))

2

j

The integral on the right side is equal to  iθ  s e 1 X e−iθ |δu j |2 − ε + h 2 j vj v¯ j 0   Z sX iθ e−iθ 1 e h dς δ u˙ j δ u¯ j − + ε 2 0 j vj v¯ j   −iθ Z sX eiθ 1 e h dς + ε δ u˙¯ j δu j − 2 0 j v¯ j vj " # Z sX iθ e−iθ v˙¯ j 1 2 e v˙ j − ε + h dς. |δu j | 2 0 j v 2j v¯ 2j

s

X

+C

i eiθ p j δ u˙¯ j + e−iθ p¯ j δ u˙ j h dς

δu j δ u˙¯ j δ u¯ j δ u˙ j eiθ + e−iθ vj v¯ j

Z

!3

s

Z

which means that there is positive τ depending on A(0), B(0), and a positive constant C depending on Ω , such that A(s) + B(s) ≤ C(A(0) + B(0))

(2.26)

when 0 ≤ s ≤ τ and as long as v j , u n exist and v j ∈ Ω . It follows from (2.1), (2.22), and (2.26) that |v j (seiθ + a) − v j (a)| Z s ≤ |δu j (ςeiθ + a)| dς ≤ Cτ (A(0) + B(0))

(2.27)

0

as long as v j ∈ Ω and s ≤ τ . Thus the combination of (2.8)– (2.11), (2.26), and (2.27) prove that there are θ0 , τ, C as in the statement of Lemma 2.3, and D as in (2.13) such that v j , u n are holomorphic in D for any N and that (2.14) and (2.16) hold. The proof of (2.15) requires several other standard energies estimates. The details in the case of spherically symmetric Navier–Stokes equations and real time are given in Hoff [6]. 

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E. Tsyganov / Physica D 229 (2007) 97–103

3. Time analytic weak solution We construct approximate analytic solutions vch , u ch as follows: vch (x j , t) = v j (t) for j = 12 , . . . , N − 12 , u ch (xn , t) = u n (t) for n = 0, . . . , N , and take vch (·, t) and u ch (·, t) to be linear on each interval [x j , x j+1 ] and [xn , xn+1 ] respectively. The values of vch (t) on [0, x 1 ] and [x N − 1 , 1] are set to be equal 2 2 v 1 and v N − 1 respectively. Throughout the rest of the section 2 2 we will omit c from subscripts. To show that v h , u h : D → L2 ([0, 1]) are analytic, we take a circle γ ⊂ D. For an arbitrary x ∈ [0, 1], there are 0 ≤ λ1 , λ2 ≤ 1 and j0 , n 0 such that v h (x, t) = λ1 v j0 + (1 − λ1 )v j0 +1 , Holomorphic functions v j , u n satisfy Z Z v j (ς ) 1 1 u n (ς ) v j (t) = dς, u n (t) = dς 2πi γ t − ς 2πi γ t − ς

1 2πi

Z γ

u h (x, ς ) dς. t −ς

+ 0

0

+ 0

2

+ a)| dx ≤ C(a);

2 Z 1 ∂ h u (x, seiθ + a) dx ∂s 0 2 Z sZ 1 ∂ h iθ + ∂s u x (x, ςe + a) dx dς ≤ C(a); 0 0 Z 1 |vxh (x, seiθ + a)|2 dx ≤ C(a)K , 0

(3.7)

strongly in L2 ([0, 1]) as h l → 0; (3.8)

strongly in L2 ([0, 1]) as h l → 0.

(3.9)

Proof. Let sequences {ak } and {τk } be such that the set D˜ = {t, t¯ : t = seiθ0 + ak , s ∈ [0, τk ]}

(3.1)

is dense in D. Firstly, due to the estimates in (3.4) and (3.5), we can find a sequence h = {h l } such that u hl (·, t) converges ˜ Our next step strongly in L2 ([0, 1]), say to u(·, t) when t ∈ D. h l is to show that u converges for any t ∈ D. For an arbitrary t ∈ D, there is a sequence {tk } ⊂ D˜ such that tk → t as k → ∞ and Im(tk ) = Im(t). Then Z 1 Z 1 |(u h 1 − u h 2 )(x, t)|2 dx ≤ |(u h 1 − u h 2 )(x, tk )|2 dx 0

0

(3.2)

1

Z +

0 Z 1

+

|u h 1 (x, t) − u h 1 (x, tk )|2 dx |u h 2 (x, t) − u h 2 (x, tk )|2 dx

0 1

Z ≤

|(u h 1 − u h 2 )(x, tk )|2 dx

0

2 + C|t − tk | ∂s (x, s) dx ds 0 t !1/2 2 Z tk Z 1 h 2 ∂u + ∂s (x, s) dx ds

(3.3)

1/2

t

1

Z |u hx (x, seiθ

0

2

(1 ∧ s)2 u˙ hx dx ds ≤ CC0 .

u hx − u x → 0

0 1

(1 ∧ s)u˙ dx ds

The proof is immediate from (2.14)–(2.16). We are now in a position to extract convergent subsequences as h → 0.

for any t ∈ D and x ∈ [0, 1]. In addition, there exists a positive function C : (0, ∞) → (0, ∞) independent of K such that Z 1 |u h (x, seiθ + a)|2 dx Z

1

+

Lemma 3.1. Time analytic function v h satisfies the following pointwise estimates: |v h (x, t)| ≤ Mc

h2

0

Z tZ

Below we state a number of regularity properties of v h and u h .

Mc−1 ≤ Re(v h (x, t)),

1

u h − u, v h − v → 0

and the same equality holds for u h : u h (x, t) =

0

0

Z tZ

Lemma 3.2. There is a sequence h = {h l } → 0 and functions v(·, t), u(·, t) ∈ L2 ([0, 1]) such that for all t ∈ D

u h (x, t) = λ2 u n 0 + (1 − λ2 )u n 0 +1 .

when t is inside γ . Therefore Z v j0 (ς ) 1 dς v h (x, t) = λ1 2πi γ t − ς Z v j0 +1 (ς ) 1 + (1 − λ1 ) dς 2π i γ t − ς Z h v (x, ς ) 1 = dς 2πi γ t − ς

when t = seiθ0 + a ∈ D and θ = ±θ0 . For positive values of t we also have that Z 1 Z 1 2 h2 2 u˙ h dx u x dx + (1 ∧ t) (1 ∧ t)

≤ (3.4)

(3.5) (3.6)

Z

tk

Z

1 ∂u h 1

0

|(u h 1 − u h 2 )(x, tk )|2 dx + C|t − tk |1/2 ,

0

which shows that u hl is Cauchy for any t ∈ D. In order to prove that v h (·, t) is also convergent, we need to exploit the properties of the effective viscous flux F. Firstly, we set F h (x j , t) = F j (t) and let F h be linear on each interval [x j , x j+1 ]. Then from (2.19), (2.21), and (3.5), Z 1 Z 1 h 2 |Fx (t, x)| dx ≤ C |u˙ h (x, t)|2 dx ≤ C(b), 0

0

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E. Tsyganov / Physica D 229 (2007) 97–103

where t = seiθ + a ∈ D and a ≥ b > 0. Also, by (2.21) and (2.22), Z sZ 1 | F˙ h (x, ςeiθ + a)|2 dx dς 0

0 s

Z

Z

1

|u˙ hx (x, ςeiθ + a)|2 dx dς

≤C 0

0 sZ 1

Z +C 0

0 s

Z

1

Z

+C 0

0

|u hx (x, ςeiθ + a)|4 dx dς |u hx (x, ςeiθ

2

+ a)| dx dς ≤ C(b),

where a ≥ b > 0, 0 < s ≤ τ , and θ = ±θ0 or θ = 0. The above estimates allow us to extract a further subsequence of {h lm } and find a function F(·, t) ∈ L2 [0, 1] such that F hlm → ˜ Then F uniformly on ([0, 1]) × X for any compact X ⊂ D. there is a positive constant C depending on D such that Z 1 |(v h 1 − v h 2 )(x, seiθ + ak )|2 dx 0 X ≤C |(ln(v h 1 ) − ln(v h 2 ))(x j , seiθ + ak )|2 h j

≤C

X

|(ln(v ) − ln(v ))(x j , ak )| h h1

h2

2

j s

Z +C

X

0

j s

Z +C

X

0 1

Z

|( p(v h 1 ) − p(v h 2 ))(x j , ςeiθ + ak )|2 h dς |(F h 1 − F h 2 )(x j , ςeiθ + ak )|2 h dς

j

|(v h 1 − v h 2 )(x, ak )|2 dx

≤C 0

Z

s

Z

s

Z

1

+C 0

Z

|(v h 1 − v h 2 )(x, ςeiθ + ak )|2 dx dς

0 1

+C 0

|(F h 1 − F h 2 )(x, ςeiθ + ak )|2 dx dς,

0

where θ = ±θ0 or θ = 0, 0 < s ≤ τk , and the sums are taken over the union of nodes for v h when h = h 1 , h 2 . We can conclude now that Z 1 |(v h 1 − v h 2 )(x, seiθ + ak )|2 dx 0 1

Z

|(v h 1 − v h 2 )(x, ak )|2 dx

≤C 0

s

Z

1

Z

+C 0

|(F h 1 − F h 2 )(x, ςeiθ + ak )|2 dx dς

0

and, therefore, Z 1 |(v h 1 − v h 2 )(x, seiθ + ak )|2 dx 0 1

Z ≤ C(k) Z

0 s

|(v h 1 − v h 2 )(x, 0)|2 dx

Z

+C 0

0

1

|(F h 1 − F h 2 )(x, ςeiθ + ak )|2 dx dς,

which shows that v hlm (·, t) is Cauchy in L2 ([0, 1]) as h lm → 0 ˜ An argument similar to the one given for u hl can for any t ∈ D. now be applied to show that v hlm (·, t) converges in L2 ([0, 1]) for any t ∈ D. This concludes the proof of (3.8). The proof of (3.9) is standard and it requires only minor modifications to the one given in Hoff [6]. We, therefore, will omit it, stressing only that it relies on the estimates (3.3)– (3.5).  We now take limits in (3.1) and (3.2) as h → 0 to show that Z v(·, ς ) 1 dς, v(·, t) = 2πi γ t − ς Z 1 u(·, ς ) u(·, t) = dς in L2 ([0, 1]). (3.10) 2π i γ t − ς Lemma 3.3. Analytic functions v, u in (3.10) is a weak solution of (1.1)–(1.5) with Cauchy data v0 ∈ L 2 ([0, 1]),

kv0 k H 1 ≤ K ,

0 < v ≤ v0 ≤ v, ¯

u 0 ∈ L ([0, 1]). 2

In addition, (v, u) satisfies the regularity estimates (1.7)–(1.9) and equalities (1.10)–(1.12). Proof. First we prove (1.12). From (2.1) and (2.2) we obtain that   v˙ j (3.11) v¨ j = δδ ε − p(v j ) . vj j Then we multiply (3.11) by the values of a test function ψ ∈ C03 ([0, 1]) at x = x j and sum over j to arrive at  X X  v˙ j v¨ j ψ(x j )h = ε − p(v j ) δδψ j h. vj j j We differentiate the last equality with respect to t n times to obtain  X dn+2 X  dn+1 v j ψ(x j )h = G v j , . . . , n+1 v j δδψ j h, dt n+2 dt j j (3.12) where G is a smooth function of its arguments. Then  Z 1  dn+1 G v h (t, x), . . . , n+1 v h (t, x) ψx x dx dt 0  X Z x j +1/2  dn+1 h h = G v (t, x), . . . , n+1 v (t, x) ψx x dx dt x j −1/2 j   X dn+1 = G v j , . . . , n+1 v j ψx x (x j )h + O(h) dt j   Z x j +1/2   dn+1 h h × G v (t, x), . . . , dt n+1 v (t, x) ψx x dx x j −1/2 x  X  dn+1 = G v j , . . . , n+1 v j ψx x (x j )h + O(h) dt j

103

E. Tsyganov / Physica D 229 (2007) 97–103

by (3.6). Therefore, the right-hand side of (3.12) converges to R1 dn+1 0 G(v(t, x), . . . , dt n+1 v(t, x))ψx x (x)dx. A similar argument R 1 n+2 shows that the left-hand side approaches 0 dtd n+2 v(t, x)ψ(x)dx as h → 0. Thus    dn+2 dn+1 v(t, x) = G v(t, x), . . . , n+1 v(t, x) dt n+2 dt xx for any n ≥ 0 which proves (1.12). A similar argument shows that constructed (v, u) is a weak solution of (1.1)–(1.5) and that (1.10) holds. Regularity estimates (1.7)–(1.9) follow immediately from (3.7) and Z 1 Z 1 iθ 2 |u x (x, seiθ + a)|2 dx ≤ C(a); |u(x, se + a)| dx + (3.13) 2 ∂ u(x, seiθ + a) dx ∂s 0 2 Z sZ 1 ∂ u x (x, ςeiθ + a) dx dς ≤ C(a); + ∂s 0 0 1

1

Z

kvk x k L 2 ≤ k,

¯ 0 < v ≤ vk ≤ v. Then we apply Lemma 3.3 to find analytic solution (v k , u k ) of (1.1)–(1.5) with initial data (vk , u 0 ). We notice that by Lemma 3.4 and Cauchy integral formula for holomorphic n functions, sequences { dtd n v k (·, t)}, n ≥ 0 and {u k (·, t)} are n Cauchy in L2 ([0, 1]) for any t ∈ D. Therefore, { dtd n v k (·, t)} n and {u k } strongly converge to time analytic functions dtd n v, u. This will prove (1.12) and that (v, u) is a weak solution of (1.1)–(1.5) with regularity estimates (1.7)–(1.9). Then the uniqueness result of Hoff and Tsyganov [8] for weak solution of Navier–Stokes equations completes the proof of Theorem 1.1.

0

0

Z

lim kvk − v0 k L 2 ([0,1]) = 0,

k→∞

|vx (x, seiθ + a)|2 dx ≤ C(a)K

(3.14) (3.15)

0

follow from (3.4)–(3.6).



Lemma 3.4. For any compact set X ⊂ D there is a constant C = C(X ) independent of K such that for any two weak solutions of (1.1)–(1.5) from Lemma 3.3 with initial data (v10 , u 10 ) and (v20 , u 20 ) respectively, we have that Z 1 Z 1 2 |v1 (t, x) − v2 (t, x)| dx + |u 1 (t, x) − u 2 (t, x)|2 dx 0

0

≤ C(X )

1

Z

|v10 (x) − v20 (x)|2 dx

0 1

Z +

! |u 10 (x) − u 20 (x)| dx 2

(3.16)

0

for t ∈ X . The proof of (3.16) for real and positive values of t can be found in Hoff and Tsyganov [8]. Then the inequalities (3.13), (3.14) together with straightforward L 2 -estimates of the differences v1 − v2 and u 1 − u 2 prove (3.16) for all values of t ∈ X. Now for an arbitrary v0 ∈ L 2 ([0, 1]), 0 < v ≤ v0 ≤ v¯ we can find a sequence {vk } such that

4. Proof of Theorem 1.2 Let (vi , u i ), i = 1, 2 be as in the statement of Theorem 1.2. If (v1 (·, t0 ), u 1 (·, t0 )) = (v2 (·, t0 ), u 2 (·, t0 )) for some t0 ≥ 0 then (v1 (·, t), u 1 (·, t)) = (v2 (·, t), u 2 (·, t)) for any t ≥ t0 . The proof of this is elementary when t0 > 0. In the case t0 = 0, the proof is given in Hoff and Tsyganov [8]. Then time analytic functions (vc1 , u c1 ) and (vc2 , u c2 ) coincide on [t0 , ∞). Therefore, they are identical in their domain of definition. Thus (v1 (·, t), u 1 (·, t)) = (v2 (·, t), u 2 (·, t)) for any t ≥ 0. References [1] K. Masuda, On the analyticity and the unique continuation theorem for solutions of the Navier–Stokes equation, Proc. Japan Acad. 43 (1967). [2] C. Foias, R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier–Stokes equations, J. Math. Pures Appl. 58 (1979) 339–368. [3] P. Constantin, C. Foias, I. Kukavica, A. Majda, Dirichlet quotients and 2D periodic Navier–Stokes equations, J. Math. Pures Appl. (9) 76 (2) (1997) 125–153. [4] E. Tsyganov, Asymptotic behavior and dynamic stability of phase mixtures for the equations of Navier–Stokes with nonmonotonic pressure, ZAMP (in press). [5] B. Ducomet, A.A. Zlotnik, Remark on the stabilization of a viscous barotropic medium with a nonmonotonic equation of state, Appl. Math. Lett. 14 (8) (2001) 921–928. [6] D. Hoff, Spherically symmetric solutions of the Navier–Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indiana Univ. Math. J. 41 (4) (1992) 1225–1302. [7] D. Hoff, Global Solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations 120 (1) (1995) 215–254. [8] D. Hoff, E. Tsyganov, Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys. 56 (5) (2005) 791–804.