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Local and Global Estimates for the Solutions of Convection–Diffusion Problems
Journal of Mathematical Analysis and Applications 229, 137–157 (1999) Article ID jmaa.1998.6150, available online at http://www.idealibrary.com on
Local and Global Estimates for the Solutions of Convection–Diffusion Problems Moulay D. Tidriri* Iowa State University, Department of Mathematics, 400 Carver Hall, Ames, Iowa 50011-2064 Submitted by John Lavery Received May 21, 1997
In this paper we develop and study local and global estimates for the solutions of convection–diffusion problems, respectively, with Dirichlet–Neumann and Dirichlet boundary conditions. This study is based on a combination of Aleksandrov, Bakelman, and Pucci theory, with more standard traces estimates. © 1999 Academic Press
1. INTRODUCTION In this paper we shall develop local and global estimates for the solution of convection–diffusion problems, respectively, with Dirichlet–Neumann and Dirichlet boundary conditions. The derivation of such estimates is based on the use of the most recent methods for elliptic equations initiated by Aleksandrov, Bakelman, and Pucci together with more standard traces estimates. These estimates are independent on the low-order, “relaxation,” term. They can be applied, for example, to the analysis of an explicit time marching algorithm. This time marching algorithm was introduced and studied by Bourgat, Le Tallec, and Tidriri [4] Le Tallec and Tidriri [5, 6], and Tidriri [7–9], in order to couple different models and/or approximations, which results in an efficient strategy for solving problems in applied sciences and engineering. For the applications of these estimates to the analysis of this algorithm we refer to [6] and [7]. * This work has been supported by the Hermes Research program under Grant Number RDAN 86.1/3. The author was also supported by the National Science Foundation under Contract Number ECS-8957475 and by the United Technologies Research Center while he was at Yale University. E-mail address: [email protected]. 137 0022-247X/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
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In the next section we develop a maximum principle for a general secondorder elliptic problem based on the Aleksandrov, Bakelman, and Pucci theory. This local estimate is independent on the low-order, relaxation, term. In Section 3, we develop local and global estimates for the solutions of convection–diffusion problems with Dirichlet–Neumann boundary conditions. Finally, in Section 4 we develop local and global estimates for the Dirichlet boundary conditions case. 2. A MAXIMUM PRINCIPLE In this section we shall establish a maximum principle for an arbitrary elliptic operator of second order. This result is central to the development of the estimates studied in the next sections. Let L be an operator of the form, Lu = aij xDij u + bi xDi u + cxu; for any u in W 2; n , with a bounded domain of n . The coefficients aij , bi , and c; i; j = 1; : : : ; n are defined on . As usual, the repeated indices indicate a summation from 1 to n. We suppose that the operator L is strictly elliptic in in the sense that the matrix A of coefficients aij is strictly positive everywhere in . Let λ and 3 denote, respectively, the smallest and the largest eigenvalue of A. Let D denote the determinant of the matrix A and D∗ = D1/n . We have 0 < λ ≤ D∗ ≤ 3: We suppose in addition that the coefficients aij , bi , and c are bounded in , and that there exist two positive real numbers γ and δ such that 3 ≤ γ; λ 2 b ≤ δ: λ
(L is uniformly elliptic);
(1) (2)
Now, we are in a position to state the principal result of this section, proved in the Appendix. Theorem 2.1. Let u ∈ W 2; n and suppose that Lu ≥ f with f ∈ L and c ≤ 0. Then for all spheres B = B2R y of center y and radius 2R included in and for all p > 0, we have 1/p R 1 Z u+ p + f n; B ; (3) sup u ≤ CR B B λ BR y n
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139
where the constant CR depends on n; γ; δR2 ; p, but is independent of c, and u+ = maxu; 0. Remark 2.1. The statement of the same theorem can be found in [3], under the additional assumption, c ≤ δ: λ
(4)
However, there the constant CR depends indirectly on c through δ. That is exactly what we want to avoid, because we would like this constant to be independent of c (see next sections).
3. DIRICHLET–NEUMANN CASE Let loc be a connected domain of n with loc ⊂ (Fig. 1). The boundaries of the two subdomains are defined as follows, 0b = ∂ ∩ ∂loc (internal boundary); 0i = ∂loc ∩ (interface), 0∞ = ∂\0b (farfield boundary): We denote by n the external unit normal vector to ∂ or ∂loc . Let V be a given velocity field of an inviscid incompressible flow such that div V = 0; in ; (5) V · n = 0; on 0b :
FIG. 1. Description of the domain and its splitting.
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We shall derive an estimate for the solution of the following Dirichlet– Neumann problem, 1 Lv = −ν 1v + V · ∇v + v = 0; τ v = 0; ∂v = g; ∂n
in ;
(6)
on 0∞ ;
(7)
on 0b ;
(8)
where the function g is given in H −1/2 0b , the coefficient τ is strictly positive, and ν is the diffusion coefficient. Let W be the subspace of H 1 defined by (9) W = w ∈ H 1 w = 0 on 0∞ : We then define the following bilinear form on W , Z Z av; w = ν ∇v ∇w + divVvw;
v; w =
Z
vw:
(10) (11)
The first basic problem associated to (6)–(8), can be written as follows: Find v ∈ W satisfying Z 1 νgw d0; ∀w ∈ W: (12) av; w + v; w = τ 0b Moreover, we assume that the coefficients ν and τ satisfy the following relation, ντ ≤ 1:
(13)
This hypothesis is not necessary but simplifies the proofs to come. Moreover, it is not restrictive (see [6]). Let d denote the distance as described in Fig. 2. Let β be a real number such that √ 3 ν ; 0<β< d and set β k= √ : ν τ The first basic result states a global H 1 estimate of the solution of the first basic problem (12) in terms of the boundary data g.
local and global estimates
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FIG. 2. Description of the domain loc and the splitting used in the majorization of the local solution.
Lemma 3.1.
There exists a constant c0 such that v1; ≤ c0 g−1/2; 0b :
Proof of Lemma 3.1. Using (5) we obtain the following equality, Z 1Z v divVv = divVv2 2 1Z V · nv2 = 2 0 = 0;
∀ v ∈ W:
(14)
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Choosing w = v in (12), we then obtain Z Z 1 νgv: ν∇v2 + v2 = τ 0b
(15)
From this equality we deduce the following estimate, v21; ≤ g−1/2; 0b v1/2; 0b : The application of the trace theorem yields the estimate (14). Let i be the subdomain of width d/3 with external boundary 0i as described in Fig. 2. Let Ky = Bd/3 y be the sphere of center y and radius d/3. There exist y1 ; : : : ; yl belonging to i such that 2i =
[ y∈i
Bd/6 y ⊂
l [ j=1
Kyj :
We then define K by setting K=
l [ j=1
Kyj :
The next lemma states a local estimate of the solution v of the first basic problem (12). Lemma 3.2.
There exists a constant c1 such that v∞; K ≤ c1 v0; ;
(16)
where c1 is a constant depending only on ν; γ; δd 2 and 3/2dn/2 . Proof of Lemma 3.2.
The operator, L = −L
satisfies the assumptions of Theorem 2.1, with c = −1/τ and f = 0. Applying this theorem with p = 2, y ∈ i we obtain v∞; Ky ≤ c1 v0; B2d/3 y : Therefore, v∞; Ky ≤ c1 v0; ; where c1 is a constant depending only on ν, γ, δd 2 , and 3/2dn/2 . Applying (17) to each Kyj we obtain v∞; K ≤ sup c1j v0; : j=1;:::;l
(17)
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Setting c1 = supj=1;:::;l c1j , we finally have v∞; K ≤ c1 v0; :
(18)
This concludes the proof of the lemma. We shall now establish other local estimates for the solution v of the first basic problem. For any Mi in i ; we introduce (see Fig. 2): • Bi = the ball centered on Mi of radius d/6, • vi = expkr 2 − d 2 /36v∞; ∂Bi : We then have: Lemma 3.3.
The solution v of the first basic problem satisfies kd 2 ∀Mi ∈ i : v∞; ∂Bi ; vMi ≤ exp − 36
(19)
Proof of Lemma 3.3. The operator L applied to vi can be written in polar coordinates (with r = Mi M), k 1 2 2 v: Lvi = 4 −k νr − kν + V · er r + 2 4τ i Therefore,
k 1 2 2 − kν vi : Lvi ≥ 4 −k νr − V · er r + 2 4τ
(20)
ϕr; k = akr 2 + bkr + ck;
(21)
We then set
with ak = −k2 ν; k bk = − V · er ; 2 1 − kν: ck = 4τ We seek to satisfy the following relation, 0 ≤ inf ϕr; k;
for 0 ≤ r ≤
d : 6
As ϕr; k decreases on + , this will be satisfied if and only if d ≥ 0; ϕ 6
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i.e., if and only if kdV 1 k2 νd 2 − + − kν ≥ 0: 36 12 4τ We then replace k by its value. Therefore, we have to satisfy −
− √
βdV βν 1 β2 d 2 − − √ ≥ 0: √ + 36ντ 4τ ν τ 12ν τ
τ, it follows that dV β2 d 2 1 ≥β 1+ : √ 1− 9ν 12ν 4 τ √ The constraint β < 3 ν/d, finally yields after division, β2 d 2 −1 dV 1 1− : ϕr; k ≥ 0 if and only if √ ≥ β 1 + 12ν 9ν 4 τ
Multiplying by
(22)
From √ the relation (20) and the previous calculation, we deduce that for β < 3 ν/d and τ satisfying the previous inequality, we have Lvi ≥ 0 = Lv: In addition, by construction, vi ≥ v;
on ∂Bi :
Consequently, using the maximum principle we obtain v ≤ vi ; In particular,
in Bi :
kd 2 v∞; ∂Bi : vMi ≤ exp − 36
Repeating the same process for −v, we finally obtain kd 2 v∞; ∂Bi ; ∀Mi ∈ i : vMi ≤ exp − 36
(23)
Let ∞ be defined by ∞ = \ loc . The next result establishes an H 1 estimate of the solution v of the first basic problem on the domain ∞ . Lemma 3.4.
There exists a constant c2 such that 1/2 p V p d/c2 : v1; ∞ ≤ v∞; i c2 /d 1 + ν
(24)
local and global estimates Let ξ ∈ H 1 be such that ξ = 1; in ∞ ; supp ξ ⊂ i ∪ ∞ :
Proof of Lemma 3.4.
We have using (12),
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Z
−ν 1v + divVv +
v 2 ξ v = 0: τ
Using Green’s formula we deduce Z Z Z −ν 1vξ2 v = ν∇ξv2 − ν∇ξ2 v2 :
On the other hand, we have 2 2 Z Z Z Vξ v 2 divVvξ v = div − V · ∇ξξv2 : 2
(25)
(26)
(27)
Using the relations (26) and (27), (25) becomes 2 2 Z Z Vξ v ξ2 v2 2 + − νv2 ∇ξ2 + v2 ξV · ∇ξ ν∇ξv + div 0= 2 τ Z 1 Z Z − ν ξ2 v2 − νv2 ∇ξ2 + v2 ξV · ∇ξ = ν∇ξv2 + ξv2 + τ Z Z 1 Z 2 2 2 2 ν∇v + v + ν∇ξv + ξv + − ν ξ2 v2 = ∞ i τ Z − νv2 ∇ξ2 + v2 ξV · ∇ξ: i
Hence, we obtain νv21; ∞ + =
Z i
Z i
ν∇ξv2 + ξv2 +
Z 1 − ν ξ2 v2 τ
νv2 ∇ξ2 + v2 ξV · ∇ξ:
The relation (13) then yields Z νv2 ∇ξ2 + v2 ξV · ∇ξ νξv21; ∞ ∪i ≤ i
≤ v2∞; i
Z i
ν∇ξ2 + ξV · ∇ξ
≤ v2∞; i νξ21; i + ξ0; i V ξ1; i ξ0; i V 2 2 : ≤ v∞; i ξ1; i ν + ξ1; i
(28) (29)
(30)
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If we take ξ such that ξ0; i ≤ 1; c ξ21; i = 2 ; d where c2 is a constant, (30) then becomes 1/2 p V p d/c2 ; v1; ∞ ≤ v∞; i c2 /d 1 + ν
(31)
and this concludes the proof. Now we are in a position to state the main result of this section. Theorem 3.1. Let v be the solution of the first basic problem (12). If τ is sufficiently small, we have 1/2 p p 1 kd 2 exp − g−1/2; 0b ; v1/2; 0i ≤ C1 C2 /d 1 + V ∞ d/C2 ν 36 where C1 and C2 are constants, with C1 depending only on d, ν, γ, and δ, but not on τ. Proof of Theorem 3.1. The proof of this theorem is based on the preceding lemmas. Because ∂Bi ⊂ K, we have v∞; ∂Bi ≤ v∞; K ;
(32)
v∞; ∂Bi ≤ c1 v0; :
(33)
Lemma 3.2 then implies Using Lemma 3.3 and the foregoing estimate we obtain kd 2 c1 v0; ; ∀Mi ∈ i : vMi ≤ exp − 36 Consequently we have
v∞; i
kd 2 ≤ exp − c1 v0; : 36
Applying Lemma 3.1 we obtain v∞; i
kd 2 g−1/2; 0b : ≤ c1 c0 exp − 36
(34)
An application of Lemma 3.4 then yields 1/2 p V p kd 2 d/c2 exp − g−1/2; 0b : v1; ∞ ≤ c0 c1 c2 /d 1 + ν 36
(35)
(36)
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To conclude we use the trace theorem which yields v1/2; 0i ≤ c3 v1; ∞ : Consequently, we have the final estimate, 1/2 p V p kd 2 g−1/2; 0b ; d/c2 exp − v1/2; 0i ≤ c0 c1 c3 c2 /d 1 + ν 36 which corresponds to our theorem with C1 = c0 c1 c3 and C2 = c2 . 4. DIRICHLET–DIRICHLET CASE In this section we shall derive an estimate of the solution of the following Dirichlet problem, 1 Lv = −ν 1v + V · ∇v + v = 0; τ
in loc ;
(37)
v = h;
on 0i ;
(38)
v = 0;
on 0b ;
(39)
where the function h is given in H 1/2 0i , the coefficient τ is strictly positive, and ν is the diffusion coefficient. The velocity field V is given by (5). Let W be the subspace of H 1 loc defined by W = w ∈ H 1 loc w = 0 on 0b : We then define the following bilinear form in W ; Z Z ∇v:∇w + divVvw; av; w = ν loc
v; w =
Z loc
loc
vw;
(40) (41)
with v and w in W . The second basic problem associated to (37)–(39) corresponds to the following problem: Find v ∈ W such that Z ∂v 1 ν w; ∀w ∈ W; (42) av; w + v; w = τ 0i ∂n v0i = h;
(43)
where h is given in H 1/2 0i . We first have the following lemma: Lemma 4.1.
For τ sufficiently small, we have ν 1 aw; w + w; w ≥ w21; loc ; τ 2
∀w ∈ W:
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Proof of Lemma 4.1. Under the hypothesis 1/τ ≥ ν/2 + 1/2νV 2∞ , and using the Cauchy–Schwarz inequality, we obtain Z Z 1Z 1 ν ∇v:∇v + V · ∇vv + v2 av; v + v; v = τ τ loc loc loc 1 ≥ ν∇v20; 2 + v20; 2 − V ∞ ∇v0; 2 v0; 2 τ 1 ν 1 ≥ ν∇v20; 2 + v20; 2 − ∇v20; 2 − V 2∞ v20; 2 τ 2 2ν ν 2 2 ≥ ∇v0; 2 + v0; 2 : 2 We will make the simplifying assumption (13). We first establish a global estimate for the solution of the second basic problem. Lemma 4.2. satisfies
The solution v of the second basic problem (42) and (43)
1 + V 2∞ 1/2 h1/2; 0i : 1+ v1; loc ≤ 21 + τ ν2 Proof of Lemma 4.2. Choosing w = v in (42) we obtain Z Z Z ∂v 1 ∇v2 + ν h: divVvv + v2 = ν τ loc loc 0i ∂n Lemma 4.1 then yields
∂v
h1/2; 0i : v21; loc ≤ 2
∂n −1/2; 0i −2 1/2
(44)
(45)
(46)
We shall now establish an estimate of ∂v/∂n−1/2; 0i . Combining (42) and (5) we obtain Z Z ∂v 1 1 w= ∇v∇w + V · ∇vw + vw : ν ντ 0i ∂n loc Therefore, for any w in W , we have Z ∂v 1 0 ∂n w ≤ ∇v0; loc ∇w0; loc + ν V ∞ ∇v0; loc w0; loc i 1 + v0; loc w0; loc ντ 1/2 1 1 2 2 2 2 ≤ ∇v0; loc + 2 V ∞ ∇v0; loc + 2 v0; loc ν ν 1/2 1 2 2 2 × ∇w0; loc + w0; loc + 2 w0; loc τ 1 + V 2∞ 1/2 ≤ 1+ v1; loc 1 + τ−2 1/2 w1; loc : ν2
local and global estimates The trace theorem then yields
∂v 1 + V 2∞ 1/2 −2 1/2
≤ 1 + τ v1; loc : 1+
∂n ν2 −1/2; 0i Combining now the relations (46) and (47) we have 1 + V 2∞ 1/2 h1/2; 0i ; v1; loc ≤ 21 + τ−2 1/2 1 + ν2
149
(47)
(48)
and hence in particular, 1 + V 2∞ 1/2 h1/2; 0i : v0; loc ≤ 21 + τ−2 1/2 1 + ν2
(49)
Let Ky = Bd/4 y be the sphere centered on y and of radius d/4, with y belonging to 0V (see Fig. 3). By construction, 0V is the center surface of loc and i is the subdomain of width d/6 centered on 0V . We have the following lemma: Lemma 4.3.
There exists a constant c1 such that v∞; K ≤ c1 v0; loc :
Proof of Lemma 4.3. Lemma 3.2 we obtain
(50)
Following the same argument as in the proof of v∞; Ky ≤ c1 v0; loc ;
(51)
where c1 is a constant depending only on d, ν, γ, and δ. On the other hand there exist y1 ; : : : ; yl in i such that 2i =
[ y∈i
Bd/6 y ⊂
l [ j=1
Kyj = K:
By applying (51) to each Kyj , we obtain v∞; K ≤ sup c1j v0; loc = c1 v0; loc : j=1;:::;l
(52)
Next we shall establish another local estimate for the solution of the second basic problems (42) and (43). For any Mi ∈ i , we introduce (see Fig. 3). • a ball Bi centered on Mi and of radius d/6, • the function vi = exp kr 2 − d 2 /36v∞; ∂Bi .
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FIG. 3. Description of the local domain loc and the splitting used in the majorization of the global solution.
We then have: Lemma 4.4. satisfies
The solution v of the second basic problem (42) and (43) −kd 2 v∞; ∂Bi : vMi ≤ exp 36
(53)
Proof of Lemma 4.4. By definition of k (see the previous section), ϕr; k is positive for all r ∈ 0; d/6. Then by following the same argument as in the proof of Lemma 3.3 we obtain the inequality (53). Let b be the subdomain of loc described in Fig. 3. We establish next an H 1 global estimate of the solution of the second basic problem.
local and global estimates Lemma 4.5. satisfies
151
The solution v of the second basic problems (42) and (43)
v1; b ∪i ≤ v∞; i Proof of Lemma 4.5.
p
V ∞ p c2 /d 1 + d/c2 ν
1/2
:
Consider ξ ∈ H 1 loc , such that ξ = 1; in b ; supp ξ ⊂ b ∪ i :
Choosing w = ξ2 v in (42) we obtain Z v 2 ξ v = 0: −ν1v + divVv + τ loc
(54)
(55)
(56)
Similarly to the proof of Lemma 3.4 we obtain Z νv2 ∇ξ2 + v2 ξV · ∇ξ: νξv1; b ∪i ≤ i
Choosing ξ such that ξ0; i ≤ 1; and ξ21; i =
c2 ; d
we finally obtain as in the proof of Lemma 3.4 the inequality (54). Finally, the main result of this section is presented in the following theorem: Theorem 4.1. For τ sufficiently small, the solution v of Problem (42) and (43) satisfies
1/2 p
∂v 1 + V 2∞ V ∞ p
≤ C1 C2 /d 1 + d/C2 1+
∂n ν2 ν −1/2; 0b 1 kd 2 h1/2; 0i ; × 1 + 2 exp − (57) τ 36 where C1 and C2 are constants with C1 depending only on d, v, ν, and δ. Proof of Theorem 4.1. and 4.4 imply
Because ∂Bi ⊂ K by construction, Lemmas 4.3
kd 2 c1 v0; loc : v∞; i ≤ exp − 36
(58)
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Furthermore, using Lemma 4.2 then yields 1 + V 2∞ 1/2 kd 2 1 1/2 c exp − h1/2; 0i : (59) 1 + v∞; i ≤ 2 1 + 1 ν2 τ2 36 By using Lemma 4.5 we then obtain 1/2 1 + V 2∞ 1/2 p V ∞ p c1 c2 /d 1 + d/c2 v1; b ∪i ≤ 2 1 + ν2 ν 1 1/2 kd 2 × 1+ 2 exp − (60) h1/2; 0i : τ 36 Before concluding we shall establish an estimate of the term,
∂v
:
∂n −1/2; 0b Choosing w such that w ∈ H 1 loc ; and using (42) we obtain Z b
with w = 0 on ∂b ∩ ∂i ;
v w = 0: −ν1v + divVv + τ
Applying Green’s formula and using (5), we obtain Z Z ∂v 1 1 w= ∇v∇w + V · ∇vw + vw : ν ντ 0b ∂n b Similarly to the proof of Lemma 4.2 we obtain the following inequality,
∂v 1 1/2 1 + V 2∞ 1/2
≤ 1+ 2 v1; b : (61) 1+
∂n τ ν2 −1/2; 0b The completion of the proof of the theorem results from the combination of the relation (60) with (61). APPENDIX In this appendix we shall give the proof of Theorem 2.1 of Section 2. This proof relies on the notion of a contact set. If u is a continuous arbitrary function on , the upper contact set, denoted 0+ or 0+ u , is the subset of , defined by 0+ = y ∈ ; ∃py ∈ n such that ux ≤ uy + p · x − y ∀x ∈ : (62)
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We see that u is a concave function on if and only if 0+ = . When u ∈ C 1 we must have p = Duy in the relation (62). In addition, when u ∈ C 2 , the Hessian matrix D2 u = Dij u is negative on 0+ . In general, 0+ is closed in . If u is a continuous arbitrary function on , we define the “normal mapping” χy = χu y at point y ∈ by (63) χy = p ∈ n ; ux ≤ uy + p · x − y ∀x ∈ : We can see that χy is nonempty if and only if y ∈ 0+ . In addition when u ∈ C 1 , we have χy = Duy on 0+ ; in other words χ is the gradient field of u on 0+ . As a particular case of the Bakelman–Alexandrov ([1] and [2]) maximum principle, we have: Lemma 1.
¯ we have For u ∈ C 2 ∩ C 0 ,
ij
a Dij u d
sup u ≤ sup u +
+; 1/n D∗ ∂ nwn n; 0
with d the diameter of and wn the volume of a unit sphere in n . For more details see [3]. We now proceed to the proof of Theorem 2.1. We take Bˆ = B1 0 and the general case is deduced by considering the coordinate transform, x → xˆ = x − y/2R. We will begin, in a first step, by showing this result for u ∈ C 2 ∩ W 2;n and then in a second step we will deduce the result for u ∈ W 2; n . Step 1: We suppose that u ∈ C 2 ∩ W 2; n . For β ≥ 1, we consider the cutoff function η defined by ηx ˆ = 1 − x ˆ 2 β : By differentiation, we obtain ˆ i η = −2βxˆ i 1 − x ˆ 2 β−1 ; D ˆ ij η = −2βδij 1 − x ˆ 2 β−1 + 4ββ − 1xˆ i xˆ j 1 − x ˆ 2 β−2 : D By setting v = ηu; we then obtain ˆ ij v = ηaˆ ij D ˆ ij u + 2aˆ ij D ˆ j u + uaˆ ij D ˆ i ηD ˆ ij η aˆ ij D ˆ i u − cu ˆ i ηD ˆ ij η: ˆ j u + uaˆ ij D ˆ + 2aˆ ij D ≥ ηfˆ − bˆ i D
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ˆ Let 0+ = 0+ v be the upper contact set v, in the sphere B; we have u > 0;
on 0+ :
If x ∈ ∂Bˆ such that p · x − y < 0 we indeed have vx = 0. Consequently, vy + p · x − y ≥ vx = 0: Moreover, using the concavity of v on 0+ , we can estimate the following quantity ˆ = Du
1 ˆ − uDη: ˆ Dv η
Indeed, 1 Du ˆ + u Dη ˆ ˆ ≤ Dv η v 1 ˆ + uDη ≤ η 1 − x ˆ ≤ 21 + βη−1/β u: In that way, we have on 0+ the following inequality, ˆ −2/β + 2βbη ˆ −1/β + cˆ v + ηfˆ: ˆ ij v ≤ 16β2 + 2ηβ3η −aˆ ij D Because cˆ ≤ 0, we deduce the inequality, ˆ −2/β + 2βbη ˆ −1/β v + ηfˆ ˆ ij v ≤ 16β2 + 2ηβ3η −aˆ ij D ˆ −2/β v + fˆ; ≤ c1 λη ˆ independent of c. ˆ with c1 = cn; β; γ; δ ˆ we obtain, for β ≥ 2, Consequently, applying Lemma 1 on B, ˆ d 1 ˆ −2/β v + fˆn; Bˆ : sup v ≤ c1 λη 1/n ∗ ˆ D nwn Bˆ Using relation (2), this becomes ˆ ˆ d d 1 −2/β η v + fˆn; Bˆ c sup v ≤ ˆ 1 n; B 1/n 1/n λˆ nwn nwn Bˆ 1 ≤ c1 dˆ η−2/β vn; Bˆ + fˆn; Bˆ λˆ 1 ˆ −2/β + ˆ v n; Bˆ + f n; Bˆ ≤ c1 d η λˆ 1 ≤ c1 dˆ sup v+ 1−2/β u+ 2/β n; Bˆ + fˆn; Bˆ ; λˆ
(64)
local and global estimates
155
ˆ Here, dˆ is the where c1 is a constant depending only on n, β, γ, and δ. ˆ ˆ d = 2. diameter of B Using Young’s inequality in the form, ab ≤ εaq + ε−r/q br ; for q = 1 − 2/β−1 and r = β/2, we obtain β/2
sup v+ 1−2/β u+ 2/β n; Bˆ ≤ ε sup v+ + ε1−β/2 u+ 2/β n; Bˆ ;
∀ε > 0:
By taking ε = 1/2c1 dˆ and plugging in our inequality for v, we obtain sup v ≤ Bˆ
ˆ 1 1 1−β/2 ˆ β/2 u+ 2/β β/2 + c1 d fˆn; Bˆ : c1 d sup v+ + ˆ n; B ˆ 2 2 λ
(65)
We want to prove the theorem for all p > 0. We will treat separately the cases p ≤ n and p > n. If p ≤ n, we set β = 2n/p. In this case we have β/2
u+ 2/β n; Bˆ = u+ p; Bˆ : Plugging this in our inequality for v, we obtain ˆ 1 1−β/2 1 ˆ β/2 u+ p; Bˆ + c1 d fˆn; Bˆ : sup v ≤ c1 d 2 Bˆ 2 λˆ Consequently, we obtain the following inequality; Z 1/p dˆ ˆ f n; Bˆ : u+ p + sup v ≤ c2 Bˆ 2λˆ Bˆ On the sphere B1/2 0, the cutoff function satisfies β 1 1 ≤ : η 2 It follows, then v η B1/2 0
sup u ≤ sup
B1/2 0
≤ 2β sup v: Bˆ
Finally we obtain the desired estimate, Z 1/p dˆ ˆ f n; Bˆ : u+ p + sup u ≤ c3 Bˆ 2λˆ B1/2 0
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¯ The constant c3 in the previous formula depends for u in W 2;n ∩ C 2 . ˆ only on n, β, γ, and δ, but is independent of c. ˆ On the other hand if p > n, we have 2n < p; β
∀β ≥ 2:
Then, it follows (by assuming β ≥ 2), ˆ −1/p u+ p; Bˆ : ˆ −1/2n/β u+ 2n/β; Bˆ ≤ B B But β/2
u+ 2n/β = u+ 2/β n; Bˆ ; and therefore, by proceeding as before, we obtain the desired estimate, Z 1/p dˆ ˆ + p f n; Bˆ ; u + sup u ≤ c4 Bˆ 2λˆ B1/2 0 ¯ The constant c4 in the previous formula depends for u in W 2; n ∩ C 2 . ˆ but is independent of c. only on n, β, γ, and δ, ˆ Transformation xˆ → x. ˆ ij = R−2 Dij , thus λˆ = R−2 λ and δˆ = δR2 . In addition, By construction, D we have B = wn 2Rn and gp; Bˆ = R−n/p gp; B . Written in term of x, the last inequality becomes 1/p 1/n n 2wn R 2 wn Z + p f n; B ; u dx + sup u ≤ c4 B B λ BRy with c4 a function of n; γ; δˆ = δR2 and p. This is the desired estimate for ¯ u ∈ W 2; n ∩ C 0 . Step 2: Now, let u ∈ W 2; n . By a density argument, let um be a se¯ converging toward u in W 2; n B. The injecquence of functions of C 2 B, 2; n 0 tion of W B in C B is continuous, consequently um converges uniformly toward u in B. We have Lum = Lum − u + Lu ≥ f + Lum − u: By setting, fm = Lum − u, we observe by construction that fm converges toward 0 in Ln . Because um ∈ W 2; n ∩ C 2 and f˜m = f + fm is in Ln , the estimate (3) is valid also for um , and then we have 1/p 1 Z R ˜ p f u+ + (66) sup um ≤ C n; B : B B m λ BR y
local and global estimates
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Using previous results and taking the limit, we obtain 1/p 1 Z R + p u + f n; B : sup u ≤ C B B λ BR y Observe also that by replacing u by −u, the theorem can be extended easily to the case of supersolutions and solutions of the equation, Lu = f: AKNOWLEDGMENTS The author would like to express his gratitude to Professor Patrick Le Tallec for the assistance he has given him throughout the development of this work. He also thanks the referees and the editor for their comments on this work.
REFERENCES 1. A. D. Aleksandrov, Majorization of solutions of second order linear equations, Vestnik Leningrad Univ. 21 (1966), 5–25 (English translation in Amer. Math. Soc. Transl. Ser. 2 68 (1968), 120–143). 2. I. Y. Bakel’man, Theory of quasilinear elliptic equations, Siberian Math. J. 2 (1961), 179–186. 3. D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order,” 2nd ed., Springer-Verlag, Berlin–Heidelberg–New York, 1983. 4. J.-F. Bourgat, P. Le Tallec, and M. D. Tidriri, Coupling Navier–Stokes and Boltzmann, J. Comput. Phys. 127 (1996), 227–245. 5. P. Le Tallec and M. D. Tidriri, “Convergence Analysis of Domain Decomposition Algorithms with Full Overlapping for the Advection–Diffusion Problems,” ICASE Report No. 96-37, 1996, also to appear in Math. Comp. 6. P. Le Tallec and M. D. Tidriri, Application of maximum principles to the analysis of a coupling time marching algorithm, to appear. 7. M. D. Tidriri, Couplage d’approximations et de mod`eles de types diff´erents dans le calcul d’´ecoulements externes, Ph.D. thesis, Universit´e de Paris IX, 1992. 8. M. D. Tidriri, “Domain Decomposition for Incompatible Nonlinear Models,” INRIA Research Report 2435, Dec. 1994. 9. M. D. Tidriri, Domain decompositions for compressible Navier–Stokes equations, J. Comput. Phys. 119 (1995), 271–282.
Local and Global Estimates for the Solutions of Convection–Diffusion Problems Moulay D. Tidriri* Iowa State University, Department of Mathematics, 400 Carver Hall, Ames, Iowa 50011-2064 Submitted by John Lavery Received May 21, 1997
In this paper we develop and study local and global estimates for the solutions of convection–diffusion problems, respectively, with Dirichlet–Neumann and Dirichlet boundary conditions. This study is based on a combination of Aleksandrov, Bakelman, and Pucci theory, with more standard traces estimates. © 1999 Academic Press
1. INTRODUCTION In this paper we shall develop local and global estimates for the solution of convection–diffusion problems, respectively, with Dirichlet–Neumann and Dirichlet boundary conditions. The derivation of such estimates is based on the use of the most recent methods for elliptic equations initiated by Aleksandrov, Bakelman, and Pucci together with more standard traces estimates. These estimates are independent on the low-order, “relaxation,” term. They can be applied, for example, to the analysis of an explicit time marching algorithm. This time marching algorithm was introduced and studied by Bourgat, Le Tallec, and Tidriri [4] Le Tallec and Tidriri [5, 6], and Tidriri [7–9], in order to couple different models and/or approximations, which results in an efficient strategy for solving problems in applied sciences and engineering. For the applications of these estimates to the analysis of this algorithm we refer to [6] and [7]. * This work has been supported by the Hermes Research program under Grant Number RDAN 86.1/3. The author was also supported by the National Science Foundation under Contract Number ECS-8957475 and by the United Technologies Research Center while he was at Yale University. E-mail address: [email protected]. 137 0022-247X/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
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In the next section we develop a maximum principle for a general secondorder elliptic problem based on the Aleksandrov, Bakelman, and Pucci theory. This local estimate is independent on the low-order, relaxation, term. In Section 3, we develop local and global estimates for the solutions of convection–diffusion problems with Dirichlet–Neumann boundary conditions. Finally, in Section 4 we develop local and global estimates for the Dirichlet boundary conditions case. 2. A MAXIMUM PRINCIPLE In this section we shall establish a maximum principle for an arbitrary elliptic operator of second order. This result is central to the development of the estimates studied in the next sections. Let L be an operator of the form, Lu = aij xDij u + bi xDi u + cxu; for any u in W 2; n , with a bounded domain of n . The coefficients aij , bi , and c; i; j = 1; : : : ; n are defined on . As usual, the repeated indices indicate a summation from 1 to n. We suppose that the operator L is strictly elliptic in in the sense that the matrix A of coefficients aij is strictly positive everywhere in . Let λ and 3 denote, respectively, the smallest and the largest eigenvalue of A. Let D denote the determinant of the matrix A and D∗ = D1/n . We have 0 < λ ≤ D∗ ≤ 3: We suppose in addition that the coefficients aij , bi , and c are bounded in , and that there exist two positive real numbers γ and δ such that 3 ≤ γ; λ 2 b ≤ δ: λ
(L is uniformly elliptic);
(1) (2)
Now, we are in a position to state the principal result of this section, proved in the Appendix. Theorem 2.1. Let u ∈ W 2; n and suppose that Lu ≥ f with f ∈ L and c ≤ 0. Then for all spheres B = B2R y of center y and radius 2R included in and for all p > 0, we have 1/p R 1 Z u+ p + f n; B ; (3) sup u ≤ CR B B λ BR y n
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139
where the constant CR depends on n; γ; δR2 ; p, but is independent of c, and u+ = maxu; 0. Remark 2.1. The statement of the same theorem can be found in [3], under the additional assumption, c ≤ δ: λ
(4)
However, there the constant CR depends indirectly on c through δ. That is exactly what we want to avoid, because we would like this constant to be independent of c (see next sections).
3. DIRICHLET–NEUMANN CASE Let loc be a connected domain of n with loc ⊂ (Fig. 1). The boundaries of the two subdomains are defined as follows, 0b = ∂ ∩ ∂loc (internal boundary); 0i = ∂loc ∩ (interface), 0∞ = ∂\0b (farfield boundary): We denote by n the external unit normal vector to ∂ or ∂loc . Let V be a given velocity field of an inviscid incompressible flow such that div V = 0; in ; (5) V · n = 0; on 0b :
FIG. 1. Description of the domain and its splitting.
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We shall derive an estimate for the solution of the following Dirichlet– Neumann problem, 1 Lv = −ν 1v + V · ∇v + v = 0; τ v = 0; ∂v = g; ∂n
in ;
(6)
on 0∞ ;
(7)
on 0b ;
(8)
where the function g is given in H −1/2 0b , the coefficient τ is strictly positive, and ν is the diffusion coefficient. Let W be the subspace of H 1 defined by (9) W = w ∈ H 1 w = 0 on 0∞ : We then define the following bilinear form on W , Z Z av; w = ν ∇v ∇w + divVvw;
v; w =
Z
vw:
(10) (11)
The first basic problem associated to (6)–(8), can be written as follows: Find v ∈ W satisfying Z 1 νgw d0; ∀w ∈ W: (12) av; w + v; w = τ 0b Moreover, we assume that the coefficients ν and τ satisfy the following relation, ντ ≤ 1:
(13)
This hypothesis is not necessary but simplifies the proofs to come. Moreover, it is not restrictive (see [6]). Let d denote the distance as described in Fig. 2. Let β be a real number such that √ 3 ν ; 0<β< d and set β k= √ : ν τ The first basic result states a global H 1 estimate of the solution of the first basic problem (12) in terms of the boundary data g.
local and global estimates
141
FIG. 2. Description of the domain loc and the splitting used in the majorization of the local solution.
Lemma 3.1.
There exists a constant c0 such that v1; ≤ c0 g−1/2; 0b :
Proof of Lemma 3.1. Using (5) we obtain the following equality, Z 1Z v divVv = divVv2 2 1Z V · nv2 = 2 0 = 0;
∀ v ∈ W:
(14)
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Choosing w = v in (12), we then obtain Z Z 1 νgv: ν∇v2 + v2 = τ 0b
(15)
From this equality we deduce the following estimate, v21; ≤ g−1/2; 0b v1/2; 0b : The application of the trace theorem yields the estimate (14). Let i be the subdomain of width d/3 with external boundary 0i as described in Fig. 2. Let Ky = Bd/3 y be the sphere of center y and radius d/3. There exist y1 ; : : : ; yl belonging to i such that 2i =
[ y∈i
Bd/6 y ⊂
l [ j=1
Kyj :
We then define K by setting K=
l [ j=1
Kyj :
The next lemma states a local estimate of the solution v of the first basic problem (12). Lemma 3.2.
There exists a constant c1 such that v∞; K ≤ c1 v0; ;
(16)
where c1 is a constant depending only on ν; γ; δd 2 and 3/2dn/2 . Proof of Lemma 3.2.
The operator, L = −L
satisfies the assumptions of Theorem 2.1, with c = −1/τ and f = 0. Applying this theorem with p = 2, y ∈ i we obtain v∞; Ky ≤ c1 v0; B2d/3 y : Therefore, v∞; Ky ≤ c1 v0; ; where c1 is a constant depending only on ν, γ, δd 2 , and 3/2dn/2 . Applying (17) to each Kyj we obtain v∞; K ≤ sup c1j v0; : j=1;:::;l
(17)
local and global estimates
143
Setting c1 = supj=1;:::;l c1j , we finally have v∞; K ≤ c1 v0; :
(18)
This concludes the proof of the lemma. We shall now establish other local estimates for the solution v of the first basic problem. For any Mi in i ; we introduce (see Fig. 2): • Bi = the ball centered on Mi of radius d/6, • vi = expkr 2 − d 2 /36v∞; ∂Bi : We then have: Lemma 3.3.
The solution v of the first basic problem satisfies kd 2 ∀Mi ∈ i : v∞; ∂Bi ; vMi ≤ exp − 36
(19)
Proof of Lemma 3.3. The operator L applied to vi can be written in polar coordinates (with r = Mi M), k 1 2 2 v: Lvi = 4 −k νr − kν + V · er r + 2 4τ i Therefore,
k 1 2 2 − kν vi : Lvi ≥ 4 −k νr − V · er r + 2 4τ
(20)
ϕr; k = akr 2 + bkr + ck;
(21)
We then set
with ak = −k2 ν; k bk = − V · er ; 2 1 − kν: ck = 4τ We seek to satisfy the following relation, 0 ≤ inf ϕr; k;
for 0 ≤ r ≤
d : 6
As ϕr; k decreases on + , this will be satisfied if and only if d ≥ 0; ϕ 6
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i.e., if and only if kdV 1 k2 νd 2 − + − kν ≥ 0: 36 12 4τ We then replace k by its value. Therefore, we have to satisfy −
− √
βdV βν 1 β2 d 2 − − √ ≥ 0: √ + 36ντ 4τ ν τ 12ν τ
τ, it follows that dV β2 d 2 1 ≥β 1+ : √ 1− 9ν 12ν 4 τ √ The constraint β < 3 ν/d, finally yields after division, β2 d 2 −1 dV 1 1− : ϕr; k ≥ 0 if and only if √ ≥ β 1 + 12ν 9ν 4 τ
Multiplying by
(22)
From √ the relation (20) and the previous calculation, we deduce that for β < 3 ν/d and τ satisfying the previous inequality, we have Lvi ≥ 0 = Lv: In addition, by construction, vi ≥ v;
on ∂Bi :
Consequently, using the maximum principle we obtain v ≤ vi ; In particular,
in Bi :
kd 2 v∞; ∂Bi : vMi ≤ exp − 36
Repeating the same process for −v, we finally obtain kd 2 v∞; ∂Bi ; ∀Mi ∈ i : vMi ≤ exp − 36
(23)
Let ∞ be defined by ∞ = \ loc . The next result establishes an H 1 estimate of the solution v of the first basic problem on the domain ∞ . Lemma 3.4.
There exists a constant c2 such that 1/2 p V p d/c2 : v1; ∞ ≤ v∞; i c2 /d 1 + ν
(24)
local and global estimates Let ξ ∈ H 1 be such that ξ = 1; in ∞ ; supp ξ ⊂ i ∪ ∞ :
Proof of Lemma 3.4.
We have using (12),
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Z
−ν 1v + divVv +
v 2 ξ v = 0: τ
Using Green’s formula we deduce Z Z Z −ν 1vξ2 v = ν∇ξv2 − ν∇ξ2 v2 :
On the other hand, we have 2 2 Z Z Z Vξ v 2 divVvξ v = div − V · ∇ξξv2 : 2
(25)
(26)
(27)
Using the relations (26) and (27), (25) becomes 2 2 Z Z Vξ v ξ2 v2 2 + − νv2 ∇ξ2 + v2 ξV · ∇ξ ν∇ξv + div 0= 2 τ Z 1 Z Z − ν ξ2 v2 − νv2 ∇ξ2 + v2 ξV · ∇ξ = ν∇ξv2 + ξv2 + τ Z Z 1 Z 2 2 2 2 ν∇v + v + ν∇ξv + ξv + − ν ξ2 v2 = ∞ i τ Z − νv2 ∇ξ2 + v2 ξV · ∇ξ: i
Hence, we obtain νv21; ∞ + =
Z i
Z i
ν∇ξv2 + ξv2 +
Z 1 − ν ξ2 v2 τ
νv2 ∇ξ2 + v2 ξV · ∇ξ:
The relation (13) then yields Z νv2 ∇ξ2 + v2 ξV · ∇ξ νξv21; ∞ ∪i ≤ i
≤ v2∞; i
Z i
ν∇ξ2 + ξV · ∇ξ
≤ v2∞; i νξ21; i + ξ0; i V ξ1; i ξ0; i V 2 2 : ≤ v∞; i ξ1; i ν + ξ1; i
(28) (29)
(30)
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If we take ξ such that ξ0; i ≤ 1; c ξ21; i = 2 ; d where c2 is a constant, (30) then becomes 1/2 p V p d/c2 ; v1; ∞ ≤ v∞; i c2 /d 1 + ν
(31)
and this concludes the proof. Now we are in a position to state the main result of this section. Theorem 3.1. Let v be the solution of the first basic problem (12). If τ is sufficiently small, we have 1/2 p p 1 kd 2 exp − g−1/2; 0b ; v1/2; 0i ≤ C1 C2 /d 1 + V ∞ d/C2 ν 36 where C1 and C2 are constants, with C1 depending only on d, ν, γ, and δ, but not on τ. Proof of Theorem 3.1. The proof of this theorem is based on the preceding lemmas. Because ∂Bi ⊂ K, we have v∞; ∂Bi ≤ v∞; K ;
(32)
v∞; ∂Bi ≤ c1 v0; :
(33)
Lemma 3.2 then implies Using Lemma 3.3 and the foregoing estimate we obtain kd 2 c1 v0; ; ∀Mi ∈ i : vMi ≤ exp − 36 Consequently we have
v∞; i
kd 2 ≤ exp − c1 v0; : 36
Applying Lemma 3.1 we obtain v∞; i
kd 2 g−1/2; 0b : ≤ c1 c0 exp − 36
(34)
An application of Lemma 3.4 then yields 1/2 p V p kd 2 d/c2 exp − g−1/2; 0b : v1; ∞ ≤ c0 c1 c2 /d 1 + ν 36
(35)
(36)
local and global estimates
147
To conclude we use the trace theorem which yields v1/2; 0i ≤ c3 v1; ∞ : Consequently, we have the final estimate, 1/2 p V p kd 2 g−1/2; 0b ; d/c2 exp − v1/2; 0i ≤ c0 c1 c3 c2 /d 1 + ν 36 which corresponds to our theorem with C1 = c0 c1 c3 and C2 = c2 . 4. DIRICHLET–DIRICHLET CASE In this section we shall derive an estimate of the solution of the following Dirichlet problem, 1 Lv = −ν 1v + V · ∇v + v = 0; τ
in loc ;
(37)
v = h;
on 0i ;
(38)
v = 0;
on 0b ;
(39)
where the function h is given in H 1/2 0i , the coefficient τ is strictly positive, and ν is the diffusion coefficient. The velocity field V is given by (5). Let W be the subspace of H 1 loc defined by W = w ∈ H 1 loc w = 0 on 0b : We then define the following bilinear form in W ; Z Z ∇v:∇w + divVvw; av; w = ν loc
v; w =
Z loc
loc
vw;
(40) (41)
with v and w in W . The second basic problem associated to (37)–(39) corresponds to the following problem: Find v ∈ W such that Z ∂v 1 ν w; ∀w ∈ W; (42) av; w + v; w = τ 0i ∂n v0i = h;
(43)
where h is given in H 1/2 0i . We first have the following lemma: Lemma 4.1.
For τ sufficiently small, we have ν 1 aw; w + w; w ≥ w21; loc ; τ 2
∀w ∈ W:
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Proof of Lemma 4.1. Under the hypothesis 1/τ ≥ ν/2 + 1/2νV 2∞ , and using the Cauchy–Schwarz inequality, we obtain Z Z 1Z 1 ν ∇v:∇v + V · ∇vv + v2 av; v + v; v = τ τ loc loc loc 1 ≥ ν∇v20; 2 + v20; 2 − V ∞ ∇v0; 2 v0; 2 τ 1 ν 1 ≥ ν∇v20; 2 + v20; 2 − ∇v20; 2 − V 2∞ v20; 2 τ 2 2ν ν 2 2 ≥ ∇v0; 2 + v0; 2 : 2 We will make the simplifying assumption (13). We first establish a global estimate for the solution of the second basic problem. Lemma 4.2. satisfies
The solution v of the second basic problem (42) and (43)
1 + V 2∞ 1/2 h1/2; 0i : 1+ v1; loc ≤ 21 + τ ν2 Proof of Lemma 4.2. Choosing w = v in (42) we obtain Z Z Z ∂v 1 ∇v2 + ν h: divVvv + v2 = ν τ loc loc 0i ∂n Lemma 4.1 then yields
∂v
h1/2; 0i : v21; loc ≤ 2
∂n −1/2; 0i −2 1/2
(44)
(45)
(46)
We shall now establish an estimate of ∂v/∂n−1/2; 0i . Combining (42) and (5) we obtain Z Z ∂v 1 1 w= ∇v∇w + V · ∇vw + vw : ν ντ 0i ∂n loc Therefore, for any w in W , we have Z ∂v 1 0 ∂n w ≤ ∇v0; loc ∇w0; loc + ν V ∞ ∇v0; loc w0; loc i 1 + v0; loc w0; loc ντ 1/2 1 1 2 2 2 2 ≤ ∇v0; loc + 2 V ∞ ∇v0; loc + 2 v0; loc ν ν 1/2 1 2 2 2 × ∇w0; loc + w0; loc + 2 w0; loc τ 1 + V 2∞ 1/2 ≤ 1+ v1; loc 1 + τ−2 1/2 w1; loc : ν2
local and global estimates The trace theorem then yields
∂v 1 + V 2∞ 1/2 −2 1/2
≤ 1 + τ v1; loc : 1+
∂n ν2 −1/2; 0i Combining now the relations (46) and (47) we have 1 + V 2∞ 1/2 h1/2; 0i ; v1; loc ≤ 21 + τ−2 1/2 1 + ν2
149
(47)
(48)
and hence in particular, 1 + V 2∞ 1/2 h1/2; 0i : v0; loc ≤ 21 + τ−2 1/2 1 + ν2
(49)
Let Ky = Bd/4 y be the sphere centered on y and of radius d/4, with y belonging to 0V (see Fig. 3). By construction, 0V is the center surface of loc and i is the subdomain of width d/6 centered on 0V . We have the following lemma: Lemma 4.3.
There exists a constant c1 such that v∞; K ≤ c1 v0; loc :
Proof of Lemma 4.3. Lemma 3.2 we obtain
(50)
Following the same argument as in the proof of v∞; Ky ≤ c1 v0; loc ;
(51)
where c1 is a constant depending only on d, ν, γ, and δ. On the other hand there exist y1 ; : : : ; yl in i such that 2i =
[ y∈i
Bd/6 y ⊂
l [ j=1
Kyj = K:
By applying (51) to each Kyj , we obtain v∞; K ≤ sup c1j v0; loc = c1 v0; loc : j=1;:::;l
(52)
Next we shall establish another local estimate for the solution of the second basic problems (42) and (43). For any Mi ∈ i , we introduce (see Fig. 3). • a ball Bi centered on Mi and of radius d/6, • the function vi = exp kr 2 − d 2 /36v∞; ∂Bi .
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FIG. 3. Description of the local domain loc and the splitting used in the majorization of the global solution.
We then have: Lemma 4.4. satisfies
The solution v of the second basic problem (42) and (43) −kd 2 v∞; ∂Bi : vMi ≤ exp 36
(53)
Proof of Lemma 4.4. By definition of k (see the previous section), ϕr; k is positive for all r ∈ 0; d/6. Then by following the same argument as in the proof of Lemma 3.3 we obtain the inequality (53). Let b be the subdomain of loc described in Fig. 3. We establish next an H 1 global estimate of the solution of the second basic problem.
local and global estimates Lemma 4.5. satisfies
151
The solution v of the second basic problems (42) and (43)
v1; b ∪i ≤ v∞; i Proof of Lemma 4.5.
p
V ∞ p c2 /d 1 + d/c2 ν
1/2
:
Consider ξ ∈ H 1 loc , such that ξ = 1; in b ; supp ξ ⊂ b ∪ i :
Choosing w = ξ2 v in (42) we obtain Z v 2 ξ v = 0: −ν1v + divVv + τ loc
(54)
(55)
(56)
Similarly to the proof of Lemma 3.4 we obtain Z νv2 ∇ξ2 + v2 ξV · ∇ξ: νξv1; b ∪i ≤ i
Choosing ξ such that ξ0; i ≤ 1; and ξ21; i =
c2 ; d
we finally obtain as in the proof of Lemma 3.4 the inequality (54). Finally, the main result of this section is presented in the following theorem: Theorem 4.1. For τ sufficiently small, the solution v of Problem (42) and (43) satisfies
1/2 p
∂v 1 + V 2∞ V ∞ p
≤ C1 C2 /d 1 + d/C2 1+
∂n ν2 ν −1/2; 0b 1 kd 2 h1/2; 0i ; × 1 + 2 exp − (57) τ 36 where C1 and C2 are constants with C1 depending only on d, v, ν, and δ. Proof of Theorem 4.1. and 4.4 imply
Because ∂Bi ⊂ K by construction, Lemmas 4.3
kd 2 c1 v0; loc : v∞; i ≤ exp − 36
(58)
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Furthermore, using Lemma 4.2 then yields 1 + V 2∞ 1/2 kd 2 1 1/2 c exp − h1/2; 0i : (59) 1 + v∞; i ≤ 2 1 + 1 ν2 τ2 36 By using Lemma 4.5 we then obtain 1/2 1 + V 2∞ 1/2 p V ∞ p c1 c2 /d 1 + d/c2 v1; b ∪i ≤ 2 1 + ν2 ν 1 1/2 kd 2 × 1+ 2 exp − (60) h1/2; 0i : τ 36 Before concluding we shall establish an estimate of the term,
∂v
:
∂n −1/2; 0b Choosing w such that w ∈ H 1 loc ; and using (42) we obtain Z b
with w = 0 on ∂b ∩ ∂i ;
v w = 0: −ν1v + divVv + τ
Applying Green’s formula and using (5), we obtain Z Z ∂v 1 1 w= ∇v∇w + V · ∇vw + vw : ν ντ 0b ∂n b Similarly to the proof of Lemma 4.2 we obtain the following inequality,
∂v 1 1/2 1 + V 2∞ 1/2
≤ 1+ 2 v1; b : (61) 1+
∂n τ ν2 −1/2; 0b The completion of the proof of the theorem results from the combination of the relation (60) with (61). APPENDIX In this appendix we shall give the proof of Theorem 2.1 of Section 2. This proof relies on the notion of a contact set. If u is a continuous arbitrary function on , the upper contact set, denoted 0+ or 0+ u , is the subset of , defined by 0+ = y ∈ ; ∃py ∈ n such that ux ≤ uy + p · x − y ∀x ∈ : (62)
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153
We see that u is a concave function on if and only if 0+ = . When u ∈ C 1 we must have p = Duy in the relation (62). In addition, when u ∈ C 2 , the Hessian matrix D2 u = Dij u is negative on 0+ . In general, 0+ is closed in . If u is a continuous arbitrary function on , we define the “normal mapping” χy = χu y at point y ∈ by (63) χy = p ∈ n ; ux ≤ uy + p · x − y ∀x ∈ : We can see that χy is nonempty if and only if y ∈ 0+ . In addition when u ∈ C 1 , we have χy = Duy on 0+ ; in other words χ is the gradient field of u on 0+ . As a particular case of the Bakelman–Alexandrov ([1] and [2]) maximum principle, we have: Lemma 1.
¯ we have For u ∈ C 2 ∩ C 0 ,
ij
a Dij u d
sup u ≤ sup u +
+; 1/n D∗ ∂ nwn n; 0
with d the diameter of and wn the volume of a unit sphere in n . For more details see [3]. We now proceed to the proof of Theorem 2.1. We take Bˆ = B1 0 and the general case is deduced by considering the coordinate transform, x → xˆ = x − y/2R. We will begin, in a first step, by showing this result for u ∈ C 2 ∩ W 2;n and then in a second step we will deduce the result for u ∈ W 2; n . Step 1: We suppose that u ∈ C 2 ∩ W 2; n . For β ≥ 1, we consider the cutoff function η defined by ηx ˆ = 1 − x ˆ 2 β : By differentiation, we obtain ˆ i η = −2βxˆ i 1 − x ˆ 2 β−1 ; D ˆ ij η = −2βδij 1 − x ˆ 2 β−1 + 4ββ − 1xˆ i xˆ j 1 − x ˆ 2 β−2 : D By setting v = ηu; we then obtain ˆ ij v = ηaˆ ij D ˆ ij u + 2aˆ ij D ˆ j u + uaˆ ij D ˆ i ηD ˆ ij η aˆ ij D ˆ i u − cu ˆ i ηD ˆ ij η: ˆ j u + uaˆ ij D ˆ + 2aˆ ij D ≥ ηfˆ − bˆ i D
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ˆ Let 0+ = 0+ v be the upper contact set v, in the sphere B; we have u > 0;
on 0+ :
If x ∈ ∂Bˆ such that p · x − y < 0 we indeed have vx = 0. Consequently, vy + p · x − y ≥ vx = 0: Moreover, using the concavity of v on 0+ , we can estimate the following quantity ˆ = Du
1 ˆ − uDη: ˆ Dv η
Indeed, 1 Du ˆ + u Dη ˆ ˆ ≤ Dv η v 1 ˆ + uDη ≤ η 1 − x ˆ ≤ 21 + βη−1/β u: In that way, we have on 0+ the following inequality, ˆ −2/β + 2βbη ˆ −1/β + cˆ v + ηfˆ: ˆ ij v ≤ 16β2 + 2ηβ3η −aˆ ij D Because cˆ ≤ 0, we deduce the inequality, ˆ −2/β + 2βbη ˆ −1/β v + ηfˆ ˆ ij v ≤ 16β2 + 2ηβ3η −aˆ ij D ˆ −2/β v + fˆ; ≤ c1 λη ˆ independent of c. ˆ with c1 = cn; β; γ; δ ˆ we obtain, for β ≥ 2, Consequently, applying Lemma 1 on B, ˆ d 1 ˆ −2/β v + fˆn; Bˆ : sup v ≤ c1 λη 1/n ∗ ˆ D nwn Bˆ Using relation (2), this becomes ˆ ˆ d d 1 −2/β η v + fˆn; Bˆ c sup v ≤ ˆ 1 n; B 1/n 1/n λˆ nwn nwn Bˆ 1 ≤ c1 dˆ η−2/β vn; Bˆ + fˆn; Bˆ λˆ 1 ˆ −2/β + ˆ v n; Bˆ + f n; Bˆ ≤ c1 d η λˆ 1 ≤ c1 dˆ sup v+ 1−2/β u+ 2/β n; Bˆ + fˆn; Bˆ ; λˆ
(64)
local and global estimates
155
ˆ Here, dˆ is the where c1 is a constant depending only on n, β, γ, and δ. ˆ ˆ d = 2. diameter of B Using Young’s inequality in the form, ab ≤ εaq + ε−r/q br ; for q = 1 − 2/β−1 and r = β/2, we obtain β/2
sup v+ 1−2/β u+ 2/β n; Bˆ ≤ ε sup v+ + ε1−β/2 u+ 2/β n; Bˆ ;
∀ε > 0:
By taking ε = 1/2c1 dˆ and plugging in our inequality for v, we obtain sup v ≤ Bˆ
ˆ 1 1 1−β/2 ˆ β/2 u+ 2/β β/2 + c1 d fˆn; Bˆ : c1 d sup v+ + ˆ n; B ˆ 2 2 λ
(65)
We want to prove the theorem for all p > 0. We will treat separately the cases p ≤ n and p > n. If p ≤ n, we set β = 2n/p. In this case we have β/2
u+ 2/β n; Bˆ = u+ p; Bˆ : Plugging this in our inequality for v, we obtain ˆ 1 1−β/2 1 ˆ β/2 u+ p; Bˆ + c1 d fˆn; Bˆ : sup v ≤ c1 d 2 Bˆ 2 λˆ Consequently, we obtain the following inequality; Z 1/p dˆ ˆ f n; Bˆ : u+ p + sup v ≤ c2 Bˆ 2λˆ Bˆ On the sphere B1/2 0, the cutoff function satisfies β 1 1 ≤ : η 2 It follows, then v η B1/2 0
sup u ≤ sup
B1/2 0
≤ 2β sup v: Bˆ
Finally we obtain the desired estimate, Z 1/p dˆ ˆ f n; Bˆ : u+ p + sup u ≤ c3 Bˆ 2λˆ B1/2 0
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¯ The constant c3 in the previous formula depends for u in W 2;n ∩ C 2 . ˆ only on n, β, γ, and δ, but is independent of c. ˆ On the other hand if p > n, we have 2n < p; β
∀β ≥ 2:
Then, it follows (by assuming β ≥ 2), ˆ −1/p u+ p; Bˆ : ˆ −1/2n/β u+ 2n/β; Bˆ ≤ B B But β/2
u+ 2n/β = u+ 2/β n; Bˆ ; and therefore, by proceeding as before, we obtain the desired estimate, Z 1/p dˆ ˆ + p f n; Bˆ ; u + sup u ≤ c4 Bˆ 2λˆ B1/2 0 ¯ The constant c4 in the previous formula depends for u in W 2; n ∩ C 2 . ˆ but is independent of c. only on n, β, γ, and δ, ˆ Transformation xˆ → x. ˆ ij = R−2 Dij , thus λˆ = R−2 λ and δˆ = δR2 . In addition, By construction, D we have B = wn 2Rn and gp; Bˆ = R−n/p gp; B . Written in term of x, the last inequality becomes 1/p 1/n n 2wn R 2 wn Z + p f n; B ; u dx + sup u ≤ c4 B B λ BRy with c4 a function of n; γ; δˆ = δR2 and p. This is the desired estimate for ¯ u ∈ W 2; n ∩ C 0 . Step 2: Now, let u ∈ W 2; n . By a density argument, let um be a se¯ converging toward u in W 2; n B. The injecquence of functions of C 2 B, 2; n 0 tion of W B in C B is continuous, consequently um converges uniformly toward u in B. We have Lum = Lum − u + Lu ≥ f + Lum − u: By setting, fm = Lum − u, we observe by construction that fm converges toward 0 in Ln . Because um ∈ W 2; n ∩ C 2 and f˜m = f + fm is in Ln , the estimate (3) is valid also for um , and then we have 1/p 1 Z R ˜ p f u+ + (66) sup um ≤ C n; B : B B m λ BR y
local and global estimates
157
Using previous results and taking the limit, we obtain 1/p 1 Z R + p u + f n; B : sup u ≤ C B B λ BR y Observe also that by replacing u by −u, the theorem can be extended easily to the case of supersolutions and solutions of the equation, Lu = f: AKNOWLEDGMENTS The author would like to express his gratitude to Professor Patrick Le Tallec for the assistance he has given him throughout the development of this work. He also thanks the referees and the editor for their comments on this work.
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