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A new proof of Lp estimates of Stokes equations
J. Math. Anal. Appl. 420 (2014) 1251–1264
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
A new proof of Lp estimates of Stokes equations ✩ Fangju Hu a,∗ , Dongsheng Li a , Lihe Wang b,c a b c
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China
a r t i c l e
i n f o
a b s t r a c t In this paper, we establish a new proof of Lp estimates of the Stokes equations, which enables us to avoid the use of potential theory and thus simplifies the traditional proof. The modified Vitali covering lemma and the maximal function technique are the main analytical tools. © 2014 Elsevier Inc. All rights reserved.
Article history: Received 19 January 2014 Available online 20 June 2014 Submitted by D. Wang Keywords: Stokes equations Lp estimates Maximal function Vitali covering lemma
1. Introduction In this paper, we consider the following Stokes equations:
ut − λΔu + ∇p = f, div u = 0,
(1.1)
in the cylindrical domain ΩT := Ω × (a, a + T ], where Ω is a domain in Rn and a ∈ R. Here u(x, t) represents the velocity field, λ > 0 is the viscosity constants, the pressure p(x, t) is a scalar function, and f is the external force. We introduce a Sobolev space Wp,2,1 (ΩT ) = u : u, Du, D2 u, ut ∈ Lp (ΩT ) . We are interested in how the Lp -regularity of the nonhomogeneous term f is reflected to the solutions of the Stokes equations (1.1). More precisely, we want to establish such an estimate as ✩
Supported by NSFC grant 11171266.
* Corresponding author. E-mail addresses: [email protected] (F. Hu), [email protected] (D. Li), [email protected] (L. Wang). http://dx.doi.org/10.1016/j.jmaa.2014.06.039 0022-247X/© 2014 Elsevier Inc. All rights reserved.
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f ∈ Lp
⇒
D2 u ∈ Lploc ,
(1.2)
and note here generally ut ∈ Lploc or ∇p ∈ Lploc is not true for our interior estimates, see Remark 1.2. Regularity estimates in Lp spaces are one of the fundamental estimates for partial differential equations. They play important roles in the regularity theory of relevant differential operators. The classical Calderón–Zygmmund estimates were established in [5] with the main argument based on explicit representation formulas involving singular integral operators and commutators, which made the proof rather intricate and skillful. However, the integral kernels and the corresponding estimates seem hard to compute in many cases, so we need an alternative method of Lp estimates to avoid the use of potential theory. The maximum function approach, which was introduced by Caffarelli and Peral in [4], enables us to obtain the Lp estimates by combining the basic tools in harmonic analysis and some regularity results in partial differential equations; that is, based on the Calderón–Zygmmund decomposition, the Hardy–Littlewood maximal function, the approximation method and some standard estimates for equations, it gives an alternative proof to the classical one. Moreover, influenced by [4], L.H. Wang [10] gives another proof from geometric intuitions, where the maximal function technique is still the main tool, while a modified Vitali covering lemma is used instead of Calderón–Zygmmund decomposition. As pointed out in [10], this method is much more applicable since Vitali covering lemma can be easily adapted to any complicated manifolds. The Stokes equations are linear, but nevertheless they deserve special attention, because of the incompressibility condition div u = 0. The study of problem (1.1) is quite useful in the analysis of the nonlinear system ⎧ n ⎪ ⎨u + ui Di u − λΔu + ∇p = f, t i=1 ⎪ ⎩ div u = 0, which describes the motion of an incompressible viscous fluid. Furthermore, the linearized system (1.1) is quite interesting also from the purely mathematical point of view, as it does not belong to any of the well-known classes of differential equations and has its own characteristics. As for the Lp -regularity of Stokes equations, V.A. Solonnikov [7] developed potential theory in great detail to express the solutions in explicit form via Oseen’s tensor Tij and the influence tensor Gij and finally obtained the Lp estimates through estimating these tensors, which made the proof rather difficult. The aim of this paper is to show a new proof of the Lp estimates for Stokes equations to simplify the traditional method in [7]. We shall adopt the ideas in [4] and [10], which enables us to completely avoid the complexity of potential theory, and thus reflects a significant simplification. Accordingly, we are able to extend the applicable range of the maximal function approach from elliptic/parabolic problems to Stokes system. The maximal function approach has been employed to obtain the Lp estimates for solutions to a large class of elliptic and parabolic problems since Caffarelli and Peral introduced it, see [1,2] (also the references therein). However, the Stokes equations are neither elliptic nor parabolic, which is one of the main difficulties of our problem. So it is necessary to consider the regularity of solutions with respect to space and time variable separately. As we will see, it is the peculiar characteristics of Stokes equations that make our results and proofs quite different from those of elliptic/parabolic ones. Moreover, since the interior Lp estimates for ut is not true in general, we will meet difficulties when treating some estimates concerning ut . So it is worth mentioning that we do not give a direct proof of D2u in Lp , instead we consider Δu at first just to circumvent the obstacle brought by ut , which will be stated in Remark 3.6. Throughout this paper, by considering the distribution of maximal function of Δu, we have to find the decay estimates of it. Finally we mention here that, because of the technique we are using, it appears possible to extend the Lp regularity for Stokes system in Rn to that on Riemannian manifolds. The development of this project will be our aim in the near future.
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Our main result is stated as follows. Theorem 1.1. Let u ∈ W2,2,1 (ΩT ) and p ∈ L2 (a, a + T ; H 1 (Ω)) be solutions of the Stokes equations (1.1). Assume f ∈ Lp (ΩT ) for some 1 < p < ∞. Then for any ΩT = Ω × (a , b ] with Ω Ω and a < a < b ≤ a + T , we have
2
D u
≤ C f Lp (ΩT ) + uL2 (ΩT ) + ut L2 (ΩT ) .
) Lp (ΩT
(1.3)
In addition, if we assume that u ∈ W2,2,1 (ΩT ) ∩ Lp (ΩT ), then
2
D u
) Lp (ΩT
≤ C f Lp (ΩT ) + uLp (ΩT ) ,
(1.4)
where C depends only on n, p, λ and dist(ΩT , ∂p ΩT ). Remark 1.2. (1) As mentioned above, we can only obtain the interior Lp estimates for D2 u and no Lp estimates for ut or ∇p can be expected in general, which is illustrated by the following counterexample. Let p > 2. Assume Ω ⊂ Rn is bounded and that g1 : (a, a + T ) → R satisfies g1 ∈ L2 (a, a + T ),
but g1 ∈ / Lp a , b for some a , b ⊂ (a, a + T ).
(1.5)
Set
t g(t) =
g1 (s) ds a
and take u(x, t) = g(t)∇H(x) and p = −g (t)H(x) = −g1 (t)H(x),
(1.6)
where H(x) ∈ C 3 (Ω) is harmonic. It is evident that u ∈ W2,2,1 (ΩT ) and p ∈ L2 (0, T ; H 1 (Ω)) satisfy Stokes equations (1.1) with f ≡ 0. One can easily verify that D2 u = g(t)D3 H(x) ∈ Lp ΩT . However, it follows from (1.5) that ut = g (t)∇H(x) = g1 (t)∇H(x) ∈ / Lp ΩT , and ∇p = −g1 (t)∇H(x) ∈ / Lp ΩT . (2) The term ut L2 (ΩT ) cannot be removed in the estimate (1.3), or, the member uLp (ΩT ) in (1.4) cannot be replaced by uL2 (ΩT ) . We can still illustrate the fact by using the counterexample (1.6) with g(t) to be determined. Otherwise, for any p > 2 and any M > 0, we can always find g(t) such that
g(t) p ≥ M g(t) 2 , L (a ,b ) L (a,a+T ) and consequently,
for some a , b ⊂ (a, a + T ),
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2
D u
) Lp (ΩT
≥
C2 M uL2 (ΩT ) , C1
where C1 = ∇HL2 (Ω) ,
C2 = D3 H Lp (Ω ) ,
which is a contradiction if we get rid of ut L2 (ΩT ) in (1.3) or replace uLp (ΩT ) by uL2 (ΩT ) in (1.4). (3) According to the Gagliardo–Nirenberg–Sobolev inequality, the additional assumption u ∈ Lp (ΩT ) n in (1.4) can be abandoned when 1 < p ≤ 2(n+2) n−2 and Ω is bounded in R (n ≥ 3). The organization of this paper will be as follows: In Section 2, we give some preliminary tools and some geometric analysis results which will be used for our approach. Section 3 will be devoted to deriving interior Lp estimates. We finish this section by introducing some notations. Note that some common notations in PDE are not mentioned here. Notation 1.3. 1. Br = {x ∈ Rn : |x| < r} is an open ball on Rn with center 0 and radius r > 0, Br (x) = Br + x. 2. Qr = Br × (−r2 , r2 ] is a middle centered parabolic cube with center point (0, 0), Qr (y, s) = Qr + (y, s), ∂p Qr = ∂Br × [−r2 , r2 ] ∪ Br × {t = −r2 } is its parabolic boundary. 3. ΩT := Ω × (a, a + T ] is a parabolic cylinder, ∂p ΩT = ∂Ω × [a, a + T ] ∪ Ω × {t = a} is its parabolic boundary. 4. For a locally integrable function f , f¯Qr =
1 |Qr |
f (x, t) dx dt Qr
is the average of f over Qr . For simplicity, we employ the letter C to denote universal constants depending only on n, p and λ. Thus the exact value denoted by C may change from line to line in a given computation. 2. Preliminary tools The following theorem provides an equivalent view of Lp functions. Theorem 2.1. (See [3].) Suppose that f is a nonnegative and measurable function in a bounded domain U ⊂ Rn × R. Let ν > 0 and M > 1 be constants. Then for any 0 < p < ∞, we have f ∈ Lp (U )
⇔
S :=
M kp (x, t) ∈ U : f (x, t) > νM k < ∞
k≥1
and 1 S ≤ f pLp (U ) ≤ c |U | + S , c where c > 0 is a constant depending only on ν, M and p.
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This theorem shows that f ∈ Lp if and only if the decay rate of |{(x, t) ∈ U : f (x, t) > νM k }| is suitably fast. Unfortunately, the measure |{(x, t) ∈ U : |f | > M }| is not stable in the iteration process, which, however, can be solved by introducing the maximal function. For a given function f defined on Rn × R, the parabolic maximal function Mf is defined as
1 Mf (x, t) = sup |Q (x, t)| r>0 r
f (y, s) dy ds,
Qr (x,t)
which satisfies strong p–p estimates and weak 1–1 estimates (see p. 5 in [8]). If f is defined only in a bounded domain U , we define its restricted maximal function as MU f (x) = M(χU f ), where χU is the standard characteristic function on U . The main technical tool for interior estimates is the following modified version of Vitali covering lemma. Lemma 2.2. (See [10].) Let 0 < ε < 1 and let E ⊂ F ⊂ Q1 be two measurable sets satisfying the following properties: 1. |E| ≤ ε|Q1 |. 2. For each (y, s) ∈ Q1 and for each r ∈ (0, 1], E ∩ Qr (y, s) ≥ εQr (y, s)
⇒
Qr (y, s) ∩ Q1 ⊂ F.
Then we have |E| ≤ (10)n+2 ε|F |. 3. Lp estimates for D 2 u This section will be devoted to obtaining interior Lp estimates for D2 u concerning the Stokes equations (1.1) in ΩT . For convenience, we will instead prove the following theorem, from which Theorem 1.1 follows easily by using proper procedures of translation, scaling, and covering. Theorem 3.1. Assume that u ∈ W2,2,1 (Q6 ) is a solution of the Stokes PDE (1.1) with f ∈ Lp (Q6 ) for some 1 < p < ∞. Then we have
2
D u
Lp (Q 1 )
≤ C f Lp (Q6 ) + uL2 (Q6 ) + ut L2 (Q6 ) .
(3.1)
2
In addition, if we assume that u ∈ W2,2,1 (Q6 ) ∩ Lp (Q6 ), then
2
D u p ≤ C uLp (Q6 ) + f Lp (Q6 ) , L (Q 1 )
(3.2)
2
where C > 0 is a constant depending only on n, p and λ. In order to prove the theorem, we need to establish several lemmas and corollaries. We start with the L2 regularity for Stokes equations.
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Lemma 3.2. (See [9].) Let f ∈ L2 (ΩT ). If u ∈ W2,2,1 (ΩT ) is a solution of ⎧ u − λΔu + ∇p = f ⎪ ⎨ t div u = 0 ⎪ ⎩ u=0
in ΩT , in ΩT ,
(3.3)
on ∂p ΩT ,
then
2
D u
L2 (ΩT )
≤ Cf L2 (ΩT ) .
To obtain the interior L2 estimates for Stokes equations, we shall first refer to the following ε-lemma, whose proof comes from Lemma 0.5 in [6]. Lemma 3.3. Let f , g, h be nonnegative functions in L1 (Q0 ), where Q0 is a cube in Rn+1 , and let α ∈ R+ . There exists ε0 such that if for some ε ≤ ε0 the following inequality
f dx dt ≤ c(ε) QR (x0 ,t0 )
g dx dt + Q2R (x0 ,t0 )
holds for all (x0 , t0 ) ∈ Q0 and R <
1 2
Q2R (x0 ,t0 )
1 f dx dt ≤ c Rα
f dx dt
Q2R (x0 ,t0 )
g dx dt + Q2R (x0 ,t0 )
1 2
dist((x0 , t0 ), ∂p Q0 ), then there exists a constant c such that
QR (x0 ,t0 )
for all (x0 , t0 ) ∈ Q0 and R <
h dx dt + ε
1 Rα
h dx dt
Q2R (x0 ,t0 )
dist((x0 , t0 ), ∂p Q0 ).
Lemma 3.4. Let f ∈ L2 (Q6 ). If u ∈ W2,2,1 (Q6 ) is a solution of ut − λΔu + ∇p = f,
(3.4)
div u = 0
(3.5)
in Q6 , then we have
2
D u 2 L (Q
1)
≤ C f L2 (Q6 ) + uL2 (Q6 )
(3.6)
and sup ∇uL2 (B1 ) ≤ C f L2 (Q6 ) + uL2 (Q6 ) + ut L2 (Q6 ) .
(3.7)
t∈[−1,1]
Proof. In view of the classical Calderón–Zygmmund estimates for p = 2,
B1
2 2 D u dx ≤ C
|u| dx + 2
B2
|Δu| dx , 2
for a.e. t ∈ (−1, 1],
B2
the proof of (3.6) can be reduced to that of ΔuL2 (Q2 ) ≤ C f L2 (Q6 ) + uL2 (Q6 ) .
(3.8)
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Consider our problem initially in Q2R (x0 , t0 ) for any (x0 , t0 ) ∈ Q6 and all R < 12 dist((x0 , t0 ), ∂p Q6 ). Abbreviate QR (x0 , t0 ) = QR , BR (x0 ) = BR . We choose a smooth cut-off function η = η(x, t) ∈ Cc∞ (Q2R ) satisfying 0 ≤ η ≤ 1,
η=1
in QR ,
near t = t0 + 4R2 ,
η=0
(3.9)
and |∇η| ≤
2 D η ≤ C , R2
C , R
|ηt | ≤
C . R2
(3.10)
Multiplying Eq. (3.4) by ϕ = Δ(uη 2 ), we obtain
ut Δ uη 2 dx −
B2R
λΔuΔ uη 2 dx −
B2R
Δp div uη 2 dx =
B2R
f Δ uη 2 dx.
B2R
Using the relation Δp = div f , which is obtained by taking divergence (in the weak sense) on both sides of (3.4), we derive from Green’s formula and Cauchy’s inequality that d dt
|∇u|2 η 2 dx − B2R
d dt
|u|2 D2 η 2 dx +
B2R
B2R
B2R
+ c(ε)
c f 2 dx + ε
|Δu|2 dx + c(ε)
≤ε
λ|Δu|2 η 2 dx
B2R
|∇u|2 |ηt | + |∇η|2 dx
B2R
2 |u|2 D2 η + |∇η|4 + Dt D2 η 2 dx
(3.11)
B2R
for arbitrary ε > 0. Substituting the conditions (3.9), (3.10) on η into (3.11), along with the interpolation inequality for ∇u, we obtain by integrating with respect to t from t0 − 4R2 to t0 + 4R2 that
|Δu|2 dx dt ≤ c(ε) QR
1 R4
|u|2 dx dt + Q2R
|f |2 dx dt + ε |Δu|2 dx dt.
Q2R
Q2R
According to Lemma 3.3, there exists a constant C such that
|Δu|2 dx dt ≤ C
1 R4
QR
|f |2 dx dt ,
|u|2 dx dt + Q2R
Q2R
which completes the proof of (3.8) for R = 2 and (x0 , t0 ) = (0, 0). We propose next to derive estimate (3.7). Note that
d dt
|u|2 D2 η 2 dx = B2R
|u|2 Dt D2 η 2 dx + 2 B2R
ut uD2 η 2 dx.
(3.12)
B2R
Returning to inequality (3.11) we now substitute the relation (3.12) and apply Cauchy’s inequality and interpolation inequality as above to obtain d dt
|∇u|2 η 2 dx ≤ C
B2R
B2R
C |Δu|2 + |ut |2 + f 2 dx + 4 R
|u|2 dx. B2R
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In particular, for (x0 , t0 ) = (0, 0) and R = 1, we integrate in time from −4 to t (−4 < t < 4), to obtain
sup
|∇u| dx ≤ C 2
t∈[−1,1] B1
|Δu|2 + |ut |2 + f 2 + u2 dx dt
|ut |2 + f 2 + u2 dx dt,
Q2
≤C Q6
where the last inequality comes from (3.8). Then we complete the proof of (3.7). 2 Lemma 3.5. Let v ∈ W2,2,1 (Q4 ) be a solution of ⎧ v − λΔv + ∇h = 0 ⎪ ⎨ t div v = 0 ⎪ ⎩ v=u
in Q4 , in Q4 ,
(3.13)
on ∂p Q4 ,
where u ∈ W2,2,1 (Q4 ) is a solution of (1.1) with f ∈ L2 (Q4 ). Then we have Δv2L∞ (Q3 )
≤C
|f | dx dt + 2
Q4
|Δu| dx dt . 2
(3.14)
Q4
Proof. Obviously v is smooth inside of Q4 . It is sufficient to prove the interior estimate (3.14) for smooth v. We start from dealing with the first equation of (3.13). Observing that div v = 0, we apply the operator div to both sides of it to obtain that Δh = 0. As a result, we have (Δv)t − λΔ(Δv) = 0
(3.15)
by imposing the Laplace operator on both sides of the first equation. Thus the interior L∞ -regularity for heat equation (3.15) implies that ΔvL∞ (Q3 ) ≤ CΔvL2 (Q4 ) ≤ C Δ(u − v)L2 (Q4 ) + ΔuL2 (Q4 ) ≤ C f L2 (Q4 ) + ΔuL2 (Q4 ) , where the proof of the last inequality is the same as that of (3.20). This completes the proof. 2 Remark 3.6. Note here, although the estimate
2
D v ∞ L (Q
3)
≤ C ut L2 (Q4 ) + D2 u L2 (Q ) 4
is also available just as the parabolic case (see Lemma 11 in [10]), it is useless for us since we have no control information about ut in the iteration steps due to lack of interior Lp regularity for ut . As a consequence, we will initially consider the Lp estimates of Δu in the following instead of that of D2 u. So the approach here differs from that of elliptic and parabolic equations.
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Lemma 3.7. There is a universal constant N1 > 1, so that for any ε > 0, there exists a small constant δ = δ(ε) > 0 such that if u is a solution of (1.1) in Q6 with Q1 ∩ (x, t) : M |Δu|2 ≤ 1 ∩ (x, t) : M |f |2 ≤ δ 2 = ∅,
(3.16)
then we have (x, t) : M |Δu|2 > N12 ∩ Q1 < ε|Q1 |. Proof. From (3.16), we see that there is a point (x0 , t0 ) ∈ Q1 such that for all r > 0,
1 |Qr |
1 |Qr |
|Δu| dx dt ≤ 1 and 2
Qr (x0 ,t0 )
|f |2 dx dt ≤ δ 2 .
(3.17)
Qr (x0 ,t0 )
Since Q4 ⊂ Q5 (x0 , t0 ) ⊂ Q6 , it follows from (3.17) that 1 |Q4 |
|f |2 dx dt ≤ Q4
|Q5 | 1 |Q4 | |Q5 |
|f |2 dx dt ≤ (5/4)n+2 δ 2 .
(3.18)
Q5 (x0 ,t0 )
By the same reason, we have 1 |Q4 |
|Δu|2 dx dt ≤ (5/4)n+2 .
(3.19)
Q4
Let (v, h) be the solution of (3.13). Then (u − v, p − h) is the solution of (3.3) in Q4 . Using (3.18) and Lemma 3.2, we have
Δ(u − v)2 dx dt ≤ C
Q4
|f |2 dx dt ≤ Cδ 2 .
(3.20)
Q4
In view of (3.18), (3.19) and by the interior L∞ regularity for Δv (Lemma 3.5), we can choose some universal constant N0 = N0 (n, λ) > 1 such that Δv2L∞ (Q3 ) ≤ N02 .
(3.21)
Denote by N1 the constant N12 = max{4N02 , 2n+2 } and assert that 2 (x, t) ∈ Q1 : M |Δu|2 > N12 ⊂ (x, t) ∈ Q1 : MQ4 Δ(u − v) > N02 .
(3.22)
To see this, fix any point (x1 , t1 ) ∈ Q1 satisfying 2 MQ4 Δ(u − v) ≤ N02 .
(3.23)
Then it suffices to show that M(|Δu|2 )(x1 , t1 ) ≤ N12 . Indeed, if 0 < r ≤ 2, then Qr (x1 , t1 ) ⊂ Q3 , whence (3.21) and (3.23) imply 1 |Qr |
2 |Δu| dx dt ≤ |Qr |
2
Qr (x1 ,t1 )
Qr (x1 ,t1 )
Δ(u − v)2 + |Δv|2 dx dt ≤ 4N02 .
(3.24)
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On the other hand, if r > 2, then (x1 , t1 ) ∈ Qr (x1 , t1 ) ⊂ Q2r (x0 , t0 ), whence (3.17) implies 1 |Qr |
|Δu|2 dx dt ≤
1 |Qr |
Qr (x1 ,t1 )
|Δu|2 dx dt ≤ 2n+2 .
(3.25)
Q2r (x0 ,t0 )
Consequently (3.24) and (3.25) imply (x1 , t1 ) ∈ (x, t) : M |Δu|2 ≤ N12 ∩ Q1 .
(3.26)
Thus assertion (3.22) follows easily from (3.23) and (3.26). Now (3.22), parabolic weak 1–1 estimate and (3.20) finally yield (x, t) : M |Δu|2 > N12 ∩ Q1 ≤ (x, t) : MQ4 Δ(u − v)2 > N02 ∩ Q1
2 C ≤ 2 Δ(u − v) dx dt N0 Q4
≤
C 2 δ < ε|Q1 |, N02
by selecting δ > 0 sufficiently small so that the last inequality above is valid. This completes the proof of the lemma. 2 The scaling invariant form of Lemma 3.7 will be the following corollary, which makes the second condition in Lemma 2.2 satisfied, and hence can be viewed as a crucial part during the first iteration for Lp estimates. Corollary 3.8. Let 0 < r ≤ 1 and (y, s) ∈ Q1 . Assume that u ∈ W2,2,1 is a solution of (1.1) in Q6 . Then there is a universal constant N1 > 1, so that for any ε > 0, there exists δ = δ(ε) > 0 such that if (x, t) ∈ Q1 : M |Δu|2 > N12 ∩ Qr (y, s) ≥ εQr (y, s), then we have Qr (y, s) ∩ Q1 ⊂ (x, t) ∈ Q1 : M |Δu|2 > 1 ∪ (x, t) ∈ Q1 : M |f |2 > δ 2 . Proof. Proof by a scaling: Since Stokes equations are invariant under a translation, we may assume (y, s) = (0, 0). We define for 0 < r ≤ 1 and (x, t) ∈ Q6 , ur (x, t) :=
u(rx, r2 t) , r2
pr (x, t) :=
p(rx, r2 t) , r
fr (x, t) := f rx, r2 t .
Under this scaling, we apply Lemma 3.7, with ur replacing u, pr replacing p, and fr replacing f , respectively, to deduce that the condition Qr ∩ (x, t) : M |Δu|2 ≤ 1 ∩ (x, t) : M |f |2 ≤ δ 2 = ∅ implies (x, t) : M |Δu|2 > N12 ∩ Qr < ε|Qr |. We then argue by contradiction to have Corollary 3.8 as the counterpart of Lemma 3.7.
2
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Now we take N1 , ε and the corresponding δ > 0 given by the corollary above, and set ε1 := (10)n+2 ε. In view of Corollary 3.8 and Lemma 2.2, we can obtain the decay estimate of (x, t) ∈ Q6 : M |Δu|2 > N12 ∩ Q1 , which, through induction, finally leads to the so-called good Λ-inequalities as follows. Corollary 3.9. Let u ∈ W2,2,1 be a solution of (1.1) in Q6 and k be a positive integer. Assume that (x, t) ∈ Q1 : M |Δu|2 > N12 < ε|Q1 |.
(3.27)
Then we have (x, t) ∈ Q1 : M |Δu|2 > N12k ≤ εk1 (x, t) ∈ Q1 : M |Δu|2 > 1 +
k
2(k−i) εi1 (x, t) ∈ Q1 : M f 2 > δ 2 N1 .
i=1
Proof. We prove by induction on k. For the case k = 1, set E = (x, t) ∈ Q1 : M |Δu|2 > N12 , F = (x, t) ∈ Q1 : M |Δu|2 > 1 ∪ (x, t) ∈ Q1 : M f 2 > δ 2 . Then in view of (3.27) and Corollary 3.8, we apply Lemma 2.2 to obtain that |E| ≤ ε1 |F |, so our conclusion is valid for k = 1. Suppose then that the conclusion is true for some positive integer k ≥ 2. We define u1 = and corresponding f1 = Nf1 . Then u1 is a solution of ⎧ ⎨ ∂u1 − λΔu1 + ∇p1 = f1 , ∂t ⎩ div u1 = 0,
u N1 ,
in Q6 , and the following inequality holds (x, t) ∈ Q1 : M |Δu1 |2 > N12 < ε|Q1 |. Now it follows from induction assumption and simple computations that (x, t) ∈ Q1 : M |Δu|2 > N 2(k+1) 1 = (x, t) ∈ Q1 : M |Δu1 |2 > N12k ≤
k
2(k−i) + εk1 (x, t) ∈ Q1 : M |Δu1 |2 > 1 εi1 (x, t) ∈ Q1 : M |f1 |2 > δ 2 N1
i=1
≤
k+1
2(k+1−i) (x, t) ∈ Q1 : M |Δu|2 > 1 . + εk+1 εi1 (x, t) ∈ Q1 : M |f |2 > δ 2 N1 1
i=1
These estimates in turn complete the induction on k. 2
p1 =
p N1 ,
F. Hu et al. / J. Math. Anal. Appl. 420 (2014) 1251–1264
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Now we are in a position to prove the main result of this section, the interior Lp estimate of D2 u, whose proof is based on Theorem 2.1 and Corollary 3.9. Proof of Theorem 3.1. We only consider the regularity issue of our solution when p > 2. The case p = 2 has been established in Lemma 3.4 and the case 1 < p < 2 can be recovered by duality. Normalizing the linear system (1.1) by multiplying an arbitrary small constant depending only on f Lp (Q6 ) and uL2 (Q6 ) , we assume f Lp (Q6 ) is small enough,
(3.28)
(x, t) ∈ Q1 : M |Δu|2 > N12 < ε|Q1 |,
(3.29)
and that
where we have used the fact that (x, t) ∈ Q1 : M |Δu|2 > N12 ≤ C
|Δu|2 dx dt Q1
≤ C u2L2 (Q6 ) + f 2L2 (Q6 ) according to the weak 2–2 estimate and Lemma 3.4. p Since f ∈ Lp (Q6 ), we have M(|f |2 ) ∈ L 2 (Q6 ) by strong p–p estimates. According to Theorem 2.1, there exists a constant C depending only on δ, p and N1 such that ∞
p N1pk (x, t) ∈ Q1 : M |f |2 > δ 2 N12k ≤ C M f 2 2 p
L 2 (Q6 )
k=1
.
Then this estimate, strong p–p estimates, and (3.28) imply ∞
N1pk (x, t) ∈ Q1 : M |f |2 > δ 2 N12k ≤ 1.
(3.30)
k=1
Now we employ Corollary 3.9 and (3.30) to estimate as follows: ∞
N1pk (x, t) ∈ Q1 : M |Δu|2 > N12k
k=1
≤
∞
∞ k 2(k−i) N1pk εk1 (x, t) ∈ Q1 : M |Δu|2 > 1 + N1pk εi1 (x, t) ∈ Q1 : M |f |2 > δ 2 N1 k=1 i=1
k=1
=
∞
N1p ε1
k (x, t) ∈ Q1 : M |Δu|2 > 1
k=1
+
∞ ∞ p i p(k−i) 2(k−i) (x, t) ∈ Q1 : M |f |2 > δ 2 N1 N1 ε1 N1 i=1
≤ |Q1 | + 1
k=i ∞ k=1
≤C
N1p ε1
k
F. Hu et al. / J. Math. Anal. Appl. 420 (2014) 1251–1264
1263
provided ε > 0 is selected small enough to have N1p ε1 = N1p 10n+2 ε < 1. This selection is possible since N1 is a universal constant depending only on n and λ. Consequently we obtain ∞
N1pk (x, t) ∈ Q1 : M |Δu|2 > N12k ≤ C
k=1 p
for some constant C = C(p, n, λ) > 0. Then it follows from Theorem 2.1 that M(|Δu|2 ) ∈ L 2 (Q1 ), which in turn implies that Δu ∈ Lp (Q1 ). Moreover, if we assume u ∈ Lp (Q6 ), we recall the normalization to obtain that ΔuLp (Q1 ) ≤ C uLp (Q6 ) + f Lp (Q6 )
(3.31)
for some constant C = C(p, n, λ) > 0. On the other hand, it follows from the classical Calderón–Zygmmund estimates that
2 p D u dx ≤ C
B1
|u| dx + B1
2
|Δu| dx
p
for a.e. t ∈ [−1, 1].
p
(3.32)
B1
Integrating with respect to t from − 14 to
1 4
and noting (3.31), we see
2
D u p ≤ C uLp (Q6 ) + f Lp (Q6 ) , L (Q 1 ) 2
which completes the proof of (3.2). In order to get (3.1) in Theorem 3.1 without the assumption u ∈ Lp (Q6 ), we have to treat the estimates above more precisely. In fact, it follows from the initial normalization that the norm uLp in the right member of (3.31) originates from the L2 norm of u. So (3.31) remains to be valid with uLp replaced by uL2 . Obviously, the same is true for (3.32), where the modified term uL2 can be further replaced by ∇uL2 according to Poincaré inequality since (3.32) is also valid for u − const.; that is, we have
2
D u
Lp (B
1 2
)
≤ C ΔuLp (B1 ) + ∇uL2 (B1 )
for a.e. t ∈ [−1, 1].
(3.33)
Finally, we integrate (3.33) with respect to t and utilize the modified (3.31) and Lemma 3.4 to deduce
2 p D u dx dt ≤ C
Q1
∇upL2 (B1 )
|Δu| dx dt +
dt
−1
Q1
2
1 p
|Δu|p dx dt +
≤C Q1
sup ∇upL2 (B1 )
t∈(−1,1)
p ≤ C uL2 (Q6 ) + f Lp (Q6 ) + ut L2 (Q6 ) , which completes the proof of Theorem 3.1.
2
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References [1] S.S. Byun, L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math. 57 (10) (2004) 1283–1310. [2] S.S. Byun, L. Wang, Parabolic equations with BMO coefficients in Lipschitz domains, J. Differential Equations 209 (2) (2005) 229–265. [3] L.A. Caffarelli, X. Cabré, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ., vol. 43, Amer. Math. Soc., Providence, RI, 1995. [4] L.A. Caffarelli, I. Peral, On W 1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1–21, MR 1486629(99c:35053). [5] A.P. Calderón, A. Zygmmund, On the existence of certain singular integrals, Acta Math. 88 (1952) 85–139. [6] M. Giaquinta, G. Modica, Nonlinear systems of the type of the stationary Navier–Stokes system, J. Reine Angew. Math. 330 (1982) 173–214. [7] V.A. Solonnikov, Estimates for solutions of a non-stationary linearized system of Navier–Stokes equations, Tr. Mat. Inst. Steklova 70 (1964) 213–317, Amer. Math. Soc. Transl. 75 (1968) 1–116. [8] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., vol. 30, Princeton University Press, Princeton, NJ, 1970. [9] R. Temam, Navier–Stokes Equations, Stud. Math. Appl., vol. 2, Elsevier North-Holland, Inc., New York, 1979. [10] Lihe Wang, A geometric approach to the Calderón–Zygmmund estimates, Acta Math. Sinica 19 (2003) 381–396.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
A new proof of Lp estimates of Stokes equations ✩ Fangju Hu a,∗ , Dongsheng Li a , Lihe Wang b,c a b c
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China
a r t i c l e
i n f o
a b s t r a c t In this paper, we establish a new proof of Lp estimates of the Stokes equations, which enables us to avoid the use of potential theory and thus simplifies the traditional proof. The modified Vitali covering lemma and the maximal function technique are the main analytical tools. © 2014 Elsevier Inc. All rights reserved.
Article history: Received 19 January 2014 Available online 20 June 2014 Submitted by D. Wang Keywords: Stokes equations Lp estimates Maximal function Vitali covering lemma
1. Introduction In this paper, we consider the following Stokes equations:
ut − λΔu + ∇p = f, div u = 0,
(1.1)
in the cylindrical domain ΩT := Ω × (a, a + T ], where Ω is a domain in Rn and a ∈ R. Here u(x, t) represents the velocity field, λ > 0 is the viscosity constants, the pressure p(x, t) is a scalar function, and f is the external force. We introduce a Sobolev space Wp,2,1 (ΩT ) = u : u, Du, D2 u, ut ∈ Lp (ΩT ) . We are interested in how the Lp -regularity of the nonhomogeneous term f is reflected to the solutions of the Stokes equations (1.1). More precisely, we want to establish such an estimate as ✩
Supported by NSFC grant 11171266.
* Corresponding author. E-mail addresses: [email protected] (F. Hu), [email protected] (D. Li), [email protected] (L. Wang). http://dx.doi.org/10.1016/j.jmaa.2014.06.039 0022-247X/© 2014 Elsevier Inc. All rights reserved.
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F. Hu et al. / J. Math. Anal. Appl. 420 (2014) 1251–1264
f ∈ Lp
⇒
D2 u ∈ Lploc ,
(1.2)
and note here generally ut ∈ Lploc or ∇p ∈ Lploc is not true for our interior estimates, see Remark 1.2. Regularity estimates in Lp spaces are one of the fundamental estimates for partial differential equations. They play important roles in the regularity theory of relevant differential operators. The classical Calderón–Zygmmund estimates were established in [5] with the main argument based on explicit representation formulas involving singular integral operators and commutators, which made the proof rather intricate and skillful. However, the integral kernels and the corresponding estimates seem hard to compute in many cases, so we need an alternative method of Lp estimates to avoid the use of potential theory. The maximum function approach, which was introduced by Caffarelli and Peral in [4], enables us to obtain the Lp estimates by combining the basic tools in harmonic analysis and some regularity results in partial differential equations; that is, based on the Calderón–Zygmmund decomposition, the Hardy–Littlewood maximal function, the approximation method and some standard estimates for equations, it gives an alternative proof to the classical one. Moreover, influenced by [4], L.H. Wang [10] gives another proof from geometric intuitions, where the maximal function technique is still the main tool, while a modified Vitali covering lemma is used instead of Calderón–Zygmmund decomposition. As pointed out in [10], this method is much more applicable since Vitali covering lemma can be easily adapted to any complicated manifolds. The Stokes equations are linear, but nevertheless they deserve special attention, because of the incompressibility condition div u = 0. The study of problem (1.1) is quite useful in the analysis of the nonlinear system ⎧ n ⎪ ⎨u + ui Di u − λΔu + ∇p = f, t i=1 ⎪ ⎩ div u = 0, which describes the motion of an incompressible viscous fluid. Furthermore, the linearized system (1.1) is quite interesting also from the purely mathematical point of view, as it does not belong to any of the well-known classes of differential equations and has its own characteristics. As for the Lp -regularity of Stokes equations, V.A. Solonnikov [7] developed potential theory in great detail to express the solutions in explicit form via Oseen’s tensor Tij and the influence tensor Gij and finally obtained the Lp estimates through estimating these tensors, which made the proof rather difficult. The aim of this paper is to show a new proof of the Lp estimates for Stokes equations to simplify the traditional method in [7]. We shall adopt the ideas in [4] and [10], which enables us to completely avoid the complexity of potential theory, and thus reflects a significant simplification. Accordingly, we are able to extend the applicable range of the maximal function approach from elliptic/parabolic problems to Stokes system. The maximal function approach has been employed to obtain the Lp estimates for solutions to a large class of elliptic and parabolic problems since Caffarelli and Peral introduced it, see [1,2] (also the references therein). However, the Stokes equations are neither elliptic nor parabolic, which is one of the main difficulties of our problem. So it is necessary to consider the regularity of solutions with respect to space and time variable separately. As we will see, it is the peculiar characteristics of Stokes equations that make our results and proofs quite different from those of elliptic/parabolic ones. Moreover, since the interior Lp estimates for ut is not true in general, we will meet difficulties when treating some estimates concerning ut . So it is worth mentioning that we do not give a direct proof of D2u in Lp , instead we consider Δu at first just to circumvent the obstacle brought by ut , which will be stated in Remark 3.6. Throughout this paper, by considering the distribution of maximal function of Δu, we have to find the decay estimates of it. Finally we mention here that, because of the technique we are using, it appears possible to extend the Lp regularity for Stokes system in Rn to that on Riemannian manifolds. The development of this project will be our aim in the near future.
F. Hu et al. / J. Math. Anal. Appl. 420 (2014) 1251–1264
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Our main result is stated as follows. Theorem 1.1. Let u ∈ W2,2,1 (ΩT ) and p ∈ L2 (a, a + T ; H 1 (Ω)) be solutions of the Stokes equations (1.1). Assume f ∈ Lp (ΩT ) for some 1 < p < ∞. Then for any ΩT = Ω × (a , b ] with Ω Ω and a < a < b ≤ a + T , we have
2
D u
≤ C f Lp (ΩT ) + uL2 (ΩT ) + ut L2 (ΩT ) .
) Lp (ΩT
(1.3)
In addition, if we assume that u ∈ W2,2,1 (ΩT ) ∩ Lp (ΩT ), then
2
D u
) Lp (ΩT
≤ C f Lp (ΩT ) + uLp (ΩT ) ,
(1.4)
where C depends only on n, p, λ and dist(ΩT , ∂p ΩT ). Remark 1.2. (1) As mentioned above, we can only obtain the interior Lp estimates for D2 u and no Lp estimates for ut or ∇p can be expected in general, which is illustrated by the following counterexample. Let p > 2. Assume Ω ⊂ Rn is bounded and that g1 : (a, a + T ) → R satisfies g1 ∈ L2 (a, a + T ),
but g1 ∈ / Lp a , b for some a , b ⊂ (a, a + T ).
(1.5)
Set
t g(t) =
g1 (s) ds a
and take u(x, t) = g(t)∇H(x) and p = −g (t)H(x) = −g1 (t)H(x),
(1.6)
where H(x) ∈ C 3 (Ω) is harmonic. It is evident that u ∈ W2,2,1 (ΩT ) and p ∈ L2 (0, T ; H 1 (Ω)) satisfy Stokes equations (1.1) with f ≡ 0. One can easily verify that D2 u = g(t)D3 H(x) ∈ Lp ΩT . However, it follows from (1.5) that ut = g (t)∇H(x) = g1 (t)∇H(x) ∈ / Lp ΩT , and ∇p = −g1 (t)∇H(x) ∈ / Lp ΩT . (2) The term ut L2 (ΩT ) cannot be removed in the estimate (1.3), or, the member uLp (ΩT ) in (1.4) cannot be replaced by uL2 (ΩT ) . We can still illustrate the fact by using the counterexample (1.6) with g(t) to be determined. Otherwise, for any p > 2 and any M > 0, we can always find g(t) such that
g(t) p ≥ M g(t) 2 , L (a ,b ) L (a,a+T ) and consequently,
for some a , b ⊂ (a, a + T ),
F. Hu et al. / J. Math. Anal. Appl. 420 (2014) 1251–1264
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2
D u
) Lp (ΩT
≥
C2 M uL2 (ΩT ) , C1
where C1 = ∇HL2 (Ω) ,
C2 = D3 H Lp (Ω ) ,
which is a contradiction if we get rid of ut L2 (ΩT ) in (1.3) or replace uLp (ΩT ) by uL2 (ΩT ) in (1.4). (3) According to the Gagliardo–Nirenberg–Sobolev inequality, the additional assumption u ∈ Lp (ΩT ) n in (1.4) can be abandoned when 1 < p ≤ 2(n+2) n−2 and Ω is bounded in R (n ≥ 3). The organization of this paper will be as follows: In Section 2, we give some preliminary tools and some geometric analysis results which will be used for our approach. Section 3 will be devoted to deriving interior Lp estimates. We finish this section by introducing some notations. Note that some common notations in PDE are not mentioned here. Notation 1.3. 1. Br = {x ∈ Rn : |x| < r} is an open ball on Rn with center 0 and radius r > 0, Br (x) = Br + x. 2. Qr = Br × (−r2 , r2 ] is a middle centered parabolic cube with center point (0, 0), Qr (y, s) = Qr + (y, s), ∂p Qr = ∂Br × [−r2 , r2 ] ∪ Br × {t = −r2 } is its parabolic boundary. 3. ΩT := Ω × (a, a + T ] is a parabolic cylinder, ∂p ΩT = ∂Ω × [a, a + T ] ∪ Ω × {t = a} is its parabolic boundary. 4. For a locally integrable function f , f¯Qr =
1 |Qr |
f (x, t) dx dt Qr
is the average of f over Qr . For simplicity, we employ the letter C to denote universal constants depending only on n, p and λ. Thus the exact value denoted by C may change from line to line in a given computation. 2. Preliminary tools The following theorem provides an equivalent view of Lp functions. Theorem 2.1. (See [3].) Suppose that f is a nonnegative and measurable function in a bounded domain U ⊂ Rn × R. Let ν > 0 and M > 1 be constants. Then for any 0 < p < ∞, we have f ∈ Lp (U )
⇔
S :=
M kp (x, t) ∈ U : f (x, t) > νM k < ∞
k≥1
and 1 S ≤ f pLp (U ) ≤ c |U | + S , c where c > 0 is a constant depending only on ν, M and p.
F. Hu et al. / J. Math. Anal. Appl. 420 (2014) 1251–1264
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This theorem shows that f ∈ Lp if and only if the decay rate of |{(x, t) ∈ U : f (x, t) > νM k }| is suitably fast. Unfortunately, the measure |{(x, t) ∈ U : |f | > M }| is not stable in the iteration process, which, however, can be solved by introducing the maximal function. For a given function f defined on Rn × R, the parabolic maximal function Mf is defined as
1 Mf (x, t) = sup |Q (x, t)| r>0 r
f (y, s) dy ds,
Qr (x,t)
which satisfies strong p–p estimates and weak 1–1 estimates (see p. 5 in [8]). If f is defined only in a bounded domain U , we define its restricted maximal function as MU f (x) = M(χU f ), where χU is the standard characteristic function on U . The main technical tool for interior estimates is the following modified version of Vitali covering lemma. Lemma 2.2. (See [10].) Let 0 < ε < 1 and let E ⊂ F ⊂ Q1 be two measurable sets satisfying the following properties: 1. |E| ≤ ε|Q1 |. 2. For each (y, s) ∈ Q1 and for each r ∈ (0, 1], E ∩ Qr (y, s) ≥ εQr (y, s)
⇒
Qr (y, s) ∩ Q1 ⊂ F.
Then we have |E| ≤ (10)n+2 ε|F |. 3. Lp estimates for D 2 u This section will be devoted to obtaining interior Lp estimates for D2 u concerning the Stokes equations (1.1) in ΩT . For convenience, we will instead prove the following theorem, from which Theorem 1.1 follows easily by using proper procedures of translation, scaling, and covering. Theorem 3.1. Assume that u ∈ W2,2,1 (Q6 ) is a solution of the Stokes PDE (1.1) with f ∈ Lp (Q6 ) for some 1 < p < ∞. Then we have
2
D u
Lp (Q 1 )
≤ C f Lp (Q6 ) + uL2 (Q6 ) + ut L2 (Q6 ) .
(3.1)
2
In addition, if we assume that u ∈ W2,2,1 (Q6 ) ∩ Lp (Q6 ), then
2
D u p ≤ C uLp (Q6 ) + f Lp (Q6 ) , L (Q 1 )
(3.2)
2
where C > 0 is a constant depending only on n, p and λ. In order to prove the theorem, we need to establish several lemmas and corollaries. We start with the L2 regularity for Stokes equations.
F. Hu et al. / J. Math. Anal. Appl. 420 (2014) 1251–1264
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Lemma 3.2. (See [9].) Let f ∈ L2 (ΩT ). If u ∈ W2,2,1 (ΩT ) is a solution of ⎧ u − λΔu + ∇p = f ⎪ ⎨ t div u = 0 ⎪ ⎩ u=0
in ΩT , in ΩT ,
(3.3)
on ∂p ΩT ,
then
2
D u
L2 (ΩT )
≤ Cf L2 (ΩT ) .
To obtain the interior L2 estimates for Stokes equations, we shall first refer to the following ε-lemma, whose proof comes from Lemma 0.5 in [6]. Lemma 3.3. Let f , g, h be nonnegative functions in L1 (Q0 ), where Q0 is a cube in Rn+1 , and let α ∈ R+ . There exists ε0 such that if for some ε ≤ ε0 the following inequality
f dx dt ≤ c(ε) QR (x0 ,t0 )
g dx dt + Q2R (x0 ,t0 )
holds for all (x0 , t0 ) ∈ Q0 and R <
1 2
Q2R (x0 ,t0 )
1 f dx dt ≤ c Rα
f dx dt
Q2R (x0 ,t0 )
g dx dt + Q2R (x0 ,t0 )
1 2
dist((x0 , t0 ), ∂p Q0 ), then there exists a constant c such that
QR (x0 ,t0 )
for all (x0 , t0 ) ∈ Q0 and R <
h dx dt + ε
1 Rα
h dx dt
Q2R (x0 ,t0 )
dist((x0 , t0 ), ∂p Q0 ).
Lemma 3.4. Let f ∈ L2 (Q6 ). If u ∈ W2,2,1 (Q6 ) is a solution of ut − λΔu + ∇p = f,
(3.4)
div u = 0
(3.5)
in Q6 , then we have
2
D u 2 L (Q
1)
≤ C f L2 (Q6 ) + uL2 (Q6 )
(3.6)
and sup ∇uL2 (B1 ) ≤ C f L2 (Q6 ) + uL2 (Q6 ) + ut L2 (Q6 ) .
(3.7)
t∈[−1,1]
Proof. In view of the classical Calderón–Zygmmund estimates for p = 2,
B1
2 2 D u dx ≤ C
|u| dx + 2
B2
|Δu| dx , 2
for a.e. t ∈ (−1, 1],
B2
the proof of (3.6) can be reduced to that of ΔuL2 (Q2 ) ≤ C f L2 (Q6 ) + uL2 (Q6 ) .
(3.8)
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Consider our problem initially in Q2R (x0 , t0 ) for any (x0 , t0 ) ∈ Q6 and all R < 12 dist((x0 , t0 ), ∂p Q6 ). Abbreviate QR (x0 , t0 ) = QR , BR (x0 ) = BR . We choose a smooth cut-off function η = η(x, t) ∈ Cc∞ (Q2R ) satisfying 0 ≤ η ≤ 1,
η=1
in QR ,
near t = t0 + 4R2 ,
η=0
(3.9)
and |∇η| ≤
2 D η ≤ C , R2
C , R
|ηt | ≤
C . R2
(3.10)
Multiplying Eq. (3.4) by ϕ = Δ(uη 2 ), we obtain
ut Δ uη 2 dx −
B2R
λΔuΔ uη 2 dx −
B2R
Δp div uη 2 dx =
B2R
f Δ uη 2 dx.
B2R
Using the relation Δp = div f , which is obtained by taking divergence (in the weak sense) on both sides of (3.4), we derive from Green’s formula and Cauchy’s inequality that d dt
|∇u|2 η 2 dx − B2R
d dt
|u|2 D2 η 2 dx +
B2R
B2R
B2R
+ c(ε)
c f 2 dx + ε
|Δu|2 dx + c(ε)
≤ε
λ|Δu|2 η 2 dx
B2R
|∇u|2 |ηt | + |∇η|2 dx
B2R
2 |u|2 D2 η + |∇η|4 + Dt D2 η 2 dx
(3.11)
B2R
for arbitrary ε > 0. Substituting the conditions (3.9), (3.10) on η into (3.11), along with the interpolation inequality for ∇u, we obtain by integrating with respect to t from t0 − 4R2 to t0 + 4R2 that
|Δu|2 dx dt ≤ c(ε) QR
1 R4
|u|2 dx dt + Q2R
|f |2 dx dt + ε |Δu|2 dx dt.
Q2R
Q2R
According to Lemma 3.3, there exists a constant C such that
|Δu|2 dx dt ≤ C
1 R4
QR
|f |2 dx dt ,
|u|2 dx dt + Q2R
Q2R
which completes the proof of (3.8) for R = 2 and (x0 , t0 ) = (0, 0). We propose next to derive estimate (3.7). Note that
d dt
|u|2 D2 η 2 dx = B2R
|u|2 Dt D2 η 2 dx + 2 B2R
ut uD2 η 2 dx.
(3.12)
B2R
Returning to inequality (3.11) we now substitute the relation (3.12) and apply Cauchy’s inequality and interpolation inequality as above to obtain d dt
|∇u|2 η 2 dx ≤ C
B2R
B2R
C |Δu|2 + |ut |2 + f 2 dx + 4 R
|u|2 dx. B2R
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In particular, for (x0 , t0 ) = (0, 0) and R = 1, we integrate in time from −4 to t (−4 < t < 4), to obtain
sup
|∇u| dx ≤ C 2
t∈[−1,1] B1
|Δu|2 + |ut |2 + f 2 + u2 dx dt
|ut |2 + f 2 + u2 dx dt,
Q2
≤C Q6
where the last inequality comes from (3.8). Then we complete the proof of (3.7). 2 Lemma 3.5. Let v ∈ W2,2,1 (Q4 ) be a solution of ⎧ v − λΔv + ∇h = 0 ⎪ ⎨ t div v = 0 ⎪ ⎩ v=u
in Q4 , in Q4 ,
(3.13)
on ∂p Q4 ,
where u ∈ W2,2,1 (Q4 ) is a solution of (1.1) with f ∈ L2 (Q4 ). Then we have Δv2L∞ (Q3 )
≤C
|f | dx dt + 2
Q4
|Δu| dx dt . 2
(3.14)
Q4
Proof. Obviously v is smooth inside of Q4 . It is sufficient to prove the interior estimate (3.14) for smooth v. We start from dealing with the first equation of (3.13). Observing that div v = 0, we apply the operator div to both sides of it to obtain that Δh = 0. As a result, we have (Δv)t − λΔ(Δv) = 0
(3.15)
by imposing the Laplace operator on both sides of the first equation. Thus the interior L∞ -regularity for heat equation (3.15) implies that ΔvL∞ (Q3 ) ≤ CΔvL2 (Q4 ) ≤ C Δ(u − v)L2 (Q4 ) + ΔuL2 (Q4 ) ≤ C f L2 (Q4 ) + ΔuL2 (Q4 ) , where the proof of the last inequality is the same as that of (3.20). This completes the proof. 2 Remark 3.6. Note here, although the estimate
2
D v ∞ L (Q
3)
≤ C ut L2 (Q4 ) + D2 u L2 (Q ) 4
is also available just as the parabolic case (see Lemma 11 in [10]), it is useless for us since we have no control information about ut in the iteration steps due to lack of interior Lp regularity for ut . As a consequence, we will initially consider the Lp estimates of Δu in the following instead of that of D2 u. So the approach here differs from that of elliptic and parabolic equations.
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Lemma 3.7. There is a universal constant N1 > 1, so that for any ε > 0, there exists a small constant δ = δ(ε) > 0 such that if u is a solution of (1.1) in Q6 with Q1 ∩ (x, t) : M |Δu|2 ≤ 1 ∩ (x, t) : M |f |2 ≤ δ 2 = ∅,
(3.16)
then we have (x, t) : M |Δu|2 > N12 ∩ Q1 < ε|Q1 |. Proof. From (3.16), we see that there is a point (x0 , t0 ) ∈ Q1 such that for all r > 0,
1 |Qr |
1 |Qr |
|Δu| dx dt ≤ 1 and 2
Qr (x0 ,t0 )
|f |2 dx dt ≤ δ 2 .
(3.17)
Qr (x0 ,t0 )
Since Q4 ⊂ Q5 (x0 , t0 ) ⊂ Q6 , it follows from (3.17) that 1 |Q4 |
|f |2 dx dt ≤ Q4
|Q5 | 1 |Q4 | |Q5 |
|f |2 dx dt ≤ (5/4)n+2 δ 2 .
(3.18)
Q5 (x0 ,t0 )
By the same reason, we have 1 |Q4 |
|Δu|2 dx dt ≤ (5/4)n+2 .
(3.19)
Q4
Let (v, h) be the solution of (3.13). Then (u − v, p − h) is the solution of (3.3) in Q4 . Using (3.18) and Lemma 3.2, we have
Δ(u − v)2 dx dt ≤ C
Q4
|f |2 dx dt ≤ Cδ 2 .
(3.20)
Q4
In view of (3.18), (3.19) and by the interior L∞ regularity for Δv (Lemma 3.5), we can choose some universal constant N0 = N0 (n, λ) > 1 such that Δv2L∞ (Q3 ) ≤ N02 .
(3.21)
Denote by N1 the constant N12 = max{4N02 , 2n+2 } and assert that 2 (x, t) ∈ Q1 : M |Δu|2 > N12 ⊂ (x, t) ∈ Q1 : MQ4 Δ(u − v) > N02 .
(3.22)
To see this, fix any point (x1 , t1 ) ∈ Q1 satisfying 2 MQ4 Δ(u − v) ≤ N02 .
(3.23)
Then it suffices to show that M(|Δu|2 )(x1 , t1 ) ≤ N12 . Indeed, if 0 < r ≤ 2, then Qr (x1 , t1 ) ⊂ Q3 , whence (3.21) and (3.23) imply 1 |Qr |
2 |Δu| dx dt ≤ |Qr |
2
Qr (x1 ,t1 )
Qr (x1 ,t1 )
Δ(u − v)2 + |Δv|2 dx dt ≤ 4N02 .
(3.24)
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On the other hand, if r > 2, then (x1 , t1 ) ∈ Qr (x1 , t1 ) ⊂ Q2r (x0 , t0 ), whence (3.17) implies 1 |Qr |
|Δu|2 dx dt ≤
1 |Qr |
Qr (x1 ,t1 )
|Δu|2 dx dt ≤ 2n+2 .
(3.25)
Q2r (x0 ,t0 )
Consequently (3.24) and (3.25) imply (x1 , t1 ) ∈ (x, t) : M |Δu|2 ≤ N12 ∩ Q1 .
(3.26)
Thus assertion (3.22) follows easily from (3.23) and (3.26). Now (3.22), parabolic weak 1–1 estimate and (3.20) finally yield (x, t) : M |Δu|2 > N12 ∩ Q1 ≤ (x, t) : MQ4 Δ(u − v)2 > N02 ∩ Q1
2 C ≤ 2 Δ(u − v) dx dt N0 Q4
≤
C 2 δ < ε|Q1 |, N02
by selecting δ > 0 sufficiently small so that the last inequality above is valid. This completes the proof of the lemma. 2 The scaling invariant form of Lemma 3.7 will be the following corollary, which makes the second condition in Lemma 2.2 satisfied, and hence can be viewed as a crucial part during the first iteration for Lp estimates. Corollary 3.8. Let 0 < r ≤ 1 and (y, s) ∈ Q1 . Assume that u ∈ W2,2,1 is a solution of (1.1) in Q6 . Then there is a universal constant N1 > 1, so that for any ε > 0, there exists δ = δ(ε) > 0 such that if (x, t) ∈ Q1 : M |Δu|2 > N12 ∩ Qr (y, s) ≥ εQr (y, s), then we have Qr (y, s) ∩ Q1 ⊂ (x, t) ∈ Q1 : M |Δu|2 > 1 ∪ (x, t) ∈ Q1 : M |f |2 > δ 2 . Proof. Proof by a scaling: Since Stokes equations are invariant under a translation, we may assume (y, s) = (0, 0). We define for 0 < r ≤ 1 and (x, t) ∈ Q6 , ur (x, t) :=
u(rx, r2 t) , r2
pr (x, t) :=
p(rx, r2 t) , r
fr (x, t) := f rx, r2 t .
Under this scaling, we apply Lemma 3.7, with ur replacing u, pr replacing p, and fr replacing f , respectively, to deduce that the condition Qr ∩ (x, t) : M |Δu|2 ≤ 1 ∩ (x, t) : M |f |2 ≤ δ 2 = ∅ implies (x, t) : M |Δu|2 > N12 ∩ Qr < ε|Qr |. We then argue by contradiction to have Corollary 3.8 as the counterpart of Lemma 3.7.
2
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Now we take N1 , ε and the corresponding δ > 0 given by the corollary above, and set ε1 := (10)n+2 ε. In view of Corollary 3.8 and Lemma 2.2, we can obtain the decay estimate of (x, t) ∈ Q6 : M |Δu|2 > N12 ∩ Q1 , which, through induction, finally leads to the so-called good Λ-inequalities as follows. Corollary 3.9. Let u ∈ W2,2,1 be a solution of (1.1) in Q6 and k be a positive integer. Assume that (x, t) ∈ Q1 : M |Δu|2 > N12 < ε|Q1 |.
(3.27)
Then we have (x, t) ∈ Q1 : M |Δu|2 > N12k ≤ εk1 (x, t) ∈ Q1 : M |Δu|2 > 1 +
k
2(k−i) εi1 (x, t) ∈ Q1 : M f 2 > δ 2 N1 .
i=1
Proof. We prove by induction on k. For the case k = 1, set E = (x, t) ∈ Q1 : M |Δu|2 > N12 , F = (x, t) ∈ Q1 : M |Δu|2 > 1 ∪ (x, t) ∈ Q1 : M f 2 > δ 2 . Then in view of (3.27) and Corollary 3.8, we apply Lemma 2.2 to obtain that |E| ≤ ε1 |F |, so our conclusion is valid for k = 1. Suppose then that the conclusion is true for some positive integer k ≥ 2. We define u1 = and corresponding f1 = Nf1 . Then u1 is a solution of ⎧ ⎨ ∂u1 − λΔu1 + ∇p1 = f1 , ∂t ⎩ div u1 = 0,
u N1 ,
in Q6 , and the following inequality holds (x, t) ∈ Q1 : M |Δu1 |2 > N12 < ε|Q1 |. Now it follows from induction assumption and simple computations that (x, t) ∈ Q1 : M |Δu|2 > N 2(k+1) 1 = (x, t) ∈ Q1 : M |Δu1 |2 > N12k ≤
k
2(k−i) + εk1 (x, t) ∈ Q1 : M |Δu1 |2 > 1 εi1 (x, t) ∈ Q1 : M |f1 |2 > δ 2 N1
i=1
≤
k+1
2(k+1−i) (x, t) ∈ Q1 : M |Δu|2 > 1 . + εk+1 εi1 (x, t) ∈ Q1 : M |f |2 > δ 2 N1 1
i=1
These estimates in turn complete the induction on k. 2
p1 =
p N1 ,
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Now we are in a position to prove the main result of this section, the interior Lp estimate of D2 u, whose proof is based on Theorem 2.1 and Corollary 3.9. Proof of Theorem 3.1. We only consider the regularity issue of our solution when p > 2. The case p = 2 has been established in Lemma 3.4 and the case 1 < p < 2 can be recovered by duality. Normalizing the linear system (1.1) by multiplying an arbitrary small constant depending only on f Lp (Q6 ) and uL2 (Q6 ) , we assume f Lp (Q6 ) is small enough,
(3.28)
(x, t) ∈ Q1 : M |Δu|2 > N12 < ε|Q1 |,
(3.29)
and that
where we have used the fact that (x, t) ∈ Q1 : M |Δu|2 > N12 ≤ C
|Δu|2 dx dt Q1
≤ C u2L2 (Q6 ) + f 2L2 (Q6 ) according to the weak 2–2 estimate and Lemma 3.4. p Since f ∈ Lp (Q6 ), we have M(|f |2 ) ∈ L 2 (Q6 ) by strong p–p estimates. According to Theorem 2.1, there exists a constant C depending only on δ, p and N1 such that ∞
p N1pk (x, t) ∈ Q1 : M |f |2 > δ 2 N12k ≤ C M f 2 2 p
L 2 (Q6 )
k=1
.
Then this estimate, strong p–p estimates, and (3.28) imply ∞
N1pk (x, t) ∈ Q1 : M |f |2 > δ 2 N12k ≤ 1.
(3.30)
k=1
Now we employ Corollary 3.9 and (3.30) to estimate as follows: ∞
N1pk (x, t) ∈ Q1 : M |Δu|2 > N12k
k=1
≤
∞
∞ k 2(k−i) N1pk εk1 (x, t) ∈ Q1 : M |Δu|2 > 1 + N1pk εi1 (x, t) ∈ Q1 : M |f |2 > δ 2 N1 k=1 i=1
k=1
=
∞
N1p ε1
k (x, t) ∈ Q1 : M |Δu|2 > 1
k=1
+
∞ ∞ p i p(k−i) 2(k−i) (x, t) ∈ Q1 : M |f |2 > δ 2 N1 N1 ε1 N1 i=1
≤ |Q1 | + 1
k=i ∞ k=1
≤C
N1p ε1
k
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provided ε > 0 is selected small enough to have N1p ε1 = N1p 10n+2 ε < 1. This selection is possible since N1 is a universal constant depending only on n and λ. Consequently we obtain ∞
N1pk (x, t) ∈ Q1 : M |Δu|2 > N12k ≤ C
k=1 p
for some constant C = C(p, n, λ) > 0. Then it follows from Theorem 2.1 that M(|Δu|2 ) ∈ L 2 (Q1 ), which in turn implies that Δu ∈ Lp (Q1 ). Moreover, if we assume u ∈ Lp (Q6 ), we recall the normalization to obtain that ΔuLp (Q1 ) ≤ C uLp (Q6 ) + f Lp (Q6 )
(3.31)
for some constant C = C(p, n, λ) > 0. On the other hand, it follows from the classical Calderón–Zygmmund estimates that
2 p D u dx ≤ C
B1
|u| dx + B1
2
|Δu| dx
p
for a.e. t ∈ [−1, 1].
p
(3.32)
B1
Integrating with respect to t from − 14 to
1 4
and noting (3.31), we see
2
D u p ≤ C uLp (Q6 ) + f Lp (Q6 ) , L (Q 1 ) 2
which completes the proof of (3.2). In order to get (3.1) in Theorem 3.1 without the assumption u ∈ Lp (Q6 ), we have to treat the estimates above more precisely. In fact, it follows from the initial normalization that the norm uLp in the right member of (3.31) originates from the L2 norm of u. So (3.31) remains to be valid with uLp replaced by uL2 . Obviously, the same is true for (3.32), where the modified term uL2 can be further replaced by ∇uL2 according to Poincaré inequality since (3.32) is also valid for u − const.; that is, we have
2
D u
Lp (B
1 2
)
≤ C ΔuLp (B1 ) + ∇uL2 (B1 )
for a.e. t ∈ [−1, 1].
(3.33)
Finally, we integrate (3.33) with respect to t and utilize the modified (3.31) and Lemma 3.4 to deduce
2 p D u dx dt ≤ C
Q1
∇upL2 (B1 )
|Δu| dx dt +
dt
−1
Q1
2
1 p
|Δu|p dx dt +
≤C Q1
sup ∇upL2 (B1 )
t∈(−1,1)
p ≤ C uL2 (Q6 ) + f Lp (Q6 ) + ut L2 (Q6 ) , which completes the proof of Theorem 3.1.
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