LOCAL FORMULAS FOR THE HYDRODYNAMIC ... - Princeton Math
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE AND APPLICATIONS PETER CONSTANTIN Abstract. We provide local formulas for the pressure of incompressible fluids. The pressure can be expressed in terms of its average and averages of squares of velocity increments in arbitrary small neighborhoods. As application, we give a brief proof of the fact that C α velocities have C 2α (or Lipschitz) pressures. We also give some regularity criteria for 3D incompressible Navier-Stokes equations.
Dedicated to the memory of Professor Mark I. Vishik. 1. Introduction We provide local formulas for the pressure of incompressible fluids. By this we mean expressions that compute a solution of 3 X
∂2 (ui uj ), −∆p = ∂xi ∂xj i,j=1 where u is a divergence-free velocity, at x ∈ Ω ⊂ R3 , from the spherical average of the pressure, 1 p(x, r) = 4πr2
p(y)dS(y), |x−y|=r
and from integrals of increments (ui (y) − ui (x))(uj (y) − uj (x)), for |y − x| ≤ r, with arbitrary small r. No knowledge of the behavior of u outside a small ball is needed. The main ingredient is a kind of monotonicity equation for a modified object 2 y−x 1 b(x, r) = p(x, r) + · u(y) dS(y). 4πr2 |x−y|=r |y − x| Date: October 11, 2013. 2000 Mathematics Subject Classification. 35Q35. Key words and phrases. Navier-Stokes equations, Euler equations, pressure, regularity criteria. 1
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PETER CONSTANTIN
This allows us to express the pressure as p(x) = β(x, r) + π(x, r) where β is just a local average of the pressure, 1 β(x, r) = r
2r
p(x, ρ)dρ, r
and π(x, r) is given by a couple of integrals (39) of squares of increments of velocity over a ball and over an annulus of radii 2r. Thus, we write the pressure as a sum of two local terms, one small, and the other sufficiently well-behaved. Indeed, β ∈ L∞ (R3 ) is bounded in space (for any r), if u ∈ L2 (R3 ) (34), and k∇βkL2 (R3 ) is bounded in terms of kuk2L4 (R2 ) (47). On the other hand, π is of the order r2 |∇u|2 for small r. Well-known criteria for regularity for the 3D incompressible NavierStokes equations in terms of the pressure ([1]), ([7]) do exist. If the pressure would obey the bounds that β obeys, then regularity of solutions of the 3D Navier-Stokes equations would easily follow. Because π(x, r) → 0 as r → 0, the suggestion that p obey the same bounds as β is not unreasonable. On the other hand, bounds on π require some smoothness of the velocity. Higher regularity in space for velocity for weak solutions of the 3D Navier-Stokes equations was obtained in ([4]) (see also ([10])). These bounds imply that π(x, r) is small for almost all time. For instance, kπkL3 (R3 ) ≤ C(t)r2 , t − a.e. (52), (59). The problem is that in general the time integrability of C(t) is too poor 1 to conclude regularity (C(t) 3 is time integrable, whereas C(t) time integrable would be sufficient for regularity.) The organization of this paper is as follows: In the next section we present the basic calculations which lead to the formulas for the pressure. In section 3 we give ensuing bounds for β and π. In section 4 we give a quick proof of the bounds of higher derivatives of solutions of the 3D Navier-Stokes equations in the whole space. These follow from the classical paper ([4]), and were well-known for decades, although, because ([4]) deals with spatially periodic solutions, a proof in the whole space of one the results (due originally to Luc Tartar, see acknowledgment in ([4])) was given only in 2001 ([2]). The 2012 preprint ([9]) contains also a proof of this result and more references. In section 5 we give two applications: the first is a simple proof of the fact that, if u ∈ C α , then p ∈ C 2α (if 2α < 1; if 2α > 1 then p is Lipschitz). This result was used recently in ([5]), with a proof based on the LittlewoodPaley decomposition. A different proof (closer to ours) was obtained before, but was not published ([8]). The 3D Navier-Stokes equations are regular if u ∈ L∞ ([0, T ], L3 (R3 )) ([3]), ([6]). We give as a second
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
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application, criteria of regularity for the 3D Navier-Stokes equations in terms of π. These essentially say that if we can find r(t) small such that in some sense, π is small, and if some integral of r(t)−1 is finite, then we have regularity. Some elementary calculations needed for the formulas are presented in the Appendix. 2. Spherical averages We denote 1 f (x, r) = 4πr2
(1) where
f (y)dS(y) = |x−y|=r
f (x + rξ)dS(ξ) |ξ|=1
denotes normalized integral. We consider solutions of − ∆p = ∇ · (u · ∇u)
(2) 3
in Ω ⊂ R . We assume ∇ · u = 0 and smoothness of u. We start by computing ∂r p(x, r) 1 = |ξ|=1 ξ · ∇x p(x + rξ)dS(ξ) = 4πr ξ · ∇ξ p(x + rξ)dS(ξ) |ξ|=1 1 r = 4πr |ξ| 0, let p solve (2) in Ω = R3 with divergence-free u ∈ (C 2 (R3 ) ∩ L2 (R3 ))3 . Then (37)
p(x) = β(x, r) + π(x, r)
with β(x, r) given by (38)
1 β(x, r) = r
2r
p(x, ρ)dρ r
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PETER CONSTANTIN
and π(x, r) given by (39)
2
y−x 1 π(x, r) = · (u(y) − u(x)) |y−x| r≤|y−x|≤2r |y−x|2 [ σ ( x −y) ij 1 w |y−x| (ui (y) − ui (x))(uj (y) − 4π |x−y|≤2r r |x−y|3 1 4πr
dy+
uj (x))dy
Remark 8. Passing to the limit r → ∞ in (37) we obtain (40) 1 σij (b z) |u(x)|2 + p(x) = (ui (x + z) − ui (x))(uj (x + z) − uj (x))dz 3 3 4π R3 |z| This can be obtained also from (13) using (16). Proof. We integrate (41)
1 r
2r r
dρ the representation (14) written as
p(x) = p(x, ρ) + |y−x|=ρ |ξ · (u(y) − u(x))|2 dS(y) ρ + 0 1l |y−x|=l [3(ξ · (u(y) − u(x)))2 − |u(y) − u(x)|2 ] dS(y)
and use the fact that ρ 2r l 1 2r f (l)dl dρ = w f (l)dl. r r r 0 0 In addition to the bounds (34) and (35) we also have bounds that follow from Morrey inequality 3 2 6 |∇u(y)| dy , |u(y)| dy ≤ C R3
R3
the representation (42)
p = Ri Rj (ui uj ) 1
of the pressure where Ri = ∂i (−∆)− 2 are Riesz transforms, and the boundedness of Riesz transforms in Lp spaces. Proposition 3. Let p the solution of (2) given by (42). For any q, 1 < q < ∞ there exist constants Cq > 0, independent of r > 0 so that, for any r > 0 (43)
kp(·, r)kLq (R3 ) ≤ Cq kuk2L2q (R3 )
and (44)
kβ(·, r)kLq (R3 ) ≤ Cq kuk2L2q (R3 ) .
For any a ∈ [0, 2) there exists Ca > 0 such that (45)
kβ(·, r)kL3 (R3 ) ≤ Ca r−a kukaL2 (R3 ) k∇uk2−a L2 (R3 ) .
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
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There exists a constant C > 0 so that k∇p(·, r)kL2 ≤ Cr−1 kuk2L4 (R3 )
(46) and
k∇β(·, r)kL2 ≤ Cr−1 kuk2L4 (R3 )
(47)
Proof. The bounds (44) for β follow from the bounds (43) for p by averaging in r. The bounds (43) follow from (42) and the boundedness of Riesz transforms in Lp spaces. The bounds (45) follow from (35), interpolation a 1− a3 kβkL3 (R3 ) ≤ kβkL3 ∞ (R3 ) kβkL3−a (R3 ) , the bound (44) for q = 3 − a, kβ(·, r)kL3−a (R3 ) ≤ Ca kuk2L6−2a (R3 ) , and interpolation combined with the Morrey inequality 6−3a
a
6−2a kukL6−2a (R3 ) ≤ CkukL6−2a 2 (R3 ) k∇ukL2 (R3 ) .
The bound (47) follows from the bound (46) by averaging in r. The bound (46) follows from k∇p(·, r)kL2 (R3 ) ≤ Cr−1 kp(·, r)kL2 (R3 )
(48)
and (43) at q = 2. The bound (48) follows from Plancherel and the observation that sin(r|ξ|) b (49) pb(ξ). p(ξ, r) = r|ξ| Indeed,
e−ix·ξ p(x, r)dx = |ω|=1 dS(ω) = pb(ξ) |ω|=1 eirξ·ω dS(ω) R3
R3
e−ix·ξ p(x + rω)dx
and the last integral is computed conveniently choosing coordinates so that ξ points to the North pole:
2π 1 dφ 4π 0 Regarding π we have
π
dθeir|ξ| cos θ sin θdθ = 0
sin(r|ξ|) . r|ξ|
Proposition 4. Let π(x, r) be defined by (39). Then
(50)
|π(x, r)| ≤ C |z|≤2r
|u(x + z) − u(x)|2 dz. |z|3
Consequently (51)
kπ(·, r)kLq ≤ Cq r2 k∇uk2L2q
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PETER CONSTANTIN
holds for all 1 < q ≤ ∞. In particular, at q = 3 we have, with Morrey’s inequality, kπ(·, r)kL3 ≤ Cr2 k∆uk2L2 .
(52) We also have
kπ(·, r)kLq (R3 ) ≤ Cq kuk2L2q (R3 ) .
(53)
Proof. The inequality (50) is immediate from definition. In order to prove (51) we write
2
1
|∇u(x + λz)|2 dλ
2
|u(x + z) − u(x)| ≤ |z|
0
and changing order of integration we have 2 |u(x + z) − u(x)| ≤ Cr2 kφk q0 k∇uk2 2q dz L L 3 φ(x)dx |z|3 |z|≤2r
R
which proves (51). The bounds (53) follow from (37), the corresponding bounds for p, and (44). 4. FGT bounds in the whole space We take the Navier-Stokes equation ∂t u + u · ∇u − ν∆u + ∇p = 0,
(54) with
∇ · u = 0,
(55)
multiply by ∂t u − ν∆u and integrate, using incompressibility:
2
|∂t u − ν∆u| dx = − R3
(u · ∇u)(∂t u − ν∆u)dx. R3
Schwartz inequality gives:
2
|u · ∇u|2 dx
|∂t u − ν∆u| dx ≤ R3
R3
and so
R3
|∂t u − ν∆u|2 dx ≤ kuk2L∞ k∇uk2L2 .
The inequality kuk2L∞ ≤ Ck∇ukL2 k∆ukL2
(56)
is easy to prove using Fourier transform. Thus
R3
|∂t u − ν∆u|2 dx ≤ Ck∆ukL2 k∇uk3L2 .
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
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On the other hand,
|∂t u − ν∆u|2 dx = k∂t uk2L2 + ν 2 k∆uk2L2 + ν
R3
d k∇uk2L2 dt
and therefore d k∇uk2L2 + νk∆uk2L2 + ν1 k∂t uk2L2 dt ≤ Cν k∆ukL2 k∇uk2L2 ≤ ν2 k∆uk2L2 + νC3 k∇uk6L2 denote y(t) = k∇u(·, t)k2L2 , pick a constant A >
Now we 0, divide by 2 (A + y) and obtain νk∆uk2L2 k∂t uk2L2 1 d C + − + ≤ 3 y. 2 2 dt A + y (A + y) ν(A + y) ν Integrating in time we obtain
T
νk∆uk2L2 dt + (A + y)2
0
Therefore
0
k∂t uk2L2 C 1 2 dt ≤ ku k 2 + 0 L ν(A + y)2 ν4 A
k∆uk2L2 C 1 dt ≤ 5 ku0 k2L2 + = Cν −4 [D + ν 3 A−1 ] 2 (A + y) ν νA
T
k∂t uk2L2 C ν 2 dt ≤ ku k = Cν −2 [D + ν −3 A−1 ] 2 + 0 L (A + y)2 ν3 A
0
T
T
(57) and
(58) 0
where we put D= Now
T
2 3
k∆ukL2 dt ≤
0
and
T
2 3
k∂t ukL2 dt ≤
0
T 0
T 0
ku0 k2L2 . ν
31 T 32 k∆uk2L2 dt (A + y)dt (A + y)2 0 31 T 23 k∂t uk2L2 dt (A + y)dt (A + y)2 0
and therefore
T 0
and
T 0
2 1 2 4 k∆ukL3 2 dt ≤ Cν − 3 D + ν 3 A−1 3 [D + AT ] 3
2 1 2 2 k∂t ukL3 2 dt ≤ Cν − 3 D + ν 3 A−1 3 [D + AT ] 3
Now A is arbitrary, but a natural explicit choice is A2 = ν 3 T −1
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PETER CONSTANTIN
and then we have
T
k∆ukL2 dt ≤ Cν
(59) 0
and
2 3
T
2 3
k∂t ukL2 dt ≤ Cν
(60) 0
− 43
− 23
ku0 k2L2 1 3 + T 2ν2 ν
ku0 k ν
2 L2
1 2
+T ν
3 2
.
Now using the inequality (56) it follows immediately that 34 14 T 2 2 ku k ku k 3 1 2 2 0 0 L L kukL∞ dt ≤ Cν −1 (61) + T 2ν2 . ν ν 0 Let us consider now the other terms in (54). We start by computing
2
|u · ∇u + ∇p| dx = ku · R3
Now
+
k∇pk2L2
(u·∇u)·(∇p)dx = −2
2
∇uk2L2
R3
2
pTr(∇u) dx = 2
R3
R3
Consequently
(u · ∇u) · (∇p)dx.
+2
p∆pdx = −2k∇pk2L2 .
0≤ R3
|u · ∇u + ∇p|2 dx = ku · ∇uk2L2 − k∇pk2L2 .
On the other hand, obviously ku · ∇ukL2 ≤ kukL∞ k∇ukL2 and in view of the previous result we have 12 12 T 2 2 2 ku k ku k 1 3 2 2 2 0 0 L L ku · ∇ukL3 2 dt ≤ Cν − 3 + T 2ν2 (62) ν ν 0 and, because of the inequality k∇pkL2 ≤ ku · ∇ukL2 , we also have 12 1 T 2 2 ku0 k2L2 2 1 3 − 23 ku0 kL2 3 (63) k∇pkL2 dt ≤ Cν + T 2ν2 ν ν 0 We have thus Theorem 2. Let u be a Leray weak solution of the Navier-Stokes equa2 tion on the interval [0, T ]. Then the quantities kukL∞ (R3 ) , k∆ukL3 2 (R3 ) , 2
2
2
k∂t ukL3 2 (R3 ) , ku · ∇ukL3 2 (R3 ) , k∇pkL3 2 (R3 ) are almost everywhere finite on the time interval [0, T ], and their time integrals are bounded uniformly, with bounds (59, 60, 61, 62, 63) depending only on T , ku0 kL2 (R3 ) and ν.
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
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The proof for Leray weak solutions follows the same pattern as the proof given above for smooth solutions, except that we mollify the advecting velocity, prove the mollification-uniform bounds and deduce the result using essentially Fatou’s lemma. For the sake of completeness, let us mention here other estimates. Interpolating
T
k∇uk2L2 dt < ∞ 0
and
T 0
2
k∇ukL3 6 dt < ∞
which comes from Morrey’s inequality and (59) we get 1
1
k∇ukL3 ≤ Ck∇ukL2 6 k∇ukL2 2 which then is integrable by H¨older
T
k∇ukL3 dt < ∞. 0
Finally, we mention that, interpolating between L∞ (dt; L2 (dx)) and 6p L2 (dt; L6 (dx)) it is easy to see that u ∈ Lp (dt, Lq (dx)) for q = 3p−4 if p ≥ 2. For p ∈ [1, 2] interpolating between L2 (dt; L6 (dx)) and 3p . L1 (dt, L∞ (dx)) we get q = p−1 5. Applications Theorem 3. Let u solve (54) and (55) in R3 and assume that u belongs to L∞ (dt; L2 (R3 )) ∩ L2q (dt; C α (R3 )) for some q ≥ 1. Then p ∈ Lq (dt; C 2α (R3 )) if α < 21 . If α = 12 then p ∈ Lq (dt; LiplogLip) where LiplogLip is the class of functions with modulus of continuity |x − y| log(|x − y|−1 ). If α > 12 then p ∈ Lq (dt; Lip) where Lip is the class of Lipschitz continuous functions. Proof. We start with two points x, y at distance |x − y| and we choose r = 8|x − y|. The representation (14) implies |p(x) − p(x, r)| ≤ Ckuk2C α r2α , (64) |p(y) − p(y, r)| ≤ Ckuk2C α r2α , so, it remains to prove that |p(x, r) − p(y, r)| ≤ Cr2α if 2α < 1 and C ∼ kuk2C α . (If 2α = 1 we obtain r log(r−1 ), and if 2α > 1, r.) In order to do so, we use (4) with v = u x+y and 2
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PETER CONSTANTIN
integrate from r to infinity. We obtain p(x, r) = − (65)
1 + 4π
|x−z|3
|x−z|≥r
and p(y, r) = −
(ξ · (u(x + rξ) − v)2 dS(ξ)
|ξ|=1 \ σij ((x−z))
(ui (z) − vi )(uj (z) − vj )dz
(ξ · (u(y + rξ) − v)2 dS(ξ)
|ξ|=1 \ σij ((y−z)) 1 + 4π |y−z|≥r |y−z|3 (ui (z)
(66)
Now clearly
− vi )(uj (z) − vj )dz
|ξ|=1
(ξ · (u(x + rξ) − v) dS(ξ) ≤ Cr2α kuk2C α
|ξ|=1
(ξ · (u(y + rξ) − v) dS(ξ) ≤ Cr2α kuk2C α ,
2
and
2
so it remains to estimate 1 4π
|x−z|≥r
\ σij ((x − z)) 1 wi wj dz − 3 |x − z| 4π
|y−z|≥r
\ σij ((y − z)) wi wj dz |y − z|3
where w = u(y) − v. Now, if |x − z| ≥ r but |y − z| ≤ r, then |x − z| ≤ |y − z| + |x − y| ≤ 89 r, and so 1 \ σij ((x − z)) wi wj dz ≤ Ckuk2C α r2α , 3 4π |x−z|≥r,|y−z|≤r |x − z| and similarly, if |y − z| ≥ r, but |x − z| ≤ r, then 1 \ σij ((y − z)) wi wj dz ≤ Ckuk2C α r2α . 3 4π |y−z|≥r,|x−z|≤r |y − z| Finally, we are left with 1 4π
(Kij (x − z) − Kij (y − z))wi wj dz |x−z|≥r,|y−z|≥r
where Kij (ζ) = 3ζi ζj |ζ|−2 − δij |ζ|−3 This is now a classical situation in singular integral theory where the smoothness of the kernel is used. We observe that
1
|z − (y + λ(x − y))|−4 dλ
|Kij (x − z) − Kij (y − z)| ≤ C|x − y| 0
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
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and that |z − (y + λ(x − y))| ≥ 78 r. Thus 1 (K (x − z) − K (y − z))w w dz 4π |x−z|≥r,|y−z|≥r ij ij i j 2 1 x+y −4 ≤ C|x − y| 0 |z−xλ |≥ 7 r |z − xλ | |u(z) − u 2 | dzdλ 8
where xλ = y + λ(x − y). Now, choosing R > 0 fixed (we could choose R = 1, but we prefer to keep dimensionally correct quantities) 2 1 |x − y| 0 |z−xλ |≥R |z − xλ |−4 |u(z) − u x+y | dzdλ 2 −1 2 ≤ C|x − y|R kukL∞ . The integral on
7r 8
1
≤ |z − xλ | ≤ R,
−4
|z − xλ | |u(z) − u
|x − y| 0
7r ≤|z−xλ |≤R 8
x+y 2
|2 dzdλ
is estimated using u(z) − u x + y ≤ Ckuk2C α (|z − xλ |2α + r2α ) 2 The resulting bound obtained by integrating on 78 r ≤ |z − xλ | ≤ R is 1 2α−1 2α−1 2 r +r CkukC α |x − y| 1 − 2α if 2α < 1, Ckuk2C α |x
− y| log
8R r
r +1− R
if 2α = 1, and R2α−1 2α−1 +r − y| 2α − 1 if 2α > 1. This concludes the proof. We state now some criteria for regularity. We will write π(x, t, r(t)) for π defined according to the formula (39) for a time dependent u(x, t) and with a time dependent r = r(t). We recall that π is small if u is regular and r is small. Ckuk2C α |x
Theorem 4. Let u be a smooth solution of the Navier-Stokes equation on the interval [0, T ). First criterion: Assume that there exists U > 0, R > 0 and 0 < r(t) ≤ R such that (67) ν2 2 |u(x, t)||∇u(x, t)|2 dx |u(x, t)||π(x, t, r(t))| dx ≤ 4 R3 {x∈R3 |u(x,t)|≥U }
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PETER CONSTANTIN
holds. Assume that there exists γ > 4 such that
T
r(t)−γ dt < ∞.
(68) 0
Then u ∈ L∞ ([0, T ], L3 (R3 )).
(69)
Second criterion: Assume that there exists r(t) such that π = π(x, r(t)) satisfies
T
kπk2L3 (R3 ) dt < ∞
(70) 0
and that, as above, there exists γ > 4 such that (68) holds. Then again (69) holds. Proof. We start with the first criterion. We consider the evolution of the L3 norm of velocity: d kuk3L3 (R3 ) + ν 3dt
2
|∇u| |u|dx + R3
|u|(u · ∇p)dx ≤ 0 R3
We represent p using the formula (37) with r = r(t). We split softly the integral involving π: |u| |u|(u · ∇π)dx = φ |u|(u · ∇π)dx 3 3 U R R + R3 1 − φ |u| |u|(u · ∇π)dx U where φ(q) is a smooth scalar function 0 ≤ φ(q) ≤ 1, supported in 0 ≤ q ≤ 1. We use the bound
1
|∇π(x)| ≤ C
dλ 0
|z|≤2r
dz (|∇u(x + z)| + |∇u(x))|∇u(x + λz)| |z|2
which follows from (39) by differentiation. It follows that |u| 2 2 3 φ U |u|(u · ∇π)dx ≤ CU rk∇ukL2 (R3 ) . R
We integrate by parts in the other piece: |u| |u| 1−φ |u|(u · ∇π)dx = − πu · ∇[|u| 1 − φ ]dx U U R3 R3 When the derivative falls on 1 − φ we are in the |u| ≤ U regime and we use (53) and the interpolation combined to Morrey’s inequality kuk2L4 (R3 ) ≤ CkukL3 (R3 ) k∇ukL2 (R3 )
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
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to deduce −1 0 |u| R3 π|u|u · ∇|u|)U φ U dx ≤ CU kπkL2 (R3 ) k∇ukL2 (R3 ) ≤ CU kukL3 (R3 ) k∇uk2L2 (R3 ) When the derivative falls on |u| we use the condition (67) and the Schwartz inequality: |u| {|u(x,t)|≥U } |u · ∇|u|(1 − φ U π|dx ≤ ν2 R3 |u||∇u|2 dx. As to the integral involving β, we integrate by parts, and use H¨older’s inequality followed by (45) q 1 3 βu · ∇|u|dx ≤ kβkL3 (R3 ) kuk 2 3 3 |u||∇u|2 dx L (R ) R3 R 1 ≤ 2ν kβk2L3 (R3 ) kukL3 (R3 ) + ν2 R3 |u||∇u|2 dx 4−2a ν 3 (R3 ) + |u||∇u|2 dx ≤ Cν −1 r−2a kuk2a k∇uk kuk 2 3 2 3 L L (R ) L (R ) 2 R3 γ By chosing a = γ−2 we have a < 2, and using Young’s inequality, we see that −γ r−2a k∇ukL4−2a + k∇uk2L2 (R3 ) ) 2 (R3 ) ≤ C(r
is time-integrable. The upshot is that the quantity y(t) = kukL3 (R3 ) obeys an ordinary differental inequality y2
dy ≤ C1 (t) + C2 (t)y + C3 (t)y dt
with C1 (t) = CU 2 rk∇uk2L2 (R3 ) , C2 (t) = CU k∇uk2L2 (R3 ) and C3 (t) = 2a Cν −1 r−2a k∇uk4−2a L2 kukL2 (R3 ) . The positive functions C1 (t), C2 (t) and C3 (t) are known to be time-integrable. The interested reader can check that the inequality above is dimensionally correct, each term has dimensions of [L]6 [T ]−4 . Then it follows that y 2 dy ≤ C1 (t) + C2 (t) + C3 (t), 1 + y dt (no longer dimensionally correct), and after an easy integration, it follows that y is bounded a priori in time. This proves the first criterion. For the proof of the second criterion we again represent p = π(x, r)+ β(x, r) with r = r(t) and bound the integral involving π using straightforward integration by parts and H¨older inequalities: 3 (u · ∇π)|u|dx = 3 π(u · ∇|u|)dx R R ≤ ν2 R3 |u||∇u|2 dx + Cν kukL3 (R3 ) kπk2L3 (R3 ) .
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PETER CONSTANTIN
We bound the contribution coming from β the same way as we did for the first criterion. The upshot is that y(t) = kukL3 (R3 ) obeys y2
dy ≤ C4 (t)y + C3 (t)y dt
with C4 (t) = Cν kπk2L3 (R3 ) which is time-integrable by assumption. It follows again that y(t) is bounded apriori in time. 6. Appendix We prove here the identities (5) and (6). We introduce polar coordinates, ξ1 = ρ cos φ sin θ = ρcS, ξ2 = ρ sin φ sin θ = ρsS, ξ3 = ρ cos θ = ρC where for simplicity of notation we abbreviate s = sin φ, S = sin θ, c = cos φ, C = cos θ. For a function on the unit sphere ρ = 1. But in general f (ξ) = f (ρcS, ρsS, ρC), and we have fθ = ∂θ f = ρ(cCf1 + sCf2 − Sf3 ), fφ = ∂φ f = ρ(−sSf1 + cSf2 ), ρfρ = ρ∂ρ f = ρ(cSf1 + sSf2 + Cf3 ) ξ where ρ∂ρ f = ξ ·∇ξ f and ∇ξ f = (f1 , f2 , f3 ). We note that ρ∂ρ ( |ξ| ) = 0, for ξ 6= 0. We have
Cfθ + Sρfρ = ρ(cf1 + sf2 ) Cρfρ − Sfθ = ρf3 and thus (71)
ρf1 = c(Cfθ + Sρfρ ) − Ss fφ , ρf2 = s(Cfθ + Sρfρ ) + Sc fφ , ρf3 = Cρfρ − Sfθ
We consider now ρ = 1 and denote for simplicity Dρ = ρ∂ρ . We compute first ξ1 ∂1 f (x + rξ)dS(ξ) |ξ|=1
using of course dS(ξ) = Sdφdθ. We have ξ1 ∂ξ1 f = cS(c(C∂θ + SDρ ) − Ss ∂φ )f = Dρ (ξ12 f ) + c2 SC∂θ f − sc∂φ f
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
21
ξ We used the fact that on the unit sphere ξ = |ξ| and Dρ (ξ) = 0. We multiply by S and integrate, integrating by parts where possible. In view of d −c2 (S 2 C) = c2 S(S 2 − 2C 2 ) = c2 S(3S 2 − 2) dθ and d S (sc) = 2c2 S − S, dφ the coefficients of f are obtained by adding
c2 S(3S 2 − 2) + 2c2 S − S = S(3ξ12 − 1), and so
ξ ∂ f dS(ξ) = |ξ|=1 [Dρ (ξ12 f ) + 3ξ12 f − f ]dS(ξ) |ξ|=1 h1 ξ1 i 2 = Dρ |ξ|=1 ξ1 f dS(ξ) + |ξ|=1 (3ξ12 − 1)f dS(ξ) which is the first relation in (5). The rest of the formulas in (5) are proved similarly. Indeed, ξ2 ∂ξ2 f = sS s(C∂θ + SDρ ) + Sc ∂θ f = [s2 S 2 Dρ + s2 SC∂θ + sc∂φ ] f Upon multiplication by S and integration by parts in the ∂θ and ∂φ terms we obtain the coefficients of f d d −s2 dθ (S 2 C) − S dφ (sc) = s2 S(3S 2 − 2) + S − 2c2 S 2 2 = s S(3S − 2) − S + 2s2 S = (3ξ22 − 1)S
and therefore
ξ ∂ f dS(ξ) = |ξ|=1 i h 2 ξ2 Dρ |ξ|=1 ξ22 f dS(ξ)
+
|ξ|=1
(3ξ22 − 1)f dS(ξ)
like above. The third term is ξ3 ∂3 f = C(CDρ − S∂θ )f. Multiplying by S and integrating by parts the ∂θ term, we compute the coefficient of f d (CS 2 ) = (3C 2 − 1)S = (3ξ32 − 1), dθ and therefore we obtain the last relation of (5)
ξ ∂ f dS(ξ) = |ξ|=1 h 3 ξ3 i Dρ |ξ|=1 ξ32 f dS(ξ)
+
|ξ|=1
(3ξ32 − 1)f dS(ξ).
22
PETER CONSTANTIN
We prove now similarly the relations (6). We start with the term corresponding to the indices (1, 3): (ξ 1 ∂ξ3 + ξ3 ∂ξ1 )f = s cS(CD − S∂ ) + C(c(C∂ + SD ) − ∂ ) f= ρ θ θ ρ φ S Cs 2 2 2cSCDρ + (cC − cS )∂θ − S ∂φ f Multiplying by S, integrating, and integrating by parts we obtain the coefficient of f via d d (S(1 − 2S 2 )) + C dφ (s) −c dθ 2 = −c(C − 6S C) + Cc = 6cSCS = 6ξ1 ξ3 S
and so
(ξ1 ∂ξ3 + ξ3 ∂ξ1 )f dS(ξ) = |ξ|=1 [2ξ1 ξ3 Dρ f + 6ξ1 ξ3 f ] dS(ξ) h i = Dρ |ξ|=1 2ξ1 ξ3 f dS(ξ) + |ξ|=1 6ξ1 ξ3 f dS(ξ) |ξ|=1
which is the (1, 3) relation in (6). At indices (1, 2) we have to compute (ξ1∂2 + ξ2 ∂1 )f = cS(sSDρ + sC∂θ + Sc ∂φ ) + sS(cSDρ + cC∂θ − Ss ∂φ ) f = 2cSsSDρ f + 2cs(SC)∂θ f + (c2 − s2 )∂φ f. Multiplying by S and integrating by parts, we obtain the coefficient of f via d d (S 2 C) − S dφ (c2 − s2 ) = −2cs dθ 2cs(S 3 − 2SC 2 ) + 4Scs = 2cs(S 3 − 2S + 2S 3 ) + 4csS = 6csS 3 = 6ξ1 ξ2 S.
We obtained thus
(ξ1 ∂ξ2 + ξ2 ∂ξ3 )f dS(ξ) = |ξ|=1 [2ξ1 ξ2 Dρ f + 6ξ1 ξ2 f ] dS(ξ) h i = Dρ |ξ|=1 2ξ1 ξ2 f dS(ξ) + |ξ|=1 6ξ1 ξ2 f dS(ξ) |ξ|=1
which is the (1, 2) relation of (6). Finally, at (2, 3) we have to compute (ξ2 ∂3 + ξ3 ∂2 )f = sS(CDρ − S∂θ )f + C(sSDρ + sC∂θ + Sc ∂φ )f = 2sSCDρ f + (s(C 2 − S 2 )∂θ + C Sc ∂φ )f. Multiplying by S and integrating by parts, the coefficient of f is computed via d d (S(C 2 − S 2 )) − C dφ c= −s dθ 2 s(6S C − C) + Cs = 6sSCS = 6ξ2 ξ3 S
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
23
and we obtain thus (ξ2 ∂ξ3 + ξ3 ∂ξ2 )f dS(ξ) |ξ|=1 = |ξ|=1 [2ξ2 ξ3 Dρ f + 6ξ2 ξ3 f ] dS(ξ) h i = Dρ |ξ|=1 2ξ2 ξ3 f dS(ξ) + |ξ|=1 6ξ2 ξ3 f dS(ξ) which is the (2, 3) relation of (6). Acknowledgment Research partially supported by grants NSF-DMS 1209394 and NSF-DMS 1265132. References [1] L. Berselli and G. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proceedings of the AMS, 130 (12), (2002), 3585-3595. [2] P. Constantin, An Eulerian-Lagrangian approach to the Navier-Stokes equations, Commun. Math. Phys., 216 (2001), 663-686. [3] L. Escauriaza, G. Seregin, V. Sverak, Backward uniqueness for parabolic equations. ARMA 169 (2) (2003), 147-157. [4] C. Foias, C. Guillop´e, R. Temam, New apriori estimates for the Navier-Stokes equations in dimension 3, Comm. PDE 6 (3), (1981)329-359. [5] P. Isett, Regularity in time along the coarse scale flow for the incompressible Euler equations, arXiv:1307.0565 [6] G. Seregin, V. Sverak, The Navier-Stokes equations and backward uniqueness, Nonlinear problems in mathematical physics abd related topics II Int. Mat. Ser (N.Y.) Kluwer/Plenum, New York (2002), 353-366. [7] G. Seregin, V. Sverak, Navier-Stokes equations with lower bounds on the pressure, ARMA 163 (1) (2002), 65-86. [8] L. Silvestre, unpublished material. [9] T. Tao, Localisation and compactness properties of the Navier-Stokes global regularity problem. arxiv 1108.1165, (31 May 2012). [10] A. Vasseur, Higher derivatives estimate for the 3D Navier-Stokes equation, Ann. Inst. Henri Poincar´e (C) Nonlinear Analysis, 27 (5) (2010), 1189-1204. Department of Mathematics, Princeton University, Princeton, NJ 08544 E-mail address: [email protected]
Dedicated to the memory of Professor Mark I. Vishik. 1. Introduction We provide local formulas for the pressure of incompressible fluids. By this we mean expressions that compute a solution of 3 X
∂2 (ui uj ), −∆p = ∂xi ∂xj i,j=1 where u is a divergence-free velocity, at x ∈ Ω ⊂ R3 , from the spherical average of the pressure, 1 p(x, r) = 4πr2
p(y)dS(y), |x−y|=r
and from integrals of increments (ui (y) − ui (x))(uj (y) − uj (x)), for |y − x| ≤ r, with arbitrary small r. No knowledge of the behavior of u outside a small ball is needed. The main ingredient is a kind of monotonicity equation for a modified object 2 y−x 1 b(x, r) = p(x, r) + · u(y) dS(y). 4πr2 |x−y|=r |y − x| Date: October 11, 2013. 2000 Mathematics Subject Classification. 35Q35. Key words and phrases. Navier-Stokes equations, Euler equations, pressure, regularity criteria. 1
2
PETER CONSTANTIN
This allows us to express the pressure as p(x) = β(x, r) + π(x, r) where β is just a local average of the pressure, 1 β(x, r) = r
2r
p(x, ρ)dρ, r
and π(x, r) is given by a couple of integrals (39) of squares of increments of velocity over a ball and over an annulus of radii 2r. Thus, we write the pressure as a sum of two local terms, one small, and the other sufficiently well-behaved. Indeed, β ∈ L∞ (R3 ) is bounded in space (for any r), if u ∈ L2 (R3 ) (34), and k∇βkL2 (R3 ) is bounded in terms of kuk2L4 (R2 ) (47). On the other hand, π is of the order r2 |∇u|2 for small r. Well-known criteria for regularity for the 3D incompressible NavierStokes equations in terms of the pressure ([1]), ([7]) do exist. If the pressure would obey the bounds that β obeys, then regularity of solutions of the 3D Navier-Stokes equations would easily follow. Because π(x, r) → 0 as r → 0, the suggestion that p obey the same bounds as β is not unreasonable. On the other hand, bounds on π require some smoothness of the velocity. Higher regularity in space for velocity for weak solutions of the 3D Navier-Stokes equations was obtained in ([4]) (see also ([10])). These bounds imply that π(x, r) is small for almost all time. For instance, kπkL3 (R3 ) ≤ C(t)r2 , t − a.e. (52), (59). The problem is that in general the time integrability of C(t) is too poor 1 to conclude regularity (C(t) 3 is time integrable, whereas C(t) time integrable would be sufficient for regularity.) The organization of this paper is as follows: In the next section we present the basic calculations which lead to the formulas for the pressure. In section 3 we give ensuing bounds for β and π. In section 4 we give a quick proof of the bounds of higher derivatives of solutions of the 3D Navier-Stokes equations in the whole space. These follow from the classical paper ([4]), and were well-known for decades, although, because ([4]) deals with spatially periodic solutions, a proof in the whole space of one the results (due originally to Luc Tartar, see acknowledgment in ([4])) was given only in 2001 ([2]). The 2012 preprint ([9]) contains also a proof of this result and more references. In section 5 we give two applications: the first is a simple proof of the fact that, if u ∈ C α , then p ∈ C 2α (if 2α < 1; if 2α > 1 then p is Lipschitz). This result was used recently in ([5]), with a proof based on the LittlewoodPaley decomposition. A different proof (closer to ours) was obtained before, but was not published ([8]). The 3D Navier-Stokes equations are regular if u ∈ L∞ ([0, T ], L3 (R3 )) ([3]), ([6]). We give as a second
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
3
application, criteria of regularity for the 3D Navier-Stokes equations in terms of π. These essentially say that if we can find r(t) small such that in some sense, π is small, and if some integral of r(t)−1 is finite, then we have regularity. Some elementary calculations needed for the formulas are presented in the Appendix. 2. Spherical averages We denote 1 f (x, r) = 4πr2
(1) where
f (y)dS(y) = |x−y|=r
f (x + rξ)dS(ξ) |ξ|=1
denotes normalized integral. We consider solutions of − ∆p = ∇ · (u · ∇u)
(2) 3
in Ω ⊂ R . We assume ∇ · u = 0 and smoothness of u. We start by computing ∂r p(x, r) 1 = |ξ|=1 ξ · ∇x p(x + rξ)dS(ξ) = 4πr ξ · ∇ξ p(x + rξ)dS(ξ) |ξ|=1 1 r = 4πr |ξ| 0, let p solve (2) in Ω = R3 with divergence-free u ∈ (C 2 (R3 ) ∩ L2 (R3 ))3 . Then (37)
p(x) = β(x, r) + π(x, r)
with β(x, r) given by (38)
1 β(x, r) = r
2r
p(x, ρ)dρ r
10
PETER CONSTANTIN
and π(x, r) given by (39)
2
y−x 1 π(x, r) = · (u(y) − u(x)) |y−x| r≤|y−x|≤2r |y−x|2 [ σ ( x −y) ij 1 w |y−x| (ui (y) − ui (x))(uj (y) − 4π |x−y|≤2r r |x−y|3 1 4πr
dy+
uj (x))dy
Remark 8. Passing to the limit r → ∞ in (37) we obtain (40) 1 σij (b z) |u(x)|2 + p(x) = (ui (x + z) − ui (x))(uj (x + z) − uj (x))dz 3 3 4π R3 |z| This can be obtained also from (13) using (16). Proof. We integrate (41)
1 r
2r r
dρ the representation (14) written as
p(x) = p(x, ρ) + |y−x|=ρ |ξ · (u(y) − u(x))|2 dS(y) ρ + 0 1l |y−x|=l [3(ξ · (u(y) − u(x)))2 − |u(y) − u(x)|2 ] dS(y)
and use the fact that ρ 2r l 1 2r f (l)dl dρ = w f (l)dl. r r r 0 0 In addition to the bounds (34) and (35) we also have bounds that follow from Morrey inequality 3 2 6 |∇u(y)| dy , |u(y)| dy ≤ C R3
R3
the representation (42)
p = Ri Rj (ui uj ) 1
of the pressure where Ri = ∂i (−∆)− 2 are Riesz transforms, and the boundedness of Riesz transforms in Lp spaces. Proposition 3. Let p the solution of (2) given by (42). For any q, 1 < q < ∞ there exist constants Cq > 0, independent of r > 0 so that, for any r > 0 (43)
kp(·, r)kLq (R3 ) ≤ Cq kuk2L2q (R3 )
and (44)
kβ(·, r)kLq (R3 ) ≤ Cq kuk2L2q (R3 ) .
For any a ∈ [0, 2) there exists Ca > 0 such that (45)
kβ(·, r)kL3 (R3 ) ≤ Ca r−a kukaL2 (R3 ) k∇uk2−a L2 (R3 ) .
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
11
There exists a constant C > 0 so that k∇p(·, r)kL2 ≤ Cr−1 kuk2L4 (R3 )
(46) and
k∇β(·, r)kL2 ≤ Cr−1 kuk2L4 (R3 )
(47)
Proof. The bounds (44) for β follow from the bounds (43) for p by averaging in r. The bounds (43) follow from (42) and the boundedness of Riesz transforms in Lp spaces. The bounds (45) follow from (35), interpolation a 1− a3 kβkL3 (R3 ) ≤ kβkL3 ∞ (R3 ) kβkL3−a (R3 ) , the bound (44) for q = 3 − a, kβ(·, r)kL3−a (R3 ) ≤ Ca kuk2L6−2a (R3 ) , and interpolation combined with the Morrey inequality 6−3a
a
6−2a kukL6−2a (R3 ) ≤ CkukL6−2a 2 (R3 ) k∇ukL2 (R3 ) .
The bound (47) follows from the bound (46) by averaging in r. The bound (46) follows from k∇p(·, r)kL2 (R3 ) ≤ Cr−1 kp(·, r)kL2 (R3 )
(48)
and (43) at q = 2. The bound (48) follows from Plancherel and the observation that sin(r|ξ|) b (49) pb(ξ). p(ξ, r) = r|ξ| Indeed,
e−ix·ξ p(x, r)dx = |ω|=1 dS(ω) = pb(ξ) |ω|=1 eirξ·ω dS(ω) R3
R3
e−ix·ξ p(x + rω)dx
and the last integral is computed conveniently choosing coordinates so that ξ points to the North pole:
2π 1 dφ 4π 0 Regarding π we have
π
dθeir|ξ| cos θ sin θdθ = 0
sin(r|ξ|) . r|ξ|
Proposition 4. Let π(x, r) be defined by (39). Then
(50)
|π(x, r)| ≤ C |z|≤2r
|u(x + z) − u(x)|2 dz. |z|3
Consequently (51)
kπ(·, r)kLq ≤ Cq r2 k∇uk2L2q
12
PETER CONSTANTIN
holds for all 1 < q ≤ ∞. In particular, at q = 3 we have, with Morrey’s inequality, kπ(·, r)kL3 ≤ Cr2 k∆uk2L2 .
(52) We also have
kπ(·, r)kLq (R3 ) ≤ Cq kuk2L2q (R3 ) .
(53)
Proof. The inequality (50) is immediate from definition. In order to prove (51) we write
2
1
|∇u(x + λz)|2 dλ
2
|u(x + z) − u(x)| ≤ |z|
0
and changing order of integration we have 2 |u(x + z) − u(x)| ≤ Cr2 kφk q0 k∇uk2 2q dz L L 3 φ(x)dx |z|3 |z|≤2r
R
which proves (51). The bounds (53) follow from (37), the corresponding bounds for p, and (44). 4. FGT bounds in the whole space We take the Navier-Stokes equation ∂t u + u · ∇u − ν∆u + ∇p = 0,
(54) with
∇ · u = 0,
(55)
multiply by ∂t u − ν∆u and integrate, using incompressibility:
2
|∂t u − ν∆u| dx = − R3
(u · ∇u)(∂t u − ν∆u)dx. R3
Schwartz inequality gives:
2
|u · ∇u|2 dx
|∂t u − ν∆u| dx ≤ R3
R3
and so
R3
|∂t u − ν∆u|2 dx ≤ kuk2L∞ k∇uk2L2 .
The inequality kuk2L∞ ≤ Ck∇ukL2 k∆ukL2
(56)
is easy to prove using Fourier transform. Thus
R3
|∂t u − ν∆u|2 dx ≤ Ck∆ukL2 k∇uk3L2 .
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
13
On the other hand,
|∂t u − ν∆u|2 dx = k∂t uk2L2 + ν 2 k∆uk2L2 + ν
R3
d k∇uk2L2 dt
and therefore d k∇uk2L2 + νk∆uk2L2 + ν1 k∂t uk2L2 dt ≤ Cν k∆ukL2 k∇uk2L2 ≤ ν2 k∆uk2L2 + νC3 k∇uk6L2 denote y(t) = k∇u(·, t)k2L2 , pick a constant A >
Now we 0, divide by 2 (A + y) and obtain νk∆uk2L2 k∂t uk2L2 1 d C + − + ≤ 3 y. 2 2 dt A + y (A + y) ν(A + y) ν Integrating in time we obtain
T
νk∆uk2L2 dt + (A + y)2
0
Therefore
0
k∂t uk2L2 C 1 2 dt ≤ ku k 2 + 0 L ν(A + y)2 ν4 A
k∆uk2L2 C 1 dt ≤ 5 ku0 k2L2 + = Cν −4 [D + ν 3 A−1 ] 2 (A + y) ν νA
T
k∂t uk2L2 C ν 2 dt ≤ ku k = Cν −2 [D + ν −3 A−1 ] 2 + 0 L (A + y)2 ν3 A
0
T
T
(57) and
(58) 0
where we put D= Now
T
2 3
k∆ukL2 dt ≤
0
and
T
2 3
k∂t ukL2 dt ≤
0
T 0
T 0
ku0 k2L2 . ν
31 T 32 k∆uk2L2 dt (A + y)dt (A + y)2 0 31 T 23 k∂t uk2L2 dt (A + y)dt (A + y)2 0
and therefore
T 0
and
T 0
2 1 2 4 k∆ukL3 2 dt ≤ Cν − 3 D + ν 3 A−1 3 [D + AT ] 3
2 1 2 2 k∂t ukL3 2 dt ≤ Cν − 3 D + ν 3 A−1 3 [D + AT ] 3
Now A is arbitrary, but a natural explicit choice is A2 = ν 3 T −1
14
PETER CONSTANTIN
and then we have
T
k∆ukL2 dt ≤ Cν
(59) 0
and
2 3
T
2 3
k∂t ukL2 dt ≤ Cν
(60) 0
− 43
− 23
ku0 k2L2 1 3 + T 2ν2 ν
ku0 k ν
2 L2
1 2
+T ν
3 2
.
Now using the inequality (56) it follows immediately that 34 14 T 2 2 ku k ku k 3 1 2 2 0 0 L L kukL∞ dt ≤ Cν −1 (61) + T 2ν2 . ν ν 0 Let us consider now the other terms in (54). We start by computing
2
|u · ∇u + ∇p| dx = ku · R3
Now
+
k∇pk2L2
(u·∇u)·(∇p)dx = −2
2
∇uk2L2
R3
2
pTr(∇u) dx = 2
R3
R3
Consequently
(u · ∇u) · (∇p)dx.
+2
p∆pdx = −2k∇pk2L2 .
0≤ R3
|u · ∇u + ∇p|2 dx = ku · ∇uk2L2 − k∇pk2L2 .
On the other hand, obviously ku · ∇ukL2 ≤ kukL∞ k∇ukL2 and in view of the previous result we have 12 12 T 2 2 2 ku k ku k 1 3 2 2 2 0 0 L L ku · ∇ukL3 2 dt ≤ Cν − 3 + T 2ν2 (62) ν ν 0 and, because of the inequality k∇pkL2 ≤ ku · ∇ukL2 , we also have 12 1 T 2 2 ku0 k2L2 2 1 3 − 23 ku0 kL2 3 (63) k∇pkL2 dt ≤ Cν + T 2ν2 ν ν 0 We have thus Theorem 2. Let u be a Leray weak solution of the Navier-Stokes equa2 tion on the interval [0, T ]. Then the quantities kukL∞ (R3 ) , k∆ukL3 2 (R3 ) , 2
2
2
k∂t ukL3 2 (R3 ) , ku · ∇ukL3 2 (R3 ) , k∇pkL3 2 (R3 ) are almost everywhere finite on the time interval [0, T ], and their time integrals are bounded uniformly, with bounds (59, 60, 61, 62, 63) depending only on T , ku0 kL2 (R3 ) and ν.
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
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The proof for Leray weak solutions follows the same pattern as the proof given above for smooth solutions, except that we mollify the advecting velocity, prove the mollification-uniform bounds and deduce the result using essentially Fatou’s lemma. For the sake of completeness, let us mention here other estimates. Interpolating
T
k∇uk2L2 dt < ∞ 0
and
T 0
2
k∇ukL3 6 dt < ∞
which comes from Morrey’s inequality and (59) we get 1
1
k∇ukL3 ≤ Ck∇ukL2 6 k∇ukL2 2 which then is integrable by H¨older
T
k∇ukL3 dt < ∞. 0
Finally, we mention that, interpolating between L∞ (dt; L2 (dx)) and 6p L2 (dt; L6 (dx)) it is easy to see that u ∈ Lp (dt, Lq (dx)) for q = 3p−4 if p ≥ 2. For p ∈ [1, 2] interpolating between L2 (dt; L6 (dx)) and 3p . L1 (dt, L∞ (dx)) we get q = p−1 5. Applications Theorem 3. Let u solve (54) and (55) in R3 and assume that u belongs to L∞ (dt; L2 (R3 )) ∩ L2q (dt; C α (R3 )) for some q ≥ 1. Then p ∈ Lq (dt; C 2α (R3 )) if α < 21 . If α = 12 then p ∈ Lq (dt; LiplogLip) where LiplogLip is the class of functions with modulus of continuity |x − y| log(|x − y|−1 ). If α > 12 then p ∈ Lq (dt; Lip) where Lip is the class of Lipschitz continuous functions. Proof. We start with two points x, y at distance |x − y| and we choose r = 8|x − y|. The representation (14) implies |p(x) − p(x, r)| ≤ Ckuk2C α r2α , (64) |p(y) − p(y, r)| ≤ Ckuk2C α r2α , so, it remains to prove that |p(x, r) − p(y, r)| ≤ Cr2α if 2α < 1 and C ∼ kuk2C α . (If 2α = 1 we obtain r log(r−1 ), and if 2α > 1, r.) In order to do so, we use (4) with v = u x+y and 2
16
PETER CONSTANTIN
integrate from r to infinity. We obtain p(x, r) = − (65)
1 + 4π
|x−z|3
|x−z|≥r
and p(y, r) = −
(ξ · (u(x + rξ) − v)2 dS(ξ)
|ξ|=1 \ σij ((x−z))
(ui (z) − vi )(uj (z) − vj )dz
(ξ · (u(y + rξ) − v)2 dS(ξ)
|ξ|=1 \ σij ((y−z)) 1 + 4π |y−z|≥r |y−z|3 (ui (z)
(66)
Now clearly
− vi )(uj (z) − vj )dz
|ξ|=1
(ξ · (u(x + rξ) − v) dS(ξ) ≤ Cr2α kuk2C α
|ξ|=1
(ξ · (u(y + rξ) − v) dS(ξ) ≤ Cr2α kuk2C α ,
2
and
2
so it remains to estimate 1 4π
|x−z|≥r
\ σij ((x − z)) 1 wi wj dz − 3 |x − z| 4π
|y−z|≥r
\ σij ((y − z)) wi wj dz |y − z|3
where w = u(y) − v. Now, if |x − z| ≥ r but |y − z| ≤ r, then |x − z| ≤ |y − z| + |x − y| ≤ 89 r, and so 1 \ σij ((x − z)) wi wj dz ≤ Ckuk2C α r2α , 3 4π |x−z|≥r,|y−z|≤r |x − z| and similarly, if |y − z| ≥ r, but |x − z| ≤ r, then 1 \ σij ((y − z)) wi wj dz ≤ Ckuk2C α r2α . 3 4π |y−z|≥r,|x−z|≤r |y − z| Finally, we are left with 1 4π
(Kij (x − z) − Kij (y − z))wi wj dz |x−z|≥r,|y−z|≥r
where Kij (ζ) = 3ζi ζj |ζ|−2 − δij |ζ|−3 This is now a classical situation in singular integral theory where the smoothness of the kernel is used. We observe that
1
|z − (y + λ(x − y))|−4 dλ
|Kij (x − z) − Kij (y − z)| ≤ C|x − y| 0
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
17
and that |z − (y + λ(x − y))| ≥ 78 r. Thus 1 (K (x − z) − K (y − z))w w dz 4π |x−z|≥r,|y−z|≥r ij ij i j 2 1 x+y −4 ≤ C|x − y| 0 |z−xλ |≥ 7 r |z − xλ | |u(z) − u 2 | dzdλ 8
where xλ = y + λ(x − y). Now, choosing R > 0 fixed (we could choose R = 1, but we prefer to keep dimensionally correct quantities) 2 1 |x − y| 0 |z−xλ |≥R |z − xλ |−4 |u(z) − u x+y | dzdλ 2 −1 2 ≤ C|x − y|R kukL∞ . The integral on
7r 8
1
≤ |z − xλ | ≤ R,
−4
|z − xλ | |u(z) − u
|x − y| 0
7r ≤|z−xλ |≤R 8
x+y 2
|2 dzdλ
is estimated using u(z) − u x + y ≤ Ckuk2C α (|z − xλ |2α + r2α ) 2 The resulting bound obtained by integrating on 78 r ≤ |z − xλ | ≤ R is 1 2α−1 2α−1 2 r +r CkukC α |x − y| 1 − 2α if 2α < 1, Ckuk2C α |x
− y| log
8R r
r +1− R
if 2α = 1, and R2α−1 2α−1 +r − y| 2α − 1 if 2α > 1. This concludes the proof. We state now some criteria for regularity. We will write π(x, t, r(t)) for π defined according to the formula (39) for a time dependent u(x, t) and with a time dependent r = r(t). We recall that π is small if u is regular and r is small. Ckuk2C α |x
Theorem 4. Let u be a smooth solution of the Navier-Stokes equation on the interval [0, T ). First criterion: Assume that there exists U > 0, R > 0 and 0 < r(t) ≤ R such that (67) ν2 2 |u(x, t)||∇u(x, t)|2 dx |u(x, t)||π(x, t, r(t))| dx ≤ 4 R3 {x∈R3 |u(x,t)|≥U }
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PETER CONSTANTIN
holds. Assume that there exists γ > 4 such that
T
r(t)−γ dt < ∞.
(68) 0
Then u ∈ L∞ ([0, T ], L3 (R3 )).
(69)
Second criterion: Assume that there exists r(t) such that π = π(x, r(t)) satisfies
T
kπk2L3 (R3 ) dt < ∞
(70) 0
and that, as above, there exists γ > 4 such that (68) holds. Then again (69) holds. Proof. We start with the first criterion. We consider the evolution of the L3 norm of velocity: d kuk3L3 (R3 ) + ν 3dt
2
|∇u| |u|dx + R3
|u|(u · ∇p)dx ≤ 0 R3
We represent p using the formula (37) with r = r(t). We split softly the integral involving π: |u| |u|(u · ∇π)dx = φ |u|(u · ∇π)dx 3 3 U R R + R3 1 − φ |u| |u|(u · ∇π)dx U where φ(q) is a smooth scalar function 0 ≤ φ(q) ≤ 1, supported in 0 ≤ q ≤ 1. We use the bound
1
|∇π(x)| ≤ C
dλ 0
|z|≤2r
dz (|∇u(x + z)| + |∇u(x))|∇u(x + λz)| |z|2
which follows from (39) by differentiation. It follows that |u| 2 2 3 φ U |u|(u · ∇π)dx ≤ CU rk∇ukL2 (R3 ) . R
We integrate by parts in the other piece: |u| |u| 1−φ |u|(u · ∇π)dx = − πu · ∇[|u| 1 − φ ]dx U U R3 R3 When the derivative falls on 1 − φ we are in the |u| ≤ U regime and we use (53) and the interpolation combined to Morrey’s inequality kuk2L4 (R3 ) ≤ CkukL3 (R3 ) k∇ukL2 (R3 )
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
19
to deduce −1 0 |u| R3 π|u|u · ∇|u|)U φ U dx ≤ CU kπkL2 (R3 ) k∇ukL2 (R3 ) ≤ CU kukL3 (R3 ) k∇uk2L2 (R3 ) When the derivative falls on |u| we use the condition (67) and the Schwartz inequality: |u| {|u(x,t)|≥U } |u · ∇|u|(1 − φ U π|dx ≤ ν2 R3 |u||∇u|2 dx. As to the integral involving β, we integrate by parts, and use H¨older’s inequality followed by (45) q 1 3 βu · ∇|u|dx ≤ kβkL3 (R3 ) kuk 2 3 3 |u||∇u|2 dx L (R ) R3 R 1 ≤ 2ν kβk2L3 (R3 ) kukL3 (R3 ) + ν2 R3 |u||∇u|2 dx 4−2a ν 3 (R3 ) + |u||∇u|2 dx ≤ Cν −1 r−2a kuk2a k∇uk kuk 2 3 2 3 L L (R ) L (R ) 2 R3 γ By chosing a = γ−2 we have a < 2, and using Young’s inequality, we see that −γ r−2a k∇ukL4−2a + k∇uk2L2 (R3 ) ) 2 (R3 ) ≤ C(r
is time-integrable. The upshot is that the quantity y(t) = kukL3 (R3 ) obeys an ordinary differental inequality y2
dy ≤ C1 (t) + C2 (t)y + C3 (t)y dt
with C1 (t) = CU 2 rk∇uk2L2 (R3 ) , C2 (t) = CU k∇uk2L2 (R3 ) and C3 (t) = 2a Cν −1 r−2a k∇uk4−2a L2 kukL2 (R3 ) . The positive functions C1 (t), C2 (t) and C3 (t) are known to be time-integrable. The interested reader can check that the inequality above is dimensionally correct, each term has dimensions of [L]6 [T ]−4 . Then it follows that y 2 dy ≤ C1 (t) + C2 (t) + C3 (t), 1 + y dt (no longer dimensionally correct), and after an easy integration, it follows that y is bounded a priori in time. This proves the first criterion. For the proof of the second criterion we again represent p = π(x, r)+ β(x, r) with r = r(t) and bound the integral involving π using straightforward integration by parts and H¨older inequalities: 3 (u · ∇π)|u|dx = 3 π(u · ∇|u|)dx R R ≤ ν2 R3 |u||∇u|2 dx + Cν kukL3 (R3 ) kπk2L3 (R3 ) .
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PETER CONSTANTIN
We bound the contribution coming from β the same way as we did for the first criterion. The upshot is that y(t) = kukL3 (R3 ) obeys y2
dy ≤ C4 (t)y + C3 (t)y dt
with C4 (t) = Cν kπk2L3 (R3 ) which is time-integrable by assumption. It follows again that y(t) is bounded apriori in time. 6. Appendix We prove here the identities (5) and (6). We introduce polar coordinates, ξ1 = ρ cos φ sin θ = ρcS, ξ2 = ρ sin φ sin θ = ρsS, ξ3 = ρ cos θ = ρC where for simplicity of notation we abbreviate s = sin φ, S = sin θ, c = cos φ, C = cos θ. For a function on the unit sphere ρ = 1. But in general f (ξ) = f (ρcS, ρsS, ρC), and we have fθ = ∂θ f = ρ(cCf1 + sCf2 − Sf3 ), fφ = ∂φ f = ρ(−sSf1 + cSf2 ), ρfρ = ρ∂ρ f = ρ(cSf1 + sSf2 + Cf3 ) ξ where ρ∂ρ f = ξ ·∇ξ f and ∇ξ f = (f1 , f2 , f3 ). We note that ρ∂ρ ( |ξ| ) = 0, for ξ 6= 0. We have
Cfθ + Sρfρ = ρ(cf1 + sf2 ) Cρfρ − Sfθ = ρf3 and thus (71)
ρf1 = c(Cfθ + Sρfρ ) − Ss fφ , ρf2 = s(Cfθ + Sρfρ ) + Sc fφ , ρf3 = Cρfρ − Sfθ
We consider now ρ = 1 and denote for simplicity Dρ = ρ∂ρ . We compute first ξ1 ∂1 f (x + rξ)dS(ξ) |ξ|=1
using of course dS(ξ) = Sdφdθ. We have ξ1 ∂ξ1 f = cS(c(C∂θ + SDρ ) − Ss ∂φ )f = Dρ (ξ12 f ) + c2 SC∂θ f − sc∂φ f
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
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ξ We used the fact that on the unit sphere ξ = |ξ| and Dρ (ξ) = 0. We multiply by S and integrate, integrating by parts where possible. In view of d −c2 (S 2 C) = c2 S(S 2 − 2C 2 ) = c2 S(3S 2 − 2) dθ and d S (sc) = 2c2 S − S, dφ the coefficients of f are obtained by adding
c2 S(3S 2 − 2) + 2c2 S − S = S(3ξ12 − 1), and so
ξ ∂ f dS(ξ) = |ξ|=1 [Dρ (ξ12 f ) + 3ξ12 f − f ]dS(ξ) |ξ|=1 h1 ξ1 i 2 = Dρ |ξ|=1 ξ1 f dS(ξ) + |ξ|=1 (3ξ12 − 1)f dS(ξ) which is the first relation in (5). The rest of the formulas in (5) are proved similarly. Indeed, ξ2 ∂ξ2 f = sS s(C∂θ + SDρ ) + Sc ∂θ f = [s2 S 2 Dρ + s2 SC∂θ + sc∂φ ] f Upon multiplication by S and integration by parts in the ∂θ and ∂φ terms we obtain the coefficients of f d d −s2 dθ (S 2 C) − S dφ (sc) = s2 S(3S 2 − 2) + S − 2c2 S 2 2 = s S(3S − 2) − S + 2s2 S = (3ξ22 − 1)S
and therefore
ξ ∂ f dS(ξ) = |ξ|=1 i h 2 ξ2 Dρ |ξ|=1 ξ22 f dS(ξ)
+
|ξ|=1
(3ξ22 − 1)f dS(ξ)
like above. The third term is ξ3 ∂3 f = C(CDρ − S∂θ )f. Multiplying by S and integrating by parts the ∂θ term, we compute the coefficient of f d (CS 2 ) = (3C 2 − 1)S = (3ξ32 − 1), dθ and therefore we obtain the last relation of (5)
ξ ∂ f dS(ξ) = |ξ|=1 h 3 ξ3 i Dρ |ξ|=1 ξ32 f dS(ξ)
+
|ξ|=1
(3ξ32 − 1)f dS(ξ).
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PETER CONSTANTIN
We prove now similarly the relations (6). We start with the term corresponding to the indices (1, 3): (ξ 1 ∂ξ3 + ξ3 ∂ξ1 )f = s cS(CD − S∂ ) + C(c(C∂ + SD ) − ∂ ) f= ρ θ θ ρ φ S Cs 2 2 2cSCDρ + (cC − cS )∂θ − S ∂φ f Multiplying by S, integrating, and integrating by parts we obtain the coefficient of f via d d (S(1 − 2S 2 )) + C dφ (s) −c dθ 2 = −c(C − 6S C) + Cc = 6cSCS = 6ξ1 ξ3 S
and so
(ξ1 ∂ξ3 + ξ3 ∂ξ1 )f dS(ξ) = |ξ|=1 [2ξ1 ξ3 Dρ f + 6ξ1 ξ3 f ] dS(ξ) h i = Dρ |ξ|=1 2ξ1 ξ3 f dS(ξ) + |ξ|=1 6ξ1 ξ3 f dS(ξ) |ξ|=1
which is the (1, 3) relation in (6). At indices (1, 2) we have to compute (ξ1∂2 + ξ2 ∂1 )f = cS(sSDρ + sC∂θ + Sc ∂φ ) + sS(cSDρ + cC∂θ − Ss ∂φ ) f = 2cSsSDρ f + 2cs(SC)∂θ f + (c2 − s2 )∂φ f. Multiplying by S and integrating by parts, we obtain the coefficient of f via d d (S 2 C) − S dφ (c2 − s2 ) = −2cs dθ 2cs(S 3 − 2SC 2 ) + 4Scs = 2cs(S 3 − 2S + 2S 3 ) + 4csS = 6csS 3 = 6ξ1 ξ2 S.
We obtained thus
(ξ1 ∂ξ2 + ξ2 ∂ξ3 )f dS(ξ) = |ξ|=1 [2ξ1 ξ2 Dρ f + 6ξ1 ξ2 f ] dS(ξ) h i = Dρ |ξ|=1 2ξ1 ξ2 f dS(ξ) + |ξ|=1 6ξ1 ξ2 f dS(ξ) |ξ|=1
which is the (1, 2) relation of (6). Finally, at (2, 3) we have to compute (ξ2 ∂3 + ξ3 ∂2 )f = sS(CDρ − S∂θ )f + C(sSDρ + sC∂θ + Sc ∂φ )f = 2sSCDρ f + (s(C 2 − S 2 )∂θ + C Sc ∂φ )f. Multiplying by S and integrating by parts, the coefficient of f is computed via d d (S(C 2 − S 2 )) − C dφ c= −s dθ 2 s(6S C − C) + Cs = 6sSCS = 6ξ2 ξ3 S
LOCAL FORMULAS FOR THE HYDRODYNAMIC PRESSURE
23
and we obtain thus (ξ2 ∂ξ3 + ξ3 ∂ξ2 )f dS(ξ) |ξ|=1 = |ξ|=1 [2ξ2 ξ3 Dρ f + 6ξ2 ξ3 f ] dS(ξ) h i = Dρ |ξ|=1 2ξ2 ξ3 f dS(ξ) + |ξ|=1 6ξ2 ξ3 f dS(ξ) which is the (2, 3) relation of (6). Acknowledgment Research partially supported by grants NSF-DMS 1209394 and NSF-DMS 1265132. References [1] L. Berselli and G. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proceedings of the AMS, 130 (12), (2002), 3585-3595. [2] P. Constantin, An Eulerian-Lagrangian approach to the Navier-Stokes equations, Commun. Math. Phys., 216 (2001), 663-686. [3] L. Escauriaza, G. Seregin, V. Sverak, Backward uniqueness for parabolic equations. ARMA 169 (2) (2003), 147-157. [4] C. Foias, C. Guillop´e, R. Temam, New apriori estimates for the Navier-Stokes equations in dimension 3, Comm. PDE 6 (3), (1981)329-359. [5] P. Isett, Regularity in time along the coarse scale flow for the incompressible Euler equations, arXiv:1307.0565 [6] G. Seregin, V. Sverak, The Navier-Stokes equations and backward uniqueness, Nonlinear problems in mathematical physics abd related topics II Int. Mat. Ser (N.Y.) Kluwer/Plenum, New York (2002), 353-366. [7] G. Seregin, V. Sverak, Navier-Stokes equations with lower bounds on the pressure, ARMA 163 (1) (2002), 65-86. [8] L. Silvestre, unpublished material. [9] T. Tao, Localisation and compactness properties of the Navier-Stokes global regularity problem. arxiv 1108.1165, (31 May 2012). [10] A. Vasseur, Higher derivatives estimate for the 3D Navier-Stokes equation, Ann. Inst. Henri Poincar´e (C) Nonlinear Analysis, 27 (5) (2010), 1189-1204. Department of Mathematics, Princeton University, Princeton, NJ 08544 E-mail address: [email protected]
