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Pressure conditions for the local regularity of solutions of the Navier–Stokes equations ∗
Mike O’Leary
Abstract
We obtain a relationship between the integrability of the pressure gra- dient and the the integrability of the velocity for local solutions of the Navier–Stokes equations with finite energy. In particular, we show that if the pressure gradient is sufficiently integrable, then the corresponding velocity is locally bounded and smooth in the spatial variables. The result is proven by using De Giorgi type estimates inLweakp spaces.
1 Introduction and statement of results
One of the most important unresolved questions in the theory of the Navier–
Stokes equations is the local behavior of solutions in three spatial dimensions, in particular, questions of their local regularity. This problem was studied by Serrin [9], who showed that if the velocity is sufficiently integrable, then it is locally bounded and smooth in the spatial variables. Indeed, let Ω⊂RN be an open set, let T > 0, and set ΩT = Ω×(0, T). Suppose that v ∈ Vloc2 (ΩT) = L2,loc(0, T;W2,loc1 (Ω))∩L∞,loc(0, T;L2,loc(Ω)) is a weak solution of the Navier–
Stokes system
vt−∆v+ (v· ∇)v+∇p= 0,
divv= 0. (1)
Serrin showed that ifv∈Lq,r(ΩT) =Lr(0, T;Lq(Ω)) for someq, rsatisfying N
q +2
r <1 (2)
thenvis locally bounded and smooth in the spatial variables. Kahane [3] later showed that this also implies that the velocity is locally analytic in the spatial variables. The case where the inequality in (2) is replaced by an equality was studied by Takahashi, [10].
In a sequence of papers including [5, 6, 7, 8], Scheffer adopted a differ- ent approach. Rather than finding conditions which guaranteed regularity, he
∗1991 Mathematics Subject Classifications: 35Q30, 76D05.
Key words and phrases: Navier-Stokes, regularity, pressure.
c1998 Southwest Texas State University and University of North Texas.
Submitted January 26, 1998. Published May 13, 1998.
1
constructed solutions to initial and initial boundary value problems that were partially regular; that is the Hausdorff dimension in space and time of the set of possible singularities could be estimated from above. Caffarelli, Kohn and Nirenberg in [2] extended these techniques to show that if ΩT ⊂ R3×Rand v is a local suitable weak solution of (1), then the one–dimensional Hausdorff measure in space and time of the set of possible singularities is zero. A suitable weak solution is a pair of functions v ∈ Vloc2 (ΩT) and p ∈ L5/4,loc(ΩT) that satisfy (1) and a generalized energy inequality [2, Equation 2.5].
Since this partial regularity result requires some regularity of the pressure, a natural question is the relationship between the regularity of the pressure and the regularity of the velocity. The purpose of this paper is to examine this relationship and to prove the following results.
Theorem 1 Let (v, p) be a local weak solution of the Navier–Stokes equations in a domainΩT ⊂R3×R, and suppose thatv∈Vloc2 (ΩT)and∇p∈Lµ,loc(ΩT).
Then v∈Lq,loc(ΩT)for any qthat satisfies q < 5
5/µ−2, (3)
where1< µ≤ 53. Furthermore, ifµ > 53, then vis locally bounded and smooth in the spatial variables.
This can be generalized to domains in an arbitrary number of spatial dimen- sions in the following fashion.
Theorem 2 Let (v, p) be a local weak solution of the Navier–Stokes equations in a domainΩT ⊂RN×R, and suppose thatv∈Vloc2 (ΩT)and∇p∈Lµ,loc(ΩT).
Then v∈Lq,loc(ΩT)for any qthat satisfies q < N+ 2
(N+ 2)/µ−2, (4)
where 1< µ≤ N+23 . Furthermore, if µ > N+23 , then v is locally bounded and smooth in the spatial variables.
A consequence of these results is the fact that singularities of the velocityv are only possible at singularities of the pressure gradient∇p.
We point out that the norm k∇pkL(N+2)/3(ΩT) is dimensionless in the same sense that the normskvkLq,r(ΩT)are dimensionless whenN/q+2/r= 1. Indeed, if v(x, t) and p(x, t) satisfy the Navier-Stokes equations (1) in a domain Ω× (0, T), then the functions vλ(x, t) = λv(λx, λ2t) and pλ(x, t) = λ2p(λx, λ2t) also satisfy (1) in the dilated domains Ω/λ×(0, T /λ2) for eachλ >0; however,
the given norms ofvλ andpλare independent ofλ. Indeed,
kvλkLq,r(Ω/λ×(0,T/λ2))=
Z T/λ2
0
Z
Ω/λ|λv(λx, λ2t)|qdx
!r/q dt
1/r
= (Z T
0
Z
Ωλq|v(x, t)|qdx λN
r/q dt λ2
)1/r
=λ1−(N/q+2/r)kvkLq,r(ΩT)=kvkLq,r(ΩT)
(5)
and
k∇pλkL(N+2)/3(Ω/λ×(0,T/λ2))
=
(Z T/λ2
0
Z
Ω/λ|λ2·λ∇p(λx, λ2t)|(N+2)/3dx dt
)3/(N+2)
= (Z T
0
Z
ΩλN+2|∇p|(N+2)/3dx λN
dt λ2
)3/(N+2)
=k∇pkL(N+2)/3(ΩT).
(6)
In this sense, Theorems 1 and 2 are analogues of the aforementioned result of Serrin [9].
The basic idea of the proof is to use as a test function a smoothed and cutoff variant of (vi∓k)±. Providedis sufficiently small, the nonlinear term is integrable and we can remove the smoothing to obtain a local energy estimate.
From this we can obtain an estimate of meas[|vi|> k] and consequently ofviin Lweakp,loc(ΩT).
Recall the definitions of the spacesLweakq (U); a measurable functionf is an element of Lweakq (U) if and only if
|f|Lweakq (U)≡sup
k>0k meas{x∈ U :|f(x)|> k}1/q
<∞. (7) The quantity |f|Lweakq (U) is not a norm, but it is a quasi-norm. The inequality
|f|Lweakq (U)≤ kfkLq(U) (8) follows immediately from
kqmeas[|f|> k]≤ Z
U|f|qχ[|f|> k]dx≤ Z
U|f|qdx (9) so thatLq(U)⊂Lweakq (U). HoweverLweakq (U)6=Lq(U), as the functionf(x) = 1/x satisfies f ∈ Lweak1 (0,1), but f /∈ L1(0,1). Finally, if q0 < q and U is
bounded, thenLweakq (U)⊂Lq0(U); indeed
kfkqL0
q0(U)=q0 Z ∞
0 kq0−1meas[|f|> k]dk
≤q0measU+q0|f|qLweak q (U)
Z ∞
1 kq0−q−1dk <∞.
(10)
For further details about the spacesLweakq (U) see [1, Chp. 1] or [4, IX.4].
Once we have the the energy inequality, we can show that v ∈Lβ,loc(ΩT) impliesvi∈Lweakα(β),locfor someα(β) and eachi; consequentlyv∈Lq,loc(ΩT) for allq < α(β). Carefully iterating the process yields the result.
Remark: The techniques of this paper exploit the structure of the nonlinear term and the fact that v is solenoidal; in particular these techniques fail for non-solenoidal solutions of the more general system
ut−∆u+ (u· ∇)u= 0. (11)
2 Proof of Theorem 1
For simplicity, suppose that ΩT ⊂R3×R, and let
QR=QR(xo, to) =BR(xo)×(to−R2, to)⊂ΩT.
Letto−R2< τ < toand defineQτR=BR(xo)×(to−R2, τ). Choose 0< σ <1, and letζ∈C∞(ΩT) be a cutoff function so thatζ(x, t) = 1 if (x, t)∈QσR, so thatζ(x, t) = 0 if either|x−xo|=Ror t=to−R2, and so that|ζt|+|∇ζ|2≤ Cσ,R. Further, let{Jη(x, t)}η>0 be a family of symmetric mollifying kernels in space and time; given a functionf(x, t), we shall denote the mollification (Jη∗f) byfη
Let k >0,ω >0, and choose 0< < 23 so small that 10−310 ≤µ. Suppose that v ∈ Vloc2 (ΩT); recall the Sobolev embedding Vloc2 (ΩT) ,→ L10/3,loc(ΩT) when ΩT ⊆R3×R. Fixi∈ {1,2,3}, and consider the function
φi(x, t) =
[(vηi(x, t)−k)++ω]ζ2(x, t) η. (12)
Ifηis sufficiently small,φiis a valid test function; multiplying theithcomponent
of the first of (1) and integrating over ΩT we obtain 0 =
ZZ
QτR
∂
∂tviη
[(viη−k)++ω]ζ2dx dt
+ X3 j=1
ZZ
QτR
∂
∂xjviη ∂
∂xj
[(vηi −k)++ω]ζ2 dx dt
+ X3 j=1
ZZ
QτR
vj∂vi
∂xj
η
[(vηi −k)++ω]ζ2dx dt +
ZZ
QτR
∂pη
∂xi[(vηi −k)++ω]ζ2dx dt
=I1+I2+I3+I4
for each fixedi.
To estimateI1, first letω↓0 so that limω↓0I1=
ZZ
QτR
∂
∂t(viη−k)+
(viη−k)+ζ2dx dt
= 1
+ 1 ZZ
QτR
∂
∂t
(vηi −k)+1+
ζ2dx dt
= 1
+ 1 Z
BR
(vηi −k)+ζ2
t=τdx− 2 + 1
ZZ
QτR(vηi −k)+1+ ζζtdx dt.
(13)
Then, sending η↓0 we obtain for almost everyτ limη↓0lim
ω↓0I1= 1 + 1
Z
BR(xo)(vi−k)+1+ ζ2
t=τdx
− 2 + 1
ZZ
QτR(vi−k)+1+ ζζtdx dt. (14) To estimate theI2 term, first rewrite the integral as
I2= X3 j=1
ZZ
QτR
∂
∂xj[(vηi −k)++ω] 2
[(vηi −k)++ω]−1ζ2dx dt
+ 2 X3 j=1
ZZ
QτR
∂vηi
∂xj[(vηi −k)++ω]ζζxj dx dt
= 4
(+ 1)2 X3 j=1
ZZ
QτR
∂
∂xi[(vηi −k)++ω]+12 2
ζ2dx dt
+ 2 X3 j=1
ZZ
QτR
∂vi
∂xj[(vηi −k)++ω]ζζxj dx dt.
(15)
Using Fatou’s lemma in the first term and the fact that≤1 in the second, we sendω↓0 to obtain
lim inf
ω↓0 I2≥C
ZZ
QτR
∇n
(vηi −k)++12 o2ζ2dx dt
−2 ZZ
QτR|∇(viη−k)+|(viη−k)+ζ|∇ζ|dx dt. (16) Using Young’s inequality and Fatou’s lemma once more, we obtain
lim inf
η↓0 lim inf
ω↓0 I2≥C
ZZ
QτR
∇n
(vi−k)++12 o2ζ2dx dt
−C
ZZ
QτR(vi−k)+1+ |∇ζ|2dx dt. (17) To estimateI3, note that because≤2/3 the integral is uniformly bounded and we can pass to the limit asω↓0 then asη↓0 to obtain
limη↓0lim
ω↓0I3= X3 j=1
ZZ
QτRvj∂vi
∂xj(vi−k)+ζ2dx dt
= 1
+ 1 X3 j=1
ZZ
QτRvj ∂
∂xj(vi−k)+1+
ζ2dx dt.
(18)
Then, becausev is solenoidal we can integrate by parts to obtain
limη↓0lim
ω↓0I3= −2 + 1
X3 j=1
ZZ
QτRvj(vi−k)+1+ ζζxjdx dt. (19) As forI4, we have
I4= ZZ
QτR
∂pη
∂xi
(vηi −k)++ω
ζ2dx dt (20)
so since∇p∈Lµ,loc(ΩT),|v| ∈L10/3,loc(ΩT) and 10−310 ≤µ, we can pass to the limit, obtaining
limη↓0lim
ω↓0I4= ZZ
QτR|∇p|(vi−k)+ζ2dx dt. (21)
Combine these results and use the arbitrariness ofτ to obtain k(vi−k)(+1)/2+ ζk2V2(QR)= ess sup
to−R2<τ<to
Z
BR
(vi−k)(+1)/2+ ζ2
τ
dx
+ ZZ
QR
∇n
(vi−k)(+1)/2+ ζo2 dx dt
≤C,σ,R
ZZ
QR
(vi−k)+1+ dx dt +C,σ,R
ZZ
QR
|v|(vi−k)+1+ dx dt +C
ZZ
QR
|∇p|(vi−k)+dx dt
(22)
which is our energy estimate.
The restriction≤23, needed to pass to the limit inI3implies that +12 <1 so we can not use this inequality to directly prove strongLq estimates. We can, however, use this technique to prove a weak-Lq estimate. Indeed, estimate the left side as follows.
k(vi−k)(+1)/2+ ζk2V2(QR)≥Ck(vi−k)(+1)/2+ k2L10/3(QσR)
≥C ZZ
QσR
(vi−k)5(+1)/3+ dx dt 3/5
≥Ck+1
meas{(x, t)∈QσR:vi(x, t)>2k}3/5 . (23) Suppose thatv∈Lβ,loc(ΩT) for someβ ≥10/3. We can then estimate the first term on the right side of (22) as
ZZ
QR
(vi−k)+1+ dx dt≤ ZZ
QR
(vi−k)β+dx dt
(+1)/β
meas[vi> k]1−(+1)/β
≤ kvk+1Lβ(QR)
|v|βLweak β (QR)
1 kβ
1−(+1)/β
≤ kvkβLβ(QR) 1
k β−−1
.
(24) Similarly,
ZZ
QR
|v|(vi−k)+1+ dx dt≤ kvkβLβ(QR) 1
k
β−−2
. (25)
Lastly ZZ
QR
|∇p|(vi−k)+dx dt≤ ZZ
QR
|∇p|µdx dt 1/µ
× ZZ
QR
|v|βdx dt /β
meas[vi > k]1−1/µ−/β
≤ k∇pkLµ(QR)kvkLβ(QR)
|v|βLweak β (QR)
1 kβ
1−1/µ−/β
≤ k∇pkLµ(QR)kvkβ(1−1/µ)Lβ(QR) 1
k
β(1−1/µ)−
.
(26)
Combining these estimates, we find
meas{(x, t)∈QσR:vi(x, t)>2k} ≤Ckvk5β/3Lβ(QR) 1
k 5β/3
+Ckvk5β/3Lβ(QR) 1
k
5(β−1)/3
+Ck∇pk5/3Lµ(QR)kvk5β(1−1/µ)/3 Lβ(QR)
1 k
5[β(1−1/µ)+1]/3
. (27) Repeating this process with test functions (vi+k)−, we obtain an estimate of the form
|vi|Lweak
α(β)(QσR)≤C(, β, R, σ, µ,k∇pkLµ(QR),kvkLβ(QR)) (28) for eachi, where
α(β) = min 5
3(β−1),5 3
β
µ−1 µ
+ 1
. (29)
Thusv∈Lβ,loc(ΩT) implies v∈Lq,loc(ΩT) for everyq < α(β).
Our result is proven by iteration. SinceVloc2 (ΩT),→L10/3,loc(ΩT), set βo=
103, and inductively define βn+1=α(βn) = min
5
3(βn−1),5 3
βn
µ−1 µ
+ 1
. (30)
Now (28) implies thatv∈Lq,loc(ΩT) for everyq < βn, for everyn. Set γ(β) =53(β−1),
δ(β) = 53
β
µ−1 µ
+ 1
.
Ifβ≥ 103, thenγ(β)≥ 76β. On the other hand,δ(β)≥β if and only if
β≤ 5
5/µ−2 (31)
so that the sequence β, δ(β), δ(δ(β)), . . . converges to 5/µ−25 independently of the choice ofβ. Thusv∈Lq,loc(ΩT) for allq < 5/µ−25 . Ifµ > 53, we can choose q > 5 and apply the regularity result of Serrin [9] which guarantees the local boundedness and smoothness of vin the spatial variables.
Remark. Theorem 2, the general result in N spatial dimensions, is proven in the same fashion as Theorem 1.
References
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[3] Kahane, C. On the spatial analyticity of solutions of the Navier–Stokes equations Arch. Rat. Mech. Anal.33(1969), 386-405.
[4] Reed, M. & Simon, B.,Fourier analysis, self-adjointness, Methods of Mod- ern Mathematical Physics, vol. 2, Academic Press, New York, 1975.
[5] Scheffer, V., Turbulence and Hausdorff dimension, Turbulence and the Navier–Stokes Equations, Lecture Notes in Mathematics, vol. 565, Springer–Verlag, 1976, pp. 94-112.
[6] Scheffer, V., Hausdorff measure and the Navier–Stokes equations, Comm.
Math. Phys.55 (1977), 97-112.
[7] Scheffer, V.,The Navier–Stokes equations in space dimension four, Comm.
Math. Phys.61 (1978), 41-68.
[8] Scheffer, V. The Navier–Stokes equations on a bounded domain, Comm.
Math. Phys.73 (1980), 1-42.
[9] Serrin, J.,On the interior regularity of weak solutions of the Navier–Stokes equations Arch. Rat. Mech. Anal.,9(1962), 187-195.
[10] Takahashi, S. On interior regularity criteria for weak solutions of the Navier–Stokes equations, Man. Math.69(1990), 237-254.
Mike O’Leary
Department of Mathematics, University of California Santa Cruz Santa Cruz, CA 95064 USA.
E-mail address: [email protected]
