On the existence of steady-state solutions to the Navier-Stokes system for large fluxes
ANTONIORUSSO ANDGIULIOSTARITA
Abstract. In this paper we deal with the stationary Navier-Stokes problem in a domainwith compact Lipschitz boundary∂and datumain Lebesgue spaces.
We prove existence of a solution for arbitrary values of the fluxes through the connected components of∂, with possible countable exceptional set, provided ais the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux forbounded.
Mathematics Subject Classification (2000):76D05 (primary); 31B10, 35Q30, 42B20 (secondary).
1.
Introduction
The boundary value problem associated with the Navier-Stokes equations is to find a solution to the system
νu−u· ∇u=∇p in, (1.1)
divu=0 in, (1.2)
u=a on∂, (1.3)
where is a bounded domain (open connected set) of
R
n (n = 2,3), u, p the unknown kinetic and pressure fields,νthe kinematical viscosity andathe boundary datum which must satisfy the condition
∂a·n=0,
where n is the outward unit normal to ∂ (see [6]). Existence of a variational solution
(u,p)∈ [Wσ1,2()∩C∞()] × [L2()∩C∞()] (1.4) Received January 29, 2007; accepted in revised form December 13, 2007.
to system (1.1)-(1.3) is known under the hypothesis of smallness of the fluxes i =
∂i
a·n
through the connected components∂i of∂, provideda ∈ W1/2,2(∂)and∂ is Lipschitz. Precisely, in [1, 5] (see also [6, Chapter VIII]) it is proved that there is a positive constant depending on such that if
|i| is suitably small, then (1.1)-(1.3) has a variational solution. These results have been extended for a in Lebesgue’s spaces in [10]. In particular, it is proved that if a∈ Lq(∂) (q ≥8/3 forn =3 andq =2 forn =2), then system (1.1)-(1.3) has aC∞solution which for q > 4 takes the boundary datum in the sense of nontangential convergence.
Moreover, making use of some regularity results in [2, 12], it is showed that if a is more regular (say H¨older continuous, with H¨older’s coefficient depending on the Lipschitz character of∂) then so doesu.
In [3] H. Fujita and H. Morimoto considered problem (1.1)-(1.3) in a regular domain with
a=µu0|∂+γ, (1.5)
µ ∈
R
, u0 = ∇β andβ harmonic function. They proved that if is regular, β|∂∈W2,2(∂),∂γ ·n=0 (1.6)
andγW1/2,2(∂)is less than a suitable constant depending onν, µ, andu0, then system (1.1)-(1.3), (1.5) admits a solution (1.4) for anyµ∈
R
\G, withGcountable subset ofR
. Moreover, ifβ ∈ W3,2()and γW3/2,2(∂) is sufficiently small, then(u,p) ∈ Wσ2,2()×W1,2(). This result is remarkable in view of the fact that, even though for special boundary data, it assures the existence of a solution to system (1.1)-(1.3) in arbitrary bounded regular domains for large fluxes. It is worth to mention that for the annulus{x ∈R
2 : R1 < |x| < R2}andβ = ∇log|x|, H.Morimoto proved thatG=
∅
[8] (see also [4, 9]).The aim of the present paper is twofold:
(i) to extend the results of [3] under more general assumptions on the domains and on the data and for anyn ≥2; in particular, forn = 3 we show that, ifis a bounded Lipschitz domain, ais given by (1.5) withu0 ∈ W1,q(),q > 3/2, γ ∈ L2(∂) satisfies (1.6) and γL2(∂) is sufficiently small, then system (1.1)-(1.3) has a solution (u,p) ∈ [Wσ1/2,2()∩C∞()] ×C∞()for any µ∈
R
\G, withGcountable subset ofR
;(ii) to prove existence of a solution for system (1.1)-(1.3), (1.5) in a Lispschitz exte- rior domain in the class[L∞σ (,r)∩C∞()] × [L∞(
SR,r2logr)∩C∞()], providedu0∈L∞(,r2)andγL∞(∂)is sufficiently small.NOTATION– We use a standard vector notation, as in [6]. Letbe a domain ofRn, n ≥2, and let{γ (ξ)}ξ∈∂be a family of circular finite (not empty) cones with vertex
atξ such thatγ (ξ)\ {ξ} ⊂(as well-known, ifis Lipschitz, such a family of cones certainly exists). Letχ be a function in;χ(x)is said to converge nontangentially at the boundary ifχ(ξ)=limx→ξ (x∈γ (ξ))χ(x)⇔χ(x)−→nt χ(ξ), for almost allξ ∈∂. As customary,Lq(),Ws,q()andLq(∂),Ws,q(∂)(q ∈ [1,+∞],s≥0)denote respectively the Lebesgue and the Sobolev-Besov spaces of (scalar, vector and tensor) fields in and∂endowed with their natural norms; W−s,q(∂)is the dual space ofWs,q(∂)andL∞(, f(r)), with f(r)positive function ofr = |x|, is the Banach space of all measurable fieldsχinsuch thatf(r)χL∞()<+∞; ifV(⊂L1loc()) is a function space, Vσ stands for the subspace ofV of all (weakly) divergence free vector fields; also, the subscriptφin the symbolWφ, whereW ⊂Ws,q(∂),s ∈ R, denotes the set of all fieldsa∈W such that
∂a·ndσ =0 ora,n =0 respectively fors≥0 ors<0, where·,·stands for the duality pairing betweenWs,q(∂)and its dual.
2.
Some results for the Stokes system
The boundary-value problem associated with the Stokes system is to find a solution to the problem
νu= ∇p in, divu=0 in, u= a on∂.
(2.1) Ifis exterior we require thatutends to zero at infinity.
The following theorems are proved in [2, 7, 10, 12].
Theorem 2.1. Letbe a Lipschitz bounded domain of
R
n and leta ∈ Lqφ(∂), q ≥2. There exists a positive constantdepending onsuch that if q∈ [2,2+), then system (2.1), admits a solution (u,p) ∈ [Wσ1/q,q()∩C∞()] ×C∞(), u−→nt aanduW1/q,q()≤caLq(∂). (2.2)
Moreover:
• (i)ifa∈W1,q(∂), q∈ [2,2+), thenu∈W1+1/q,q().
• For n=3there are two positive constantsandα0, depending on the Lipschitz character of∂such that:
(ii)ifa∈Lq(∂), q∈ [2,+∞], thenu∈W1/q,q()and(2.2)holds;
(iii)ifa∈W1−1/q,q(∂), q∈ [2,3+), then
uW1,q()≤caW1−1/q,q(∂); (2.3)
(iv)ifa∈C0,α(∂),α∈ [0, α0), then
uC0,α() ≤caC0,α(∂).
• If n =4anda∈L3(∂), then
uL4() ≤caL3(∂). (2.4)
If is of class C1, then properties (i)-(iv)are satisfied for all n ≥ 2 with q ∈ (1,+∞),= +∞andα0=1. In particular, if n≥5anda∈Ln−1(∂), then
uLn() ≤caLn−1(∂). (2.5)
Theorem 2.2. Let be a bounded domain of
R
n of class C2,1 and let a ∈ Wφ−1/q,q(∂), with q ∈ (1,+∞). Then system(2.1), admits a solution(u,p) ∈ Cσ∞()×C∞()such thatutakes the boundary valueain the sense of the space W−1/q,q(∂)1anduLq()≤caW−1/q,q(∂). (2.6)
Theorem 2.3. Let be an exterior domain of
R
3. If a ∈ L∞(∂), then system (2.1)admits a solution(u,p)∈ [L∞σ (,r)∩C∞()] × [L∞(SR,r2)∩C∞()], u−→nt aanduL∞(,r) ≤caL∞(∂). (2.7) Moreover, property(iv)in Theorem2.1holds unchanged.
3.
Existence theorems for the Navier-Stokes system
Thanks to the results just recalled concerning the Stokes problem (2.1), we are in a position to extend the existence results of Fujita-Morimoto [3] to data inLqφ(∂) for Lipschitz domain and inWφ−1/q,q(∂)for domains of classC2,1, for suitableq.
Moreover, we also prove an existence theorem in Lipschitz exterior domains with data inL∞(∂).
Theorem 3.1. Letbe a Lipschitz bounded domain of
R
3and leta=µu0|∂+γ, (3.1)
whereµ ∈
R
,u0 = ∇β, withβ ∈ W2,q() (q > 3/2)harmonic function, and γ ∈ L2φ(∂). Then, for everyµ/ν ∈R
\G, with G countable subset ofR
, there exists a constantκ=κ(, ν,u0, µ)such that, ifγL2(∂)≤κ then system(1.1)-(1.3)has a solution
(u,p)∈ [Wσ1/2,2()∩C∞()] ×C∞().
1See [10].
Proof. Recall that the fundamental solution to the Stokes equation is expressed by
U
(x−y)= 18πν 1
|x−y| +(x−y)⊗(x−y)
|x−y|3
,
P
(x−y)= 14π x−y
|x−y|3, where1denotes the unit second-order tensor.
Letu∈L3σ(). By classical results the linear operator
L
ˆ[u](x)= −
U
(x−y)[u0· ∇u+u· ∇u0](y)maps L3σ()into W1,t()for some t > 3/2. Let
L
0[u] be the solution to the Stokes problem with boundary datum −trL
ˆ[u] ∈ W1−1/t,t(∂). Since by (2.3) and the trace theoremL
0[u]W1,t()≤ctrL
ˆ[u]W1−1/t,t(∂)≤ ˆL
[u]W1,t()we see that also the linear operator
L
0maps L3σ()into W1,t(). Therefore, by Rellich’s compactness theorem, the operatorL
= ˆL
+L
0maps compactly L3σ()into itself and tr
L
[u] =0on∂. SetF
=I
− µ νL
,where
I
denotes the identity map. By classical results there is a countable subsetG ofR
, with a possible accumulation at 0, such thatF
is invertible for allµ/ν∈G.The nonlinear operator
N
ˆ[u](x)= −1 νU
(x−y)(u· ∇u)(y) mapsL3σ()intoWσ1,3/2()and it holdsN
ˆ[u]W1,3/2()≤cu2L3(). (3.2)
Let
N
0[u]be the solution to the Stokes problem with boundary datum −trN
ˆ[u]. By the trace theorem and (3.2) we haveN
0[u]L3() ≤ctr
N
ˆ[u]W1/3,3/2(∂)≤N
ˆ[u]W1,3/2()≤cu2L3().
Of course, the operator
N
= ˆN
+N
0 mapsL3σ()into itself and trN
[u] =0on∂.Forµ/ν∈Gconsider the map
u=−
F
1uγ +N
[u](3.3) fromL3σ()into itself, whereuγ is the solution to the Stokes problem with bound- ary datumγ. Since
−
F
1N
[u]L3()≤c0u2L3(), taking into account (2.2), ifγL2(∂)is chosen such that
−
F
1[uγ]L3() < 1 4c0, then (3.3) is a contraction in the ball
S
= u∈L3σ(): uL3()≤ 1 2c0.
Therefore, by a classical theorem of S. Banach, there is a unique fieldu∈
S
such thatu=−
F
1uγ +N
[u] . Hence it follows thatuis a solution to the equationu=uγ + µ
ν
L
[u] +N
[u].Sinceu0 is a solution to both Stokes and Navier-Stokes equations, by taking also into account standard regularity theory we see that the fieldµu0+u∈C∞()is a solution to equations (1.1)-(1.2) for a suitable pressure field p ∈ C∞(). This solution assumes the boundary datum in the sense thatu0−→nt u0|∂,uγ−→nt γ and
N
[u],L
[u]have zero trace on∂as elements of the Sobolev spaceW01,3/2()(see also Remark 3.2).Remark 3.2. Assume for simplicity thatβ is a regular harmonic function. By the regularity results for the Stokes problem we have, in particular, that if the norm of γ is small in the corresponding function space, then
• ifγ ∈Lq(∂),q >4, andu0∈W1,t(),t >3, thenu−→nt a;
• ifγ ∈L∞(∂), thenu∈ L∞()
and there are two positive constantsandα0(<1)depending onsuch that
• ifγ ∈Ls(∂),s∈ [2,2+), thenu∈W1/s,s(),
• ifγ ∈W1−1/s,s(∂),s ∈ [3/2,3+), thenu∈W1,s(),
• ifγ ∈C0,α(∂),α∈ [0, α0), thenu∈C0,α().
Ifis of classC1, then the above constantsandα0can be taken arbitrarily large and equal to 1 respectively. Standard regularity results also hold for the pressure field p.
Remark 3.3. In virtue of Theorem 2.1 and estimates (2.3), (2.4), (2.5), existence theorems like the above one can also be established for alln≥2. Ifn=2 we can take a ∈ L2−(∂)withdepending on(a ∈ Lq(∂),q > 1, forof class C1); ifn=4, we have to requirea∈L3(∂); ifn>4,must be of classC1and a∈Ln−1(∂).
Taking into account Theorem 2.2 and estimate (2.2), following the argument in the proof of the above Theorem it is not difficult to get:
Theorem 3.4. Letbe a bounded domain of
R
n of class C2,1and letabe given by(3.1)with µ ∈R
, u0 = ∇β, β ∈ W2,q() (q > n)harmonic function and γ ∈ Wφ−1/n,n(∂). There is a countable subset G ⊂R
such that, forµ/ν ∈ G, there is a constantκ =κ(, ν,u0, µ)such that, ifγW−1/n,n(∂)≤κ then(1.1)-(1.3)has a solution
(u,p)∈ [Lnσ()∩C∞()] ×C∞().
Taking into account the result of H. Morimoto recalled in the introduction, we also have:
Theorem 3.5. Letbe the annulus
= {x∈
R
2: R1<|x|< R2},and letabe given by(3.1), withu0 ∈W1,q(), q >1,γ ∈ Lqφ(). IfγLq(∂)
is sufficiently small, then(1.1)-(1.3)has a solution
(u,p)∈ [Wσ1/q,q()∩C∞()] ×C∞().
Let us pass to treat the case of an exterior domain. The following theorem holds.
Theorem 3.6. Letbe an exterior Lipschitz domain of
R
3and letabe expressed by(3.1)whereu0|∂, γ ∈L∞(∂)andu0= O(r−2). There is a countable subset G⊂R
such that, forµ/ν∈G, there is a constantξ =ξ(, ν,u0, µ)such that, if γL∞(∂)≤ξ, then the Navier-Stokes problem admits a solution(u,p)∈ [L∞σ (,r)∩C∞()] × [L∞(
SR,r2logr)∩C∞()]andu−→nt a.
Proof. By well-known results about the behavior at infinity of volume potential (see,e.g., [6] Lemma II.7.2), the operator
V
ˆ[u](x)= −∇
U
(x−y)(u⊗u0+u0⊗u)(y)maps boundedlyL∞(,r)into L∞(,r2−η)for everyη∈(0,1). Hence
V
ˆ[u] ∈ Lq(), for everyq >3/2. Moreover, by Calder´on-Zygmund’s theorem∇ ˆV
maps boundedlyL∞(,r)intoLq(), for everyq>3/2.Let {uk}k∈N be a bounded sequence in L∞σ (,r). By what we said above
V
ˆ[uk]is bounded inW1,q()for everyq > 3/2 so that we can extract from it a subsequence, we denote by the same symbol, which converges uniformly to a field uinCloc(). On the other handV
ˆ[uk−uh]L∞(,r)
≤ R
V
ˆ[uk−uh]C(∩SR)+Rη−1r2−η
V
ˆ[uk−uh]L∞(SR)
≤ R
V
ˆ[uk−uh]C(∩SR)+c Rη−1.
Let > 0 and let m be such that for every h,k > m,
V
ˆ[uk −uh]C(∩SR) <
/(2R), with R1−η > 2c/. Therefore, from the above relation it follows that ˆ
V
[uk−uh]L∞(,r) < for allh,k > m so thatV
ˆ[uk]is a Cauchy sequence in L∞σ (,r)and the operatorV
ˆ is compact fromL∞σ (,r)into itself. LetV
0[u]be the solution to the Stokes problem with boundary datum−trV
ˆ[u]. It is not difficult to see thatV
0maps compactly L∞(,r)into itself. SetG
=I
−µ νV
with
V
= ˆV
+V
0. SinceG
is compact, there is a countable subsetG ⊂R
such thatG
is invertible for allµ/ν∈G.The operators
W
ˆ [u] = −1 ν∇
U
(x−y)(u⊗u)(y)and
W
0[u], solution to the Stokes problem with datum−trW
ˆ [u], mapL∞(,r) into itself. Consider the mapu=−
G
1uγ +W
[u](3.4) forµ∈G, where
W
= ˆW
+W
0anduγ is the solution to the Stokes problem with boundary datumγ. It is not difficult to see that−
G
1W
[u]L∞(,r)≤c0u2L∞(,r).
Therefore, taking into account (2.7), ifγL∞(∂)is chosen such that −
G
1[uγ]L3() < 1 4c0,
then the map (3.4) has a fixed point u and the field u+µu0 is a solution to the Navier-Stokes problem.
Remark 3.7. Of course, if u0 andγ are more regular, then so does the solution (u,p). In particular ifu0,γ ∈C(∂), then isu∈C().
Remark 3.8. In virtue of the result in [13] the derivatives ofuhave the following behavior at infinity
∇. . .∇
ktimes
u= O(r−1−k) and
p=O(r−2logr), ∇ . . .∇
ktimes
p=O(r−2−k),
withk∈
N
.Remark 3.9. As far as uniqueness of the solutions in the above theorems are con- cerned, we quote [10], and [6, Chapter IX]. The existence of a solution to system (1.1)-(1.3) in a Lipschitz exterior domain, witha ∈ L∞(∂), which converges at infinity to an assigned nonzero constant vector has been recently proved in [11].
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Dipartimento di Matematica
Seconda Universit`a degli Studi di Napoli Via Vivaldi, 43
I-81100 Caserta, Italy a.russo@unina2.it giulio.starita@unina2.it