11 A Stokes-Like System
11.2 The Solution and Regularity Theorems
With the general theorems from Sections 9 and 10 now available, we can easily derive unique solvability and regularity theorems for our generalized Stokes-like system.
Theorem 11.1. Let m ∈ N, G ⊂⊂ Rn with ∂G ∈ Cm+3 and let F ∈
Hm,q0 0(G)?
. Then there is exactly one pair (u, p) ∈Hm,q0 (G)×H0,0m−1,q(G) such that
Bm[u,Φ] +Bm−1[p,div Φ] =F(Φ) for all Φ∈Hm,q0 0(G) and
divu= 0.
Furthermore, we find a constant C =C(m, q, G)>0 with kukm,q +kpkm−1,q ≤CkFk“
Hm,q0 0(G)”?. Proof. Let’s first show existence of (u, p). Given F ∈
Hm,q0 0(G) ?
, we find a v ∈Hm,q0 (G) such that
Bm[v,Φ] =F(Φ) for all Φ∈Hm,q0 0(G) by Theorem 9.1 and we find kvkm,q ≤ CkFk“
Hm,q0 0(G)
”?. As v ∈ Hm,q0 (G), we see that divv ∈ H0,0m−1,q(G) and find by Theorem 9.17 an unique w ∈ M(m)q (G) = T(m)q H0,0m−1,q(G)
with divw = divv. With Theorem 9.17 we also see that we have
kwkm,q ≤C1kdivvkm−1,q ≤C2kvkm,q ≤C2CkFk“
Hm,q0 0(G)”?.
So, for u:=v−w∈Hm,q0 (G), we have divu= 0 and kukm,q ≤ kvkm,q+kwkm,q ≤C2CkFk“
Hm,q0 0(G)”? +CkFk“
Hm,q0 0(G)”? =
= (C2C+C)kFk“
Hm,q0 0(G)”?.
For p ∈ H0,0m−1,q(G) with T(m)q (p) = w, we also have by Theorems 9.1 and 9.17 that T(m)q : H0,0m−1,q(G) → M(m)q (G) is a homeomorphism and thus we find a constant ˜C such that kpkm−1,q ≤C˜kwkm,q and thus we also have with a constant C0
kpkm−1,q ≤C0kFk“
Hm,q0 0(G)”?
and the desired estimate is shown for (u, p). (u, p) is indeed a solution, for we see that for Φ∈Hm,q0 0(G) we have:
Bm[u,Φ] +Bm−1[p,div Φ] =Bm[v−w,Φ] +Bm−1[p,div Φ] =
=Bm[v,Φ]
| {z }
=F(Φ)
−Bm[w,Φ] +Bm−1[p,div Φ]
| {z }
=Bm[w,Φ]
=F(Φ)
For uniqueness of the solution, we note that if we have two solution pairs (u1, p1), (u2, p2), the pair (u := u1 −u2, p := p1 −p2) is a solution to the problem with F = 0. This means divu = 0 and u =−T(m)q (p), so it follows Zq(m)(p) = div(−u) = 0 and thus p= 0 and u= 0 by injectivity ofZq(m). By Theorem 9.1, we can represent an element F ∈
Hm,q0 0(G)?
by Bm[v,Φ] =F(Φ) ∀Φ∈Hm,q0 0(G)
with av ∈Hm,q0 (G). We will show a regularity theorem stating the following:
The regularity of the v belonging to F carries over to the regularities of u and p, the solutions of our problem:
Theorem 11.2. Let m ∈ N, G ⊂⊂ Rn with ∂G ∈ Cm+k+3 and let v ∈ Hm,q0 (G)∩Hm+k,q(G) be given. Then the (by Theorem 11.1 unique) pair (u, p)∈Hm,q0 (G)×H0,0m−1,q(G) satisfying
Bm[u,Φ] +Bm−1[p,div Φ] =Bm[v,Φ] for all Φ∈Hm,q0 0(G) and
divu= 0
satisfies even(u, p)∈Hm+k,q(G)×H0,0m+k−1,q(G)and we get the two estimates kwkm+k,q ≤C1kvkm+k,q
and
kpkm+k−1,q ≤C2kvkm+k,q,
where C1, C2 >0 are constants depending on m, k, q and G.
Proof. For the proof, we simply try to go through the construction of the solution in the proof of Theorem 11.1 and show regularity at each step, using our already established regularity theorems from Section 10. We will give the corresponding objects the same name as in the proof of Theorem 11.1. Asv ∈ Hm,q0 (G)∩Hm+k,q(G), we can define r:= divv ∈ H0,0m−1,q(G)∩Hm+k−1,q(G) and represent
r= ∆s+t,
according to Theorem 9.2 with s ∈ H0m+1,q(G) ∩ Hm+k+1,q(G) and t ∈ B0m−1,q(G). Further we have the estimate
k∆skm,q +ktkm,q ≤Ckrkm,q.
The proof of this decomposition is nothing but use of the solvability statement 9.1, solving the problem
Bm+1[s,Φ] =Bm−1[r,∆Φ] for all Φ∈H0m+1,q0(G), resulting at
Bm[∆s−r,Φ] = 0 for all Φ∈ C0∞(G)
and thus (∆s−r) ∈ B0m−1,q(G). So, regularity of s is simply again elliptic regularity from Theorems 6.1 and 10.1 leading to
s∈H0m+1,q(G)∩Hm+k+1,q(G)
and then we have alsot =r−∆s ∈B0m−1,q(G)∩Hm+k−1,q(G) and a constant C with
kskm+k+1,q ≤Ckrkm+k−1,q ≤Ckvkm+k,q and we also get
ktkm+k−1,q =kr−∆skm+k−1,q ≤ krkm+k−1,q+k∆skm+k−1,q ≤(1+C)kvkm+k,q. Now we can find due to Theorem 9.17 a vector field x ∈ M(m)q (G) with divx =t ∈ B0m−1,q(G)∩Hm+k−1,q(G). As further we have by x∈ M(m)q (G) an f ∈H0,0m−1,q(G) withT(m)q (f) = x, we find easily that f ∈B0m−1,q(G):
As we have Zq(m)(f) = divx=t∈B0m−1,q(G), we see by Theorem 9.4 that f must also be in B0m−1,q(G). Now we can apply Theorem 10.7 and conclude:
f ∈Hm+k−1,q(G) and x∈Hm+k,q(G) and
kxkm+k,q ≤C0kdivxkm+k−1,q =C0ktkm+k−1,q ≤C0(1 +C)kvkm+k,q, kfkm+k−1,q ≤C00kdivxkm+k−1,q ≤C00(1 +C)kvkm+k,q.
The w from Theorem 11.1 must by uniqueness be equal to ∇s+x and is thus also in Hm+k,q(G) and
kwkm+k,q =k∇s+xkm+k,q ≤ k∇skm+k,q+kxkm+k,q ≤
≤ kskm+k+1,q+kxkm+k,q ≤(C+C0(1 +C))kvkm+k,q
and the p from Theorem 11.1 must be equal to ∆s+f ∈Hm+k−1,q(G) and with an analogous calculation as above we get the estimate
kpkm+k−1,q ≤(C+C00(1 +C))kvkm+k,q.
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