A The divergence problem and the Bogowski˘i operator

ρ₁(w²_t +|u|²) +ρ₂(|v_t|²+|v|²) + ^γτ_κ⁰(|v|²+|q|²)+

ρ₁(w²_t +w²) +ρ₂ρ₃(|Brotθ|²+|v_t|²) +τ₀ρ₃(|q|²+|Brotθ|²)

dx

≤¹₂

2ρ₁kw_tk²_L²_(Ω)+ρ₁kwk²_H¹_(Ω)+ρ₂kv_tk²_(L²_(Ω)²₎+ (ρ₂+ ^γτ_κ⁰)kvk(H¹(Ω))²

+C_Brot(ρ₂ρ₃+τ₀ρ₃)kθk²L²(Ω)+ (^γτ_κ⁰ +τ₀ρ₃)kqk²(L²(Ω))²

≤¹₂

2ρ₁kw_tk²_L²_(Ω)+ρ₂kv_tk²_(L²_(Ω)²₎+

max{ρ1,(ρ2+γτ0

κ )}

CK (Kk∇w+vk²_(L²_(Ω))² +k√

SDvk²_(L²_(Ω))²) +C_Brot(ρ₂ρ₃+τ₀ρ₃)kθk²_L²_(Ω)+ (^γτ_κ⁰ +τ₀ρ₃)kqk²_(L²_(Ω))²

≤CˆE(t).

Letting nowα₁:=N−^{max{ρ1,ρ2,C}

−1 K }

min{ρ1,ρ2,ρ3} andα₂ :=N+^{max{ρ1,ρ2,C}

−1 K }

min{ρ1,ρ2,ρ3} , we obtain the following equivalence between E andF

α₁E(t)≤ F(t)≤α₂E(t) for t≥0.

If necessary, we increase the constant N to assure for the positivity of α₁. Thus, both C,α₁ and α₂ are positive. Exploiting Gronwall’s inequality, we obtain the following estimate for E

E(t)≤ _α¹₁F(t)≤ _α¹₁E(0)e⁻

C α2t

=:CE(0)e^−2αt for t≥0 meaning an exponential decay of E.

Appendices

A The divergence problem and the Bogowski˘i operator

In various applications of partial differential equations, e.g., when studying Navier-Stokes equations, there arises a so-called “divergence problem”: For a given functionf, determine a vector fieldusuch that

its divergence coincides withf. We refer to [9] for a rather general solution of this problem in bounded domains. It has namely been shown that the solution map B:f 7→u, called the Bogowski˘i-operator, is a bounded linear operator between W₀^s,p(Ω)and W₀^s+1,p(Ω)forp∈(0,∞),s∈ −2 +¹_p,∞

. For our application, we want to additionally guarantee that the solution uis irrotational. To this end, we exploit the following result from [13].

Theorem 8. Let Ω ⊂Rⁿ be a domain with a smooth boundary and let ν: Ω → Rⁿ denote the outer unit normal vector on ∂Ω. There exists then a function u∈H¹(Ω,Rⁿ) satisfyingν⊗u=u⊗ν on ∂Ω and

k∇uk²_L²_(Ω) =kdivuk²_L²_(Ω)+¹₂k∇u−(∇u)^′k²_L²_(Ω)+ (n−1) Z

∂Ω|u|²H_ndS, (A.1) where Hn:∂Ω→ R, x 7→ Hn(x) denotes the mean curvature of ∂Ω with respect to the outer normal vector. In n= 2,3, Equation (A.1) reduces to

k∇uk²_L²_(Ω)=kdivuk²_L²_(Ω)+krotuk²_L²_(Ω)+ (n−1) Z

∂Ω|u|²H_ndS, (A.2) where

rotu=





∂_x2u₃−∂_x3u₂

∂_x3u₁−∂_x1u₃

∂x1u2−∂x2u1



 for n= 3 and rotu=∂_x1u₂−∂_x2u₁ for n= 2.

For u ∈ H₀¹(Ω,Rⁿ), the second term in (A.1) and (A.2) vanishes and no assumptions on ∂Ω are required:

k∇uk²_L²_(Ω)=kdivuk²_L²_(Ω)+k∇u−(∇u)^′k²_L²_(Ω). (A.3) In the following, we assume n= 2. We define the space

H_0,rot¹ (Ω) =

u∈(H₀¹(Ω))²| ∇u= (∇u)^′ =

u∈(H₀¹(Ω))²|rotu= 0

equipped with the standard inner product of (H₀¹(Ω))². Since H_0,rot¹ (Ω) is a closed subspace of (H₀¹(Ω))²,H_0,rot¹ (Ω)is a Hilbert space. We prove the following theorem.

Theorem 9. The mapping

div :H_0,rot¹ (Ω)→L²(Ω)/{1} is an isomorphism with an inverse div⁻¹ =Brot

Brot:L²(Ω)/{1} →H_0,rot¹ (Ω) in the sense

divBrot= id_L2(Ω)/{1} andBrotdiv = id_H1 0,rot(Ω). Furthermore, the exists C_B >0 such that

kBrotfk(H¹(Ω))² ≤C_BrotkfkL²(Ω)

holds true for all f ∈L²_∗(Ω).

Proof. The linearity ofdiv is obvious For each u∈H_0,rot¹ (Ω), we havedivu∈L²(Ω)and thus Z

Ω

divudx= Z

Γ

u·νdΓ = 0,

meaning divu∈L²(Ω)/{1}. The continuity is also trivial since kdivukL²(Ω) ≤√

2k∇ukL²(Ω)≤√

2kukH¹(Ω).

The operatordiv is injective. Indeed, letu₁, u₂∈H_0,rot¹ (Ω). Letdivu₁ = divu₂. Then, using Poincaré inequality,

0 =kdivu₁−divu₂kL²(Ω)≥ k∇u₁− ∇u₂kL²(Ω)≥ _C¹_Pku₁−u₂kL²(Ω), i.e., u₁=u₂.

To explicitely construct the operator Brot, we follow the variational approach. For f, g ∈L²(Ω)/{1}, we consider a boundary value problem for ϕ, ψ∈H¹(Ω)/{1}:

−div (∇ϕ+ rot^′ψ) =f inΩ,

−rot(∇ϕ+ rot^′ψ) =g inΩ, ν·(∇ϕ+ rot^′ψ) = 0on Γ, ν^⊥·(∇ϕ+ rot^′ψ) = 0on Γ,

(A.4)

where ν^⊥ := (ν₂,−ν₁)^′, rot^′ := (∂_x2,−∂_x1)^′. We multiply the equations with ϕ,˜ ψ˜∈H¹(Ω)/{1}, sum up the resulting identities, take into account the boundary conditions and apply a partial integration to find

− Z

Ω

div (∇ϕ+ rot^′ψ) ˜ϕdx− Z

Ω

rot(∇ϕ+ rot^′ψ) ˜ψdx= Z

Ω

(∇ϕ+ rot^′ψ)·(∇ϕ˜+ rot ˜ψ)dx This lead to the following operator equation

A(ϕ, ψ)^′ = (f, g)^′, (A.5)

where

A:D(A)⊂ H → H, (ϕ, ψ)^′ 7→

−div (∇ϕ+ rot^′ψ)

−rot(∇ϕ+ rot^′ψ)

and

D(A) =n

(ϕ, ψ)^′ ∈ V

∃(f₁, f₂)^′ ∈ H ∀( ˜ϕ,ψ)˜ ^′∈ V :B(ϕ, ψ; ˜ϕ,ψ) =˜ Z

Ω

f₁ϕ˜+f₂ψdx˜ o with the bilinear form

B:V × V →R, (φ, ψ,φ,˜ ψ)˜ ^′ 7→

Z

Ω

(∇ϕ+ rot^′ψ)·(∇ϕ˜+ rot ˜ψ)dx.

Here, we introduced the Hilbert spaces

H:= (L²(Ω)/{1})×(L²(Ω)/{1}), V := (H¹(Ω)/{1})×(H¹(Ω)/{1})

equipped with the standard inner products of L²(Ω)×L²(Ω)andH¹(Ω)×H¹(Ω), respectively. Since A has a nontrivial kernel, we consider the operator given as its restriction onto the closed subspace

V˜={(ϕ, ψ)^′ ∈ V | ∀( ˜ϕ,ψ)˜ ^′ ∈ V : Z

Ω∇ϕ·rot^′ψdx˜ = Z

Ω∇ϕ˜·rot^′ψdx= 0} of V and denote it as

A˜:D( ˜A) :=D(A)∩V ⊂ H → H˜ . Equation (A.5) reduces then to

A˜(ϕ, ψ)^′ = (f, g)^′. (A.6)

We multiply Equation (A.6) scalar in Hwith ( ˜ϕ,ψ)˜ ^′ ∈V˜ to find after a partial integration the weak formulation of (A.6): Determine an element (ϕ, ψ)^′ ∈V˜ such that

B(ϕ, ψ; ˆϕ,ψ) =ˆ F( ˆϕ,ψ)ˆ for all( ˆϕ,ψ)ˆ ^′ ∈V˜, (A.7) where

B: ˜V ×V →˜ R, (φ, ψ,φ,ˆ ψ)ˆ ^′ 7→

Z

Ω

(∇ϕ+ rot^′ψ)·(∇ϕˆ+ rot ˆψ)dx, F: ˜V →R, ( ˆφ,ψ)ˆ ^′ 7→

Z

Ω

ˆ ϕfdx+

Z

Ω

ψgdx.ˆ

The bilinear form B and the linear functional F are continuous on V ט V˜ and V˜, respectively. The bilinear form B is symmetrical. By the virtue of second Poincaré’s inequality, we obtain

B(ϕ, ψ) =k∇ϕk²_L²_(Ω)+ 2h∇ϕ,rot^′ψi+krot^′ψk²_L²_(Ω)

=k∇ϕk²_L²_(Ω)+krot^′ψk²_L²_(Ω) =k∇ϕk²_L²_(Ω)+k∇ψk²_L²_(Ω)

≥ ¹₂(1 + _C¹_P)(kϕk²H¹(Ω)+kψk²H¹(Ω)) = ¹₂(1 + _C¹_P)k(ϕ, ψ)^′k²_V =:bk(ϕ, ψ)^′k²V˜,

i.e., B is coercive. The lemma of Lax & Milgram yields the existence of a unique solution(ϕ, ψ)^′ ∈V˜ to Equation (A.7). There further holds

bk(ϕ, ψ)^′k²V˜ ≤B(ϕ, ψ) ≤ ^b₂k(ϕ, ψ)^′k²_H+_2b¹k(f, g)^′k²_H≤ ₂^bk(ϕ, ψ)^′k²V˜ +_2b¹k(f, g)^′k²_H, i.e.,

k(ϕ, ψ)^′k²V˜ ≤ ¹_bk(f, g)^′k²_H. Exploiting the trivial identities

div rot^′ϕ= 0, rot∇ϕ= 0, etc., in(C0^∞(Ω))^′ and the definition of V, we find

Z

Γ

νϕ·rotˆ ^′ψdx= Z

Ω∇ϕ·rot^′ψdxˆ = 0, Z

Γ

ν^⊥ψ· ∇ϕdΓ =ˆ Z

Γ

rot^′ψ· ∇ϕdxˆ = 0, etc.

for all (ϕ, ψ)^′ ∈V˜ and( ˆϕ,ψ)ˆ ^′ ∈ V. Hence,

− Z

Ω

div (∇ϕ+rot^′ψ) ˆϕ+ rot(∇ϕ+ rot^′ψ) ˆψdx

=B(ϕ, ψ; ˆϕ,ψ)ˆ − Z

Γ

ν·(∇ϕ+ rot^′ψ) ˆϕ+ν^⊥·(∇ϕ+ rot^′ψ) ˆϕdΓ

holds true for all ( ˆϕ,ψ)ˆ ^′ ∈ V and, in particular, the solution(ϕ, ψ)^′ ∈V˜ of (A.7). Therefore, (ϕ, ψ)^′ ∈ D( ˜A). Thus, we have shown that A˜is invertible and its inverse A˜⁻¹:H →D( ˜A) is continuous:

kA˜⁻¹(f, g)^′kV ≤ ¹_bk(f, g)^′k²_H

Let f ∈L²(Ω)/{1}. We define (φ, ψ) := ˜A⁻¹(f,0)^′,u:=∇ϕ+ rot^′ψ and obtain by construction divu=△ϕ=f inΩ,

rotu= rot0 = 0 inΩ, u=∇ϕ+ rotψ= 0 on Γ,

(A.8)

i.e., u∈H_rot¹ (Ω)withdivu=f. Thus, there exists a continuous inverse Brot:L²(Ω)/{1} →H_0,rot¹ (Ω), f 7→u of div such that

kBrotfk(H¹(Ω))² =kBrotfk²_(L²_(Ω))² +k∇Brotfk²_(L²_(Ω))^2×2

=k∇ϕ+ rot^′ψk²(L²(Ω))² +kdivBrotfk²L²(Ω)

≤2k∇ϕk²_(L²_(Ω))² + 2krot^′ψk²_(L²_(Ω))² +kfk²_L²_(Ω)

≤(²_b + 1)kfk²L²(Ω)=:C_BrotkfkL²(Ω). This finishes the proof.

Corollary 10. The operator Brot can be extended to a linear continuous operator Brot: (H¹(Ω))^′ →(L²(Ω))².

(Cp. also [4, 9] for the rotational case.)

Proof. Due to the coercivity of the bilinear formB, the operatorA˜defined in the proof of Theorem 9 strictly positive. According to [31, Section 3.4], it is possible to define square roots

A˜^−1/2 ∈L(H,H) andA˜^1/2:D( ˜A^1/2) := im ˜A^−1/2→ H of A˜⁻¹ and A˜, respectively. Further, there exists a continuous continuation ofA˜⁻¹

A˜⁻¹ ∈L(D( ˜A^−1/2), D( ˜A^1/2)), where D( ˜A^−1/2) =D( ˜A^1/2)^′. Hence,

B˜rot:D( ˜A^−1/2)→(L²(Ω))², f 7→ ∇ϕ+ rot^′ψ with(ϕ, ψ)^′ := ˜A⁻¹(f,0)^′ ∈V˜

represents a continuous continuation of Brot onto D( ˜A^−1/2). Since (H¹(Ω))^′ ⊂ D( ˜A^−1/2) and the norms of (H¹(Ω))^′ undD( ˜A^−1/2) are equivalent, the claim follows.

Let us now consider a vector field u∈(H¹(Ω))² withu·ν= 0 on Γ. Unfortunately, the identity Brotdivu=u

does not hold in general since u is not necessarily an element ofH_0,rot¹ (Ω). Nevertheless, the following estimate holds true.

Theorem 11. Let u∈H¹(Ω)satisfy u·ν= 0 on Γ. There exists then a constant C_B^′_rot >0 such that kBrotdivukL²(Ω)≤C_B^′_rotkuk(L²(Ω))²

for any u∈(H¹(Ω))². Proof. We can estimate

kBrotdivuk(L²(Ω))² ≤C_BrotkdivukH⁻¹(Ω). Further, we find

Z

Ω

divufdx=− Z

Ω

u∇fdx+ Z

∂Ω

u·νfdΓ = − Z

Ω

u∇fdx (A.9)

for all f ∈H¹(Ω)and therefore

kdivukH⁻¹(Ω)= sup

kfkH1(Ω)=1

Z

Ω

divufdx

= sup

kfkH1(Ω)=1

Z

Ω

u∇fdx

≤ sup

kfkH1(Ω)=1kuk(L²(Ω))²kfkH¹(Ω)=kuk(L²(Ω))². This yields

kBrotdivukL^p(Ω) ≤C_B^′_rotkukL^p(Ω) for allu∈H¹(Ω) (A.10) withC_B^′_rot =C_Brot.

Acknowledgment

The present work is dedicated to the honorable Mr. Urs Schaubhut, J.D. (Konstanz, Germany) in deep gratitude for his invaluable support in the author’s struggle for justice.

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