Proof of main results

Lemma 4.1.8. Let p∈(1,∞) and s >0. Then

µ(µ−A_p)⁻¹: H_per^s,p(Ω)²∩L^p_σ(Ω) →H_per^2+s,p(Ω)∩L^p_σ(Ω), µ >0,

is a well-defined family of uniformly bounded linear operators, i.e., (µ−A_p)⁻¹ maps H_per^s,p(Ω)² ∩L^p_σ(Ω) into H_per^2+s,p(Ω) ∩ L^p_σ(Ω) for all µ > 0 and there exists a constant C =C_p,s>0 such that

µk(µ−Ap)⁻¹fk_H^2+s,p_(Ω) ≤Ckfk_H^s,p_(Ω)

for all f ∈H_per^s,p∩L^p_σ(Ω). If in addition we have Γ_D 6=∅, then this also holds for A⁻¹_p . In particular, the spectrum of A_p does not depend on p∈(1,∞), consists of countably many negative eigenvalues of finite multiplicity, and all eigenvectors belong to C^∞(Ω)². Remark 4.1.9. Our proof of Theorem 4.1.1 utilizes the fact that −A_p is sectorial of angle 0 with 0 ∈ ρ(A_p) for Γ_D 6= ∅, which was established in [49]. However, we are also able to derive this property from the intermediate result (4.1.7) and the mapping properties of the resolvent. This allows us to stay self-contained.

4.1 L^p-theory using perturbation arguments Since the restriction operator given by

R: L^p(R³)→L^p(Ω), Rv :=v Ω

defines a retraction for E, the interpolation result follows via [90, Theorem 1.2.4].

2. The fact that the trace operator is a bounded mapping from H^s+1/p,p into B_p,p^s for s > 0 is established in [89, Theorem 2.7.2] for the case where the underlying domain is a half space. Given f ∈ H^s+1/p,p(Ω), one can then simply consider the restrictions ofEf ∈H^s+1/p,p(R³) ontoR²×(−∞,0) andR²×(−h,∞), respectively, and apply the result on the half space to obtain the desired result.

We are now able to prove the main result of this section.

Proof of Theorem 4.1.1. We take Bv= 1

h(1−Q)∂_zv Γ_b− 1

h(1−Q)∂_zv Γu

as in (4.0.6) and consider the operators Bp: D(Bp)→L^p(Ω)² given by

B_pv :=Bv, D(B_p) :=H^1+1/p+s,p(Ω)², s∈(0,1−1/p), (4.1.4) as well as Ap: D(Ap)→L^p(Ω)² defined via

Ap := ∆_p+B_p, D(Ap) :=D(∆_p), (4.1.5) where ∆_p is the L^p(Ω)²-realization of the operator defined in (3.4.5). We first observe that Bp: D(Bp)→L^p(Ω)² is bounded. In detail, one has that the mappings

D(B_p)3v 7→∂_zv ∈H^1/p+s,p(Ω)², H^1/p+s,p(Ω)² 3∂zv 7→∂zv

Γu, ∂zv

Γb ∈B_p,p^s (G)²,

are bounded, see Lemma 4.1.10 for the trace term. Since it further holds that B_p,p^s (G)² ∼=W^s,p(G)² ,→L^p(G)²,

whereW^s,p(G) denotes the Sobolev-Slobodeckij space onGof orders, and the projection Q is bounded on L^p(G)², the claim follows.

Observe that by (4.0.5) through (4.0.7), we have that Ap is an extension of A_p. It further holds that, whenever λ ∈ ρ(Ap), we have that v = (λ−Ap)⁻¹Pf is the unique solution to the problem

λv−∆v+∇_Hπ=f, ∂_zπ= 0, div_Hv = 0,

with boundary conditions (3.4.3). In particular it holds that (λ−Ap)⁻¹ maps L^p_σ(Ω) into itself as well as ρ(Ap)⊂ρ(A_p).

Perturbations of operators possessing a bounded H^∞-calculus have been studied by many others, compare, e.g., [8, 25, 26, 53, 62, 81]. Here we will utilize the results of [62, Proposition 13.1] and [53, Proposition 6.10].

Note that we have −∆_p ∈H^∞(L^p(Ω)²) with φ^∞_−∆p = 0 as well as 0∈ρ(∆_p) whenever Γ_D 6=∅ by Lemma 3.4.1. For v ∈D(∆_p)⊂D(B_p) we also have

kB_pvk_L^p_(Ω) ≤Ckvk_H1+1/p+s,p(Ω) ≤Ck(−∆_p)^1−δvk_L^p_(Ω),

where s ∈ (0,1−1/p) is arbitrary and δ ∈(0,1) is chosen in such a way that we have 2δ+s <1−1/p. Here we used the boundedness ofB_p in the first step, and Lemma 3.4.2 as well as 0∈ρ(∆_p) in the second step.

It now follows from [62, Proposition 13.1] that for an arbitrarily small angleφ ∈(0, π) there exists sufficiently largeµ=µ_φ≥0 such that the translated perturbation−Ap+µ satisfies

−Ap+µ∈H^∞(L^p(Ω)²), φ^∞_−Ap+µ≤φ. (4.1.6) This property is retained under the restriction on the invariant subspaceL^p_σ(Ω), yielding

−A_p+µ∈H^∞(L^p_σ(Ω)), φ^∞_−Ap+µ≤φ. (4.1.7) We now show that−Ap−εis sectorial with spectral angle 0 wheneverε >0 is sufficiently small. Since it was established in [49, Section 3 and 4] that the operator−A_p is invertible and sectorial with spectral angle φ−Ap = 0 whenever Γ_D 6=∅, it follows that σ(−A_p) is contained in the interval (δ,∞) for some δ >0. Taking ε >0 such that 2ε < δ we thus have for all φ∈(0, π) that Σπ−φ⊂ρ(A_p+ε) as well as

λ(λ−ε−A_p)⁻¹ =λ(λ−A_p)⁻¹ 1 +ε(λ−ε−A_p)⁻¹

, λ ∈Σπ−φ

by an elementary calculation. Since the family of operators {λ(λ−A_p)⁻¹ : λ ∈ Σπ−φ} is uniformly bounded on L^p_σ(Ω) by the sectoriality of −A_p, it remains to show that {ε(λ−ε−Ap)⁻¹ :λ ∈Σπ−φ} is uniformly bounded. Consider arbitrary f ∈L^p_σ(Ω) and λ∈Σπ−φ. Taking an angleψ ∈(0, π) such that

{λ−ε:λ∈Σπ−φ} ⊂Σπ−ψ∪B2ε(0) ⊂ρ(Ap), we distinguish between the two following cases.

(i) If we have|λ−ε| ≤2ε, we use the fact that the resolvent mappingλ7→(λ−A_p)⁻¹ is analytic on ρ(Ap) and thus bounded onB2ε(0), yielding

kε(λ−ε−A_p)⁻¹fk_L^p_(Ω) ≤Ckfk_L^p,

whereC > 0 is a constant only depending on ε >0 and p∈(1,∞).

(ii) If we have |λ−ε|>2ε, then it holds that kε(λ−ε−A_p)⁻¹fk_L^p_(Ω) ≤C_ψ ε

|λ−ε|kfk_L^p_(Ω)≤ 1

2C_ψkfk_L^p_(Ω).

4.1 L^p-theory using perturbation arguments It follows that −A_p −ε is sectorial with spectral angle 0 and so by (4.1.7) and [53, Proposition 6.10] we conclude that

−A_p ∈H^∞(L^p_σ(Ω)), φ^∞_−Ap = 0.

In the case Γ_D = ∅ we have that Bv = 0 for all v ∈ D(∆_p) and thus Ap = ∆_p, so the property (4.1.6) was already obtained in Lemma 3.4.1 for arbitrary µ > 0 which is inherited by −A_p+µvia the same restriction argument.

The arguments through which corollaries 4.1.2 through 4.1.5 are obtained are all straightforward as previously stated. We now turn to derivatives of the resolvent.

Proof of Corollary 4.1.6. By Corollary 4.1.5 the operator ∂_i(−A_p)^−1/2 is bounded from L^p_σ(Ω) intoL^p(Ω)² for∂_i ∈ {∂_x, ∂_y, ∂_z}and thus∂_i(−A_p)^−1/2Pis bounded on L^p(Ω)² for p∈(1,∞). Now suppose that∂_i ∈ {∂_x, ∂_y}is a horizontal derivative. Since it commutes with both the horizontal divergence div_H and the vertical average ·, the space L^p_σ(Ω) is left invariant under ∂_i(−A_p)^−1/2 for ∂_i ∈ {∂_x, ∂_y}. By [49, Remark 4.5], the adjoint operator of A_p is given by A_q where 1/p+ 1/q = 1 and so it follows that (−A_p)^−1/2∂_i is likewise bounded on L^p_σ(Ω) for p ∈ (1,∞) for ∂_i ∈ {∂_x, ∂_y} and so (−A_p)^−1/2∂_iP is bounded on L^p(Ω)². Since ∂_i also commutes with P, we obtain the L^p-boundedness of (−A_p)^−1/2P∂_i for ∂_i ∈ {∂_x, ∂_y}.

In the case ∂_i = ∂_z we have that ∂_z(−A)^−1/2 maps L^p_σ(Ω) into L^p(Ω)² and thus (−A_p)^−1/2∂_z is a bounded mapping from L^p(Ω)² into L^p_σ(Ω). Since smooth functions with compact support are dense in L^p(Ω)², we may assume without loss of generality that

∂_zf =f

Γu −f

Γb = 0, f ∈L^p(Ω)²,

and thus it follows that P∂zf = ∂zf and (−Ap)^−1/2P∂z = (−Ap)^−1/2∂z is bounded on L^p(Ω)². The resolvent estimates then follow from the fact that the families of operators

{|λ|^1/2(−Ap)^1/2(λ−Ap)⁻¹ :λ∈Σθ}, {(−Ap)(λ−Ap)⁻¹ :λ∈Σθ}, are uniformly bounded on L^p_σ(Ω) for all θ∈(0, π), together with

|λ|^1/2∂_i(λ−A_p)⁻¹P=∂_i(−A_p)^−1/2|λ|^1/2(−A_p)^1/2(λ−A_p)⁻¹P,

|λ|^1/2(λ−A_p)⁻¹P∂_i =|λ|^1/2(−A_p)^1/2(λ−A_p)⁻¹(−A_p)^−1/2P∂_i,

∂_i(λ−A_p)⁻¹P∂_j =∂_i(−A_p)^−1/2(−A_p)(λ−A_p)⁻¹(−A_p)^−1/2P∂_j. This concludes the proof.

We now prove theL^p-L^q-smoothing properties for the hydrostatic Stokes semigroup.

Proof of Theorem 4.1.7. Using the result of Lemma 4.1.10 we may proceed analogously to the proof of [33, Proposition 3.1] for n = 3. Since S is bounded analytic on L^p_σ(Ω) there exists a constant C =CΩ,p >0 such that

kS(t)Pfk_L^p_(Ω)+tkA_pS(t)Pfk_L^p_(Ω) ≤Ckfk_L^p_(Ω),

for allt > 0, and since A_p has a bounded inverse we have kvk_H^2,p_(Ω) ≤CkA_pvk_L^p_(Ω) for allv ∈D(A_p). Lemma 4.1.10 then yields the estimate

kS(t)Pfk_H^2ϑ,p_(Ω) ≤Ct^−ϑkfk_L^p_(Ω), ϑ ∈[0,1],

Settingα:= 3(1/p−1/q) we have the embeddingH^α,p(Ω),→L^q(Ω). Assume now that α≤2. Then the first inequality follows from

kS(t)Pfk_L^q_(Ω)² ≤CkS(t)Pfk_H^α,p_(Ω)

≤Ct^−α/2kfk_L^p_(Ω)

=Ct⁻³²(¹p−¹_q)kfk_L^p_(Ω), t >0.

In the caseα >2 we have p <2< q and so the estimate follows via

kS(t)Pfk_L^q_(Ω) ≤Ct⁻³²(¹2−¹_q)kS(t/2)Pfk_L²_(Ω) ≤Ct⁻³²(¹2−¹_q)t⁻³²(¹p−¹₂)kfk_L^p_(Ω). The remaining inequalities are obtained analogously using Corollary 4.1.5 and 4.1.6 by writing

∂_i =∂_i(−A_p)^−1/2(−A_p)^1/2, P∂_i = (−A_p)^1/2(−A_p)^−1/2P∂_i as well as S(t)P=PS(t)P, compare the proof of Corollary 4.1.6 above.

It remains to prove the elliptic regularity of the hydrostatic Stokes operator. For this purpose we also require the following lemma.

Lemma 4.1.11. Let p ∈ (1,∞] and s > 0. Then the two-dimensional Helmholtz pro-jection with periodic boundary conditions Q is bounded on B_p,p,per^s (G)².

Proof. Recall that the operator 1−Q is given by f 7→ ∇_Hπ where

∆_Hπ= div_Hf, π periodic on∂G.

We identify G with periodic boundary conditions with the two-dimensional torus T². Then 1−Q agrees with the Fourier multiplier with the discrete symbol

m(k) =k⊗k|k|⁻², k ∈Z²\ {0},

where we used the notation x⊗x := (x_ix_j)1≤i,j≤2 for x ∈C². On the whole-space R², the Fourier multiplier with the symbol

m(ξ) = ξ⊗ξ|ξ|⁻², ξ∈R²\ {0},

is bounded onB_p,p^s (R²)² by the theory of Fourier multipliers on Besov spaces, see, e.g., [5, Theorem 6.2], including the casep=∞fors >0. These arguments can then be adapted to the case of the torus, compare, e.g., [47, Proposition 4.5].

4.1 L^p-theory using perturbation arguments Proof of Lemma 4.1.8. As in the proof of Theorem 4.1.1 we have that

Bp :H_per^s,p(Ω)² 3v 7→ 1

h(1−Q)∂zv Γb− 1

h(1−Q)∂zv

Γu ∈H_per^{s−1−1/p−ε}(Ω)²

defines a bounded linear operator for all s > 1 + 1/p+ε and ε > 0. In detail, we have that the mappings

H_per^s,p(Ω)² 3v 7→∂_zv ∈H_per^s−1,p(Ω)², H_per^s−1,p(Ω)² 3∂zv 7→∂zv

Γu, ∂zv

Γb ∈B_p,p,per^{s−1−1/p,p}(G)²

are bounded, whereas 1 − Q is bounded on Bp,p,per^{s−1−1/p,p}(G)² by Lemma 4.1.11. The embedding

B_p,p,per^{s−1−1/p,p}(G)² ,→Fs−1−1/p−ε,p

p,2,per (G)² =Hs−1−1/p−ε,p

per (G)²

then yields the boundedness. We set v := (µ−A_p)⁻¹f ∈ D(A_p). Then it holds that (µ−∆_p)v =f +B_pv and so by Lemma 3.4.4 we have for all r >0 that v ∈ H^2+r,p(Ω) if f+B_pv ∈H^r,p(Ω). Since v ∈H_per^2,p(Ω)² we have f +B_pv ∈H_per^r0,p(Ω)² where

r0 = min{s, δ}, δ= 1−1/p−ε, ε ∈(0,1−1/p).

Iterating this argument yields v ∈H_per^2+rn,p(Ω)² and f +B_pv ∈ H_per^rn,p(Ω)² for a recursive sequence (r_n)n∈N given by

r_n+1 := min{s, r_n+δ}.

Since this sequence either increases by δ > 0 or terminates at r_n = s, we thus obtain v ∈H_per^2+s,p(G) after finitely many steps.

Due to the compactness of the embedding H^2,p(Ω) ,→ L^p(Ω), see [89, Section 4.3.2, Remark 1], we have that D(A_p) ,→ L^p_σ(Ω) is compact as well and thus (λ−A_p)⁻¹ is a compact mapping for all λ ∈ ρ(A_p), see [57, Chapter III, Theorem 6.29]. We further have ρ(Ap)6=∅ by (4.1.7). This implies that σ(Ap) consists of only a discrete sequence of eigenvalues of finite multiplicity.

If v ∈D(A_p) is an eigenvector with eigenvalue λ, we then have A_pv =λv∈H_per^2,p(Ω)² and thus v ∈ H^2n,p(Ω)² for all n ∈ N by induction. Sobolev embedding theory then implies that v ∈C^∞(Ω)².

Now observe that div_Hv = 0 implies that R

Ω∇_Hπ ·v^∗dµ = 0 for all π ∈ W_per^1,p(G).

Since Av= ∆v+Bv and the perturbation term is of the form Bv=∇Hπ for some such π ∈W_per^1,p(G), it follows that

Z

Ω

Av·v^∗dµ= Z

Ω

∆v·v^∗dµ=− Z

Ω

|∇v|²dµ≤0,

and thus Av = λv for v 6= 0 implies that λ ≤ 0. If in addition it holds that Γ_D 6= ∅, then we further have λ6= 0. This completes the proof.

Remark 4.1.12. We deliberately chose to present a proof based on bootstrapping argu-ments to highlight the close ties between the hydrostatic Stokes operator and the Laplace operator, as well as the applicability of the theory of elliptic operators. A different proof of this result can be performed using the concept of Banach scales, see [4, Chapter V.1].

4.2 L

^∞

-L

^p

-theory for Neumann boundary conditions

In this section, we will deviate from 4.1 in several ways. On the one hand, we will consider the unbounded layer domainL=R²×(−h,0) without boundary conditions in the horizontal variables. On the other hand, we will impose pure Neumann boundary conditions on ∂L, meaning that here we will consider the hydrostatic Stokes equations (4.0.2) in the form











∂_tv−∆v+∇_Hπ = f in L×(0, T), div_Hv = 0 in L×(0, T),

∂_zv = 0 on ∂L×(0, T), v(0) = a in L.

(4.2.1)

Finally, we will be moving fromL^p-spaces for p∈(1,∞) to spaces with a norm resem-bling that of L^∞ and L¹. For this purpose we will be making use of the anisotropic L^p-spaces defined in Section 2.4.2. The results of this sections have been previously published in [40, Section 2-5].

Let p ∈ [1,∞] and consider the space L^∞_HL^p_z(L) = L^∞(R²;L^p(−h,0)) as defined in Section 2.4.2, as well as its closed subspace

L^∞,p_σ (L) :=

v ∈L^∞_HL^p_z(L)² : Z

R²

v ∇_Hϕ dx= 0 for all ϕ∈Wc^1,1(R²)

. (4.2.2) Here Wc^1,1(R²) = {ϕ ∈ L¹_loc(R²) : ∇Hϕ ∈ L¹(R²)} is a homogeneous Sobolev space, meaning thatL^∞,p_σ (L) is the space of allv ∈L^∞_HL^p_z(L)² such that div_Hv = 0 in the sense of distributions.

In this setting, there is again a hydrostatic Helmholtz projection, again denoted by P, given by the mapping

Pf :=f− ∇_Hπ, ∆_Hπ= div_Hf on R²,

compare (4.0.3). Observing that the solution operator of the weak problem above is related to the Riesz transform, we find that

Pf =f + (R⊗R)f , R⊗R := (R_iR_j)1≤i,j≤2, (4.2.3) where R_i = ∂_i(−∆_H)^−1/2 for i = 1,2 denotes the two-dimensional Riesz transforms in the horizontal variables. Note that since the Riesz transforms fail to be bounded on L^p(R²) forp= 1,∞, the projection Pis likewise unbounded.

By applying P to the problem (4.2.1) one again obtains a Cauchy problem for v, compare (4.0.7). However, since Neumann boundary conditions are imposed on both the top and bottom part of the boundary in (4.2.1), the new problem is simply given by











∂_tv −∆v = Pf in L×(0, T), divHv = 0 in L×(0, T),

∂_zv = 0 on ∂L×(0, T), v(0) = a in L,

(4.2.4)

4.2 L^∞-L^p-theory for Neumann boundary conditions since we have Av = P∆v = ∆v, compare (4.0.6). In particular, the heat semigroup generated by ∆ with Neumann boundary conditions on L leaves the space L^∞,p_σ (L) invariant and the hydrostatic Stokes semigroup is simply given by the restriction of the heat semigroup. Given f = 0, the solution to (4.2.4) is given via

v(t) = (S_H(t)⊗S_N(t))a=S_H(t)S_N(t)a, t ≥0,

where SH denotes the heat semigroup on R² given by the convolution with the two-dimensional Gaussian kernel G_t and S_N is the vertical heat semigroup on (−h,0) from Lemma 3.3.2. The tensor notation means that the operators are applied successively, commute, and preserve product structures, i.e., if we have

a(x, y, z) = (a_H ⊗a_z)(x, y, z) := a_H(x, y)a_z(z) (4.2.5) for functions a_h: R² →C², a_z: (−h,0)→C, then

(S_H(t)⊗S_N(t))a=S_H(t)a_H ⊗S_N(t)a_z.

We will not be distinguishing between the heat semigroup on L^∞_HL^p_z(L) and the hydro-static Stokes semigroup on L^∞,p_σ (L) and simply denote both via

S(t) := S_H(t)⊗S_N(t) :=S_H(t)S_N(t), t ≥0. (4.2.6) Since these semigroups operate in different variables, the tensor product is simply the composition of these operators, applied first in the vertical variable and then in the horizontal one, yielding the representation

(S(t)f)(x, y, z) = Z

R²

G_t(x−x⁰, y−y⁰)(S_N(t)f)(x⁰, y⁰, z)d(x⁰, y⁰) (4.2.7) for all (x, y, z) ∈ L. Since S_H and S_N are contraction semigroups on L^∞(R²) and L^p(−h,0), respectively, see Lemma 3.3.2, it follows that S is a contraction semigroup L^∞_HL^p_z(L). However, since the two-dimensional heat semigroup fails to be strongly con-tinuous on L^∞(R²), S is not strongly continuous on L^∞,p_σ (L). However, it is strongly continuous, and even bounded analytic, on the space BU C(R²;L^p(−h,0))², as well as its invariant subspace

X_σ^∞,p(L) :=BU C(R²;L^p(−h,0))²∩L^∞,p_σ (L). p∈[1,∞). (4.2.8) The results of this sections are similar to Theorem 4.1.7 for the case p = q. Here we will consider more general estimates, namely ones involving fractional derivatives. Due to the tensor structure of the semigroup, we distinguish between those in horizontal and vertical direction, using fractional powers of the horizontal Laplace operator −∆_H, compare (3.1.1), as well as Caputo derivatives defined in (3.3.7).

Due to the fact that the hydrostatic Stokes semigroup is merely the restriction of the heat semigroup and allows for the representation (4.2.7), the proof of the following the-orem only requires estimates for the heat-semigroups S_H and S_N. Using the shorthand notation k·k_∞,p :=k·k_L^∞

HL^pz(L), we have the following.

Theorem 4.2.1. Let p∈[1,∞] and α∈(0,1). Then the following holds.

(a) The hydrostatic Stokes semigroup S is a contraction semigroup on L^∞_HL^p_z(L)² with invariant subspaces L^∞,p_σ (L) and X_σ^∞,p(L). If p∈[1,∞), it is strongly continuous on BU C(R²;L^p(−h,0))².

(b) The operator S(t) maps L^∞_HL¹_z(L)² into L^∞_HL^p_z(L)² for all t > 0 and there exists a constant C >0 such that for all f ∈L^∞_HL¹_z(L)² and t >0 it holds that

kS(t)fk∞,p ≤C 1 +t^{−(1−1/p)/2}

kfk∞,1

for all t >0.

(c) There exists a constant C =C_α >0, such that for all f ∈L^∞_HL^p_z(L)² and t >0 it holds that

k∇S(t)fk_∞,p ≤Ct^−1/2kfk∞,p, (i) kS(t)∇_H ·fk_∞,p ≤Ct^−1/2kfk∞,p, (ii) kS(t)∂zfk_∞,p ≤Ct^−1/2kfk∞,p, (iii) S(t)P(−∆H)^α/2f

_∞,p ≤Ct^−α/2kfk∞,p, (iv) kS(t)P∇_H ·fk_∞,p ≤Ct^−1/2kfk∞,p, (v) as well as

kS(t)∂zI^αfk_∞,p ≤Ct^{−(1−α)/2}kfk∞,p (vi) whenever (I^αf)(0) = 0.

(d) If p∈[1,∞), then for any f ∈BU C(R²;L^p(−h,0))² it holds that

t→0+lim t^1/2k∇S(t)fk∞,p = 0.

Proof. In the following we use the notation k·k_p := k·k_L^p_(−h,0) for simplicity. For (a), the contraction property follows from the fact that S_N is a contraction semigroup on L^p(−h,0) by Lemma 3.3.2 and the fact that Gt > 0 and kGtk_L¹_(R²₎ = 1 for all t > 0, yielding

k(S(t)f)(x, y,·)k_L^p(−h,0) =k(S_H(t)S_N(t)f)(x⁰, y⁰,·)k_p

= Z

R²

G_t(x−x⁰, y−y⁰)(S_N(t)f)(x⁰, y⁰,·)d(x⁰, y⁰) p

≤ Z

R²

G_t(x−x⁰, y−y⁰)k(S_N(t)f)(x⁰, y⁰,·)k_p d(x⁰, y⁰)

≤ Z

R²

G_t(x−x⁰, y−y⁰)kf(x⁰, y⁰,·)k_p d(x⁰, y⁰)

≤ kG_tk_L¹_(R²₎kfk∞,p

=kfk∞,p

4.2 L^∞-L^p-theory for Neumann boundary conditions for all (x, y) ∈ R² and thus kS(t)fk∞,p ≤ kfk∞,p for all t > 0. Here we also used the Minkowski inequality in the third step.

The strong continuity forp∈[1,∞) follows from the fact that S_H andS_N are strongly continuous onBU C(R²)² and L^p(−h,0), respectively, and thusS is strongly continuous on the set

{f⊗g :f ∈BU C(R²)², g ∈L^p(−h,0)},

the linear hull of which is dense inBU C(R²;L^p(−h,0))². Here we used the notation from (4.2.5). The space BU C(R²;L^p(−h,0))² is invariant since S_H preserves continuity and L^∞,p_σ (L) is invariant since Lemma 3.3.2.(vi) implies thatS(t)f =S_Hf andS_H commutes with divH. Thus, X_σ^∞,p(L) is invariant as well. For (b), we similarly have

S(t)f =S_H(t)S_N(t)f =S_H(t)I¹f, where I¹f := R0

−hf(·, z)dz denotes the vertical integral. Since S_H is contractive on L^∞(R²) it follows that

kS(t)I¹fk∞,p ≤ kI¹fk_L^∞_(R²₎ ≤ kfk∞,1. We now set g :=f −I¹f and recall from the proof of (a) that

k(S(t)g)(x, y,·)k_L^p(−h,0) ≤ Z

R²

G_t(x−x⁰, y−y⁰)k(S_N(t)g)(x⁰, y⁰,·)k_p d(x⁰, y⁰)

≤Ct^{−(1−1/p)/2} Z

R²

G_t(x−x⁰, y−y⁰)kg(x⁰, y⁰,·)k₁ d(x⁰, y⁰)

≤Ct^{−(1−1/p)/2}kgk∞,1,

where we used the L¹-L^p-smoothing of S_N from Lemma 3.3.2.(v) in the second step.

The estimate then follows from kgk∞,1 ≤ Ckfk∞,1. Estimate (i) in (c) follows from kG_tk₁ = 1 for allt >0 and

k(∂_zS(t)f)(x, y,·)k_p =kS_H(t)∂_zS_N(t)f(x, y,·)k_p

= Z

R²

G_t(x−x⁰, y−y⁰)(∂_zS_N(t)f)(x⁰, y⁰,·)d(x⁰, y⁰) p

≤ Z

R²

G_t(x−x⁰, y−y⁰)k(∂_zS_N(t)f)(x⁰, y⁰,·)k_pd(x⁰, y⁰)

≤Ct^−1/2 Z

R²

Gt(x−x⁰, y−y⁰)kf(x⁰, y⁰,·)kpd(x⁰, y⁰),

≤Ct^−1/2kfk∞,p,

where we again used the Minkowski inequality in the third and Lemma 3.3.2.(ii) in the fourth step, as well as

k(∂_iS(t)f)(x, y,·)k_p =k∂_iS_H(t)S_N(t)f(x, y,·)k_p

≤Ct^−1/2 Z

R²

G_2t(x−x⁰, y−y⁰)kf(x⁰, y⁰,·)k_pd(x⁰, y⁰)

for∂_i ∈ {∂_x, ∂_y}via Lemma 3.1.3 and the contraction property for the vertical semigroup.

Estimate (ii) follows from (i) via S(t)∂_if = ∂_iS(t)f, whereas estimates (iii) and (vi) follow from Lemma 3.3.7 via

k(S(t)∂_zI^αf)(x, y,·)k_p ≤ Z

R²

G_t(x−x⁰, y−y⁰)k(S_N(t)∂_zI^αf)(x⁰, y⁰,·)k_pd(x⁰, y⁰)

≤Ct^{−(1−α)/2} Z

R²

G_t(x−x⁰, y−y⁰)kf(x⁰, y⁰,·)k_pd(x⁰, y⁰)

≤Ct^{−(1−α)/2}kfk∞,p.

In order to obtain estimate (iv) we use (4.2.3) to write

S(t)P(−∆_H)^α/2f =S(t)(−∆_H)^α/2f +S(t)(R⊗R)(−∆_H)^α/2f

and by the contraction property of the vertical semigroup and estimate (1) and (2) from Lemma 3.1.3 we have using S(t) =S_H(t)S_N(t) = S_N(t)S_H(t) that

kS(t)(−∆_H)^α/2f(x, y,·)k_p ≤ kS_H(t)(−∆_H)^α/2f(x, y,·)k_p

≤t^−α/2 Z

R²

H_t^α(x−x⁰, y−y⁰)|f(x⁰, y⁰,·)|d(x⁰, y⁰) p

≤t^−α/2 Z

R²

H_t^α(x−x⁰, y−y⁰)kf(x⁰, y⁰,·)k_p d(x⁰, y⁰)

≤C_αt^−α/2kfk∞,p, as well as

kS(t)(R⊗R)(−∆_H)^α/2f(x, y)k_p ≤ kS_H(t)(R⊗R)(−∆_H)^α/2f(x, y)k_p

=h^1/p|S_H(t)(R⊗R)(−∆_H)^α/2f(x, y)|

≤h^1/pt^−α/2( ˜H_t^α∗H f)(x, y).

The estimate then follows from kH_t^αk₁ +kH˜_t^αk₁ ≤ C_α for all t > 0. Estimate (v) is obtained analogously via estimates (ii) and (3) from Lemma 3.1.3. Finally, for (d), we take ε > 0 and use the fact that C_c^∞(−h,0) is dense in L^p(−h,0) and that functions belonging toBU C(R²) can be uniformly approximated by smooth functions. This allows us to approximatef ∈BU C(R²;L^p(−h,0))² by

g ∈C^∞(R²;C_c^∞(−h,0))², kf−gk∞,p≤ ε

2C, k∇gk∞,p<∞, where C >0 is as in estimate (i). This yields

t^1/2k∇S(t)fk∞,p ≤ ε

2+t^1/2k∇S(t)gk∞,p.

Since∇_H commutes with the two-dimensional heat semigroup S_H as well as the vertical semigroup SN, we have ∇HS(t)g = S(t)∇Hg and due to ∂zSN(t) = SD(t)∂z for SD as in Lemma 3.3.2 we further have∂_zS(t)g =S_H(t)S_D(t)∂_zg. This yields

k∇S(t)gk∞,p≤ k∇gk∞,p, which implies the desired result.

Im Dokument On the primitive equations and the hydrostatic Stokes operator (Seite 70-81)