On off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients

We investigate off-diagonal decay properties of the generalized Stokes semigroup with bounded measurable coefficients on $\mathrm{L}^2_{\sigma} (\mathbb{R}^d)$. Such estimates are well-known for elliptic equations in the form of pointwise heat kernel bounds and for elliptic systems in the form of integrated off-diagonal estimates. On our way to unveil this off-diagonal behavior we prove resolvent estimates in Morrey spaces $\mathrm{L}^{2 , \nu} (\mathbb{R}^d)$ with $0 \leq \nu<2$.


Introduction
In this note we study decay properties of the resolvent as well as the associated semigroup of the generalized Stokes operator A on L 2 σ (R d ). This operator is formally given by Here, the function u denotes a fluid velocity and φ denotes the to the generalized Stokes equations associated pressure function. The matrix of coefficients is merely supposed to be essentially bounded and ellipticity is enforced by a Gårding type inequality. If the elliptic counterpart Lu = − div(µ∇u) is considered, then certain off-diagonal decay properties of the corresponding heat semigroup are well-known. For example, if L represents an elliptic equation with real coefficients, then the kernel k t (·, ·) of the associated heat semigroup (e −tL ) t≥0 satisfies heat kernel bounds It is well-known that if L represents an elliptic system with real/complex coefficients these heat kernel bounds seize to be valid [6,9,11]. The natural substitute for heat kernel bounds for elliptic systems are so-called off-diagonal estimates. The simplest version are L 2 off-diagonal estimates for the heat semigroup, its gradient, or also for L applied to the heat semigroup and are of the form where E, F ⊂ R d are closed subsets and f ∈ L 2 (R d ) has its support in E. Such estimates build the foundation for many deep results in the harmonic analysis of elliptic operators with rough coefficients as can be seen, e.g., in the seminal works on the Kato square root problem [5] as well as on mapping properties of Riesz transforms on L p -spaces [3] or the well-posedness results of Navier-Stokes like equations with initial data in BMO −1 [4] in the spirit of Koch and Tataru [10].
The spirit of how these off-diagonal estimates (1.1) are used is as follows. For example, one might be interested in estimating an expression that involves e −tL f in some sense. One then decomposes R d into carefully chosen disjoint sets, e.g., into annuli of the form C k := B(x 0 , 2 k+1 r)\ Date: March 5, 2021. B(x 0 , 2 k r), k ∈ N, and C 0 := B(x 0 , 2r). Then one would estimate by virtue of (1.1) and proceed with the proof in a certain manner, depending on the particular situation. The question, whose study we want to initiate here, is whether or not the generalized Stokes semigroup (e −tA ) t≥0 satisfies off-diagonal decay estimates and if so, how they look like. The main problem is already, that in a calculation of the form (1.2) one multiplies f by a characteristic function. This in general destroys the solenoidality of the function f . Thus, if one wants to perform such an operation, one is urged to think about how to extend e −tA to all of L 2 (R d ). In many situations, the gold standard is to extend e −tA to all of L 2 (R d ) by studying e −tA P, where P denotes the Helmholtz projection on L 2 (R d ). Thus, in order to imitate the calculation performed in (1.2) one would need that off-diagonal bounds for e −tA P are valid. However, estimates of the form = 0 and f being supported in E are in general wrong. The reason is simple: fix any closed subset E ⊂ R d and let F ⊂ R d denote any other closed set that satisfies dist(E, F ) > 0. On the one hand, since (e −tA ) t≥0 is strongly continuous On the other hand (1.3) together with the condition on g implies that Pf L 2 (F ) = 0. This implies that supp(Pf ) ⊂ E whenever f ∈ L 2 (R d ) with supp(f ) ⊂ E. As a consequence, the Helmholtz projection would be a local operator, which is known to be wrong. Thus, in order to establish off-diagonal bounds for the generalized Stokes semigroup, one either needs to find the correct extension of the generalized Stokes semigroup to all of L 2 (R d ) or one needs to avoid arguments that destroy the solenoidality of f . In particular, this rules out standard proofs of off-diagonal estimates that are used in the elliptic situation as, e.g., Davies' trick [7].
The main result of this note is an estimate of the type (1.2). Let us introduce some notation to state this in a precise form: : div(f ) = 0}. Define the sesquilinear form The main result of this note is the following theorem: Moreover, for all F ∈ L 2 (R d ; C d×d ) it holds In both estimates, the constant C only depends on µ • , µ • , d, and ν.
As a corollary of Theorem 1.2 one derives the following off-diagonal estimates.

Moreover, for all
In both estimates, the constant C only depends on µ • , µ • , d, and ν.

A non-local resolvent estimate
To establish Theorem 1.2 we prove analogous estimates for the resolvent of A. More precisely, we are going to estimate the solution u to the generalized Stokes resolvent problem for λ in some complex sector S ω := {z ∈ C \ {0} : |arg(z)| < ω}. Using Assumption 1.1 together with the lemma of Lax-Milgram, one finds some ω ∈ (π/2, π) depending on µ • , µ • , and d such that (2.1) is uniquely solvable for all f ∈ L 2 σ (R d ) and all F ∈ L 2 (R d ; C d×d ). In the follwing, let us denote the solution operator to (2.1) by (λ + A) −1 . The solution u to (2.1) then lies in the space , and all λ ∈ S θ it satisfies the resolvent estimates and The next lemma was proven in [12,Lem. 5.3] and combines different types of Caccioppoli inequalities to account for the non-local pressure.
, and λ ∈ S θ the following holds: for Moreover, u 1 and φ 1 satisfy for some C > 0
This lemma can be used to prove the following non-local resolvent estimate.
Proof. We use the decomposition of u from Lemma 2.1 as follows. Fix k ∈ N 0 and let ℓ 0 ∈ N to be determined. Let u 1,k , u 2,k , and φ 1,k be the functions determined by Lemma 2.1 with r 0 := 2 k+ℓ0+1 r. Now, we proceed by applying Hölder's inequality, then increase the domain of integration, and use Sobolev's embedding to obtain for q > 1 with Notice that the constant C > 0 in the previous estimate only depends on d and q. Now, use this estimate together with u 2,k = u − u 1,k and (2.4) and (2.5) to deducê Now, multiply this inequality by 2 −νk and sum with respect to k ∈ N 0 . This then delivers Now, in order to conclude that the exponent d − d q − ν is positive, we need to require further restrictions to q. One immediately verifies that the positivity of this exponent as well as (2.6) are fulfilled, whenever q satisfies Since ν < 2, such a choice is possible. Thus, fixing q subject to (2.7) allows to choose ℓ 0 large enough so as to absorb the λu-term on the right-hand side to the left-hand side. Thus, there exists C > 0 such that As a corollary we get that the generalized Stokes operator satisfies resolvent estimates with respect to the Morrey space norm of L 2,ν (R d ; C d ) for all 0 ≤ ν < 2. The definition of this Morrey space is the following: .
Proof. Fix x 0 ∈ R d and r > 0. The estimate in Theorem 2.2 readily gives for some ν < ν ′ < 2 B(x0,r) Division by r ν then delivers the desired estimate.

L 2 off-diagonal decay for the resolvent
This section is dedicated to prove a counterpart of Theorem 1.2 for the resolvent of A. For this purpose, we introduce another sesquilinear form, which is connected to the Stokes problem in a ball but with Neumann boundary conditions. Let B ⊂ R d denote a ball and let We abuse the notation and denote the same sesquilinear form but with domain H 1 (B; An application of Assumption 1.1 and the lemma of Lax-Milgram implies the existence of ω ∈ (π/2, π) such that for all λ ∈ S ω , f ∈ L 2 σ (B), and F ∈ L 2 (B; C d×d ) the equation is uniquely solvable for some u ∈ H 1 σ (B). Moreover, by [12, Rem. 5.2], there exists a pressure function φ ∈ L 2 (B) such that holds. Furthermore, for all θ ∈ (0, ω) there exists C > 0 depending only on d, θ, µ • , and µ • such that for all λ ∈ S ω , f ∈ L 2 σ (B), and F ∈ L 2 (B; C d×d ) it holds To proceed, we cite some results from [12]. The first result is a non-local Caccioppoli inequality for the generalized Stokes resolvent and can be found in [12, Thm. 1.2].
Theorem 3.1. Let µ satisfy Assumption 1.1 for some constants µ • , µ • > 0. Then there exists ω ∈ (π/2, π) such that for all θ ∈ (0, ω) and all 0 < ν < d + 2 there exists C > 0 such that for all satisfies for all balls B = B(x 0 , r) and all sequences ( The constant ω only depends on µ • , µ • , and d and C depends on µ • , µ • , d, θ, and ν. The second result is an estimate on the pressure function φ that appears in (2.1) and can be found in [12,Lem. 2.1]. To formulate this lemma, we adopt the notation C k := B(x 0 , 2 k r) \ B(x 0 , 2 k−1 r) for k ∈ N and write φ C k for the mean value of φ on the set C k . Lemma 3.2. Let µ satisfy Assumption 1.1 for some constants µ • , µ • > 0. Let λ ∈ C and let for f ∈ L 2 σ (R d ) and F ∈ L 2 (R d ; C d×d ) the functions u ∈ H 1 σ (R d ) and φ ∈ L 2 loc (R d ) solve in the sense of distributions. Let x 0 ∈ R d and r > 0 let C 0 denote the ball B(x 0 , r). Then there exists a constant C > 0 depending only on µ • and d such that for all k ∈ N we have The final preparatory result we need is a local Caccioppoli inequality that includes the pressure function.
Proof. Let η ∈ C ∞ c (B(x 0 , 2r)) with η ≡ 1 in B(x 0 , r), 0 ≤ η ≤ 1, and ∇η L ∞ ≤ 2/r. Applying [12, Lem. 5.1] with c 1 = c and c 2 = 0 implies that |λ|ˆB (x0,2r) The lemma follows by absorbing the uη-term to the left-hand side and by using the properties of η. Finally, we would like to mention that the proof of [12, Lem. 5.1] follows the standard proof that is used to establish the Caccioppoli inequality for elliptic systems and this is well-known.
The following theorem presents L 2 off-diagonal type estimates for the resolvent operators.
, and λ ∈ S θ . Define u := (λ + A) −1 (f + P div(F )) and let φ ∈ L 2 loc (R d ) be the associated pressure such that u and φ solve (2.1). Let x 0 ∈ R d and r > 0. In the following, we consider two cases.

Estimates on the generalized Stokes semigroup
Since A satisfies the resolvent estimates for some ω ∈ (π/2, π) the generalized Stokes operator −A is the infinitesimal generator of a bounded analytic semigroup (e −tA ) t≥0 which is represented via the Cauchy integral formula Here, the path γ t runs through ∂(B(0, t −1 ) ∪ S ϑ ) for some ϑ ∈ (π/2, ω) in a counterclockwise manner. This representation by the Cauchy integral formula allows to transfer estimates on the resolvent to estimates on the semigroup. For example, it is well-known that the estimates (2.2) and (2.3) used within (4.1) directly yield for all f ∈ L 2 σ (R d ), F ∈ L 2 (R d ; C d×d ), and t > 0 the semigroup estimates and t 1 2 e −tA P div(F ) L 2 + t ∇e −tA P div(F ) L 2 ≤ C F L 2 . The following proof of Theorem 1.2 shows that this transfer of estimates is also valid for the resolvent estimates established in Theorem 3.4.
Remark 4.1. If we assume that r 2 /t > 1, then all λ ∈ γ t satisfy |λ|r 2 > 1 so that in this case the estimates from Remark 3.5 together with the proof of Theorem 1.2 yield the following gradient estimate on the generalized Stokes semigroup: there exists a constant C > 0 such that for all f ∈ L 2 σ (R d ), F ∈ L 2 (R d ; C d×d ), and all t > 0 we have and ∇e −tA F L 2 (B(x0,r)) ≤ C F L 2 (B(x0,2r)) + C ∞ k=2 2 2k r 2 t − ν 4 F L 2 (B(x0,2 k r)) .
Proof of Corollary 1.3. We distinguish two cases. Assume first that 2 2k0 r 2 /t < 1. Then by using the global L 2 -estimates (4.2), we find that e −tA f L 2 (B(x0,r)) + t Ae −tA f L 2 (B(x0,r)) ≤ C f L 2 (R d ) To estimate the terms involving e −tA P div(F ) proceed similarly, but by employing (4.3) in the first case and Theorem 1.2 in the second case. We omit further details.