WELL-POSEDNESS OF EVOLUTIONARY NAVIER-STOKES EQUATIONS WITH FORCES OF LOW REGULARITY ON

https://doi.org/10.1051/cocv/2021058 www.esaim-cocv.org

WELL-POSEDNESS OF EVOLUTIONARY NAVIER-STOKES EQUATIONS WITH FORCES OF LOW REGULARITY ON

TWO-DIMENSIONAL DOMAINS

Eduardo Casas

and Karl Kunisch

Abstract. Existence and uniqueness of solutions to the Navier-Stokes equations in dimension two with forces in the spaceL^q((0, T);W^−1,p(Ω)) forpandqin appropriate parameter ranges are proven.

The case of spatially measured-valued forces is included. For the associated Stokes equation the well- posedness results are verified in arbitrary dimensions for any 1< p, q <∞.

Mathematics Subject Classification. 35B40, 35Q30, 76D07, 76N10.

Received April 17, 2020. Accepted May 26, 2021.

Dedicated to Prof. Dr. Enrique Zuazua on the occasion of his 60th birthday.

1. Introduction

In this paper we investigate the following Navier-Stokes system







∂y

∂t −ν∆y+ (y· ∇)y+∇p=f inQ= Ω×I, divy= 0 inQ, y= 0 on Σ = Γ×I, y(0) =y0 in Ω,

(1.1)

with focus on low regularity assumptions on the inhomogeneityf. Here,I= (0, T) with 0< T <∞, and Ω⊂R^d denotes a connected bounded domain with aC³ boundary Γ.

Our interest in this problem is two-fold. First, it has received very little attention in the literature so far.

Indeed the only result which we are aware of is given in [24], where f is chosen in W^1,∞(I;W^−1,p(Ω)), with W^−1,p(Ω) = Nd

i=1W^−1,p(Ω), d∈ {2,3}, and p∈(^d₂,2]. It is mentioned there, that likely the result is not optimal, and the natural question arises whether, and how, it can be improved. Secondly we are interested in control problems with sparsity constraints, subject to (1.1) as constraint. In this case it is natural to demand that

∗The first author was partially supported by Spanish Ministerio de Econom´ıa y Competitividad under research project MTM2017-83185-P. The second was supported by the ERC advanced grant 668998 (OCLOC) under the EU’s H2020 research program.

Keywords and phrases:Evolution Navier-Stokes equations, weak solutions, uniqueness clasess, sensitivity analysis, asymptotic stability.

1 Departmento de Matem´atica Aplicada y Ciencias de la Computaci´on, E.T.S.I. Industriales y de Telecomunicaci´on, Universidad de Cantabria, 39005 Santander, Spain.

2 Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria.

**Corresponding author:[email protected]

c

The authors. Published by EDP Sciences, SMAI 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

for almost everyt∈I the forcing function is a vector valued Borel measure,i.e.f(t,·)∈M(Ω) =Nd

i=1M(Ω), where M(Ω) is the space of real and regular Borel measures in Ω, see e.g. [7] and the references on sparse control given there. To treat (1.1) with spatially measure valued controls it is natural to consider the space W^−1,p(Ω) with p∈[1,_d−1^d ), since in this caseM(Ω)⊂W^−1,p(Ω). In dimension 3 this requires to consider the spaceW^−1,p(Ω) withp∈[1,³₂). However, even in the stationary case the existence of a solution is an open issue for d= 3 and this range of values forp, see [18] or [24]. For this reason we restrict our attention to the case d= 2 throughout the paper, unless specifically mentioned otherwise. For the two-dimensional case the result in [24] guarantees the existence of a solution to (1.1) forf ∈W^1,∞(I;M(Ω)). But this regularity requirement with respect to time is not practical for control theory purposes.

Thus the focus of our work is the investigation of (1.1) forf ∈L^q(I;W^−1,p(Ω)) in the case Ω⊂R², andp <2.

For this purpose we also require results on the Stokes equation associated to (1.1). Surprisingly, even this case has not yet been analysed for f ∈L^q(I;W^−1,p(Ω)). We carry out such an analysis which will be independent of the spatial dimensiond.

Before we start, let us summarize, very selectively, some relevant literature. In the stationary case the investi- gation of the Navier-Stokes system with data less regular thanL²(Q) dates back to [20], who considers the case f ∈W^−1,2(Ω). In [24] the range of admissible forcing functions is increased tof ∈W^−1,p(Ω), with p∈(^d₂,2) For more recent results we refer to [10,12,18], and the references there.

For the evolutionary system well-posedness for forcing functions in the Hilbert spaces L²(I;L²(Ω)), and L²(I;W^−1,2(Ω)), with d∈ {2,3}, is well understood, see e.g. [5] or [28]. The analysis of the Stokes problem associated to (1.1) with forcing functions in the Bochner spacesL^q(I;L^p(Ω)), with 1< p, q <∞,has attracted much attention. We refer to [17] for an informative summary, including the development of the maximal regu- larity techniques for this scenario. Well-posedness of the Navier-Stokes system with f ∈L^q(I;L^p(Ω)) has been investigated in [16] or [30]. As mentioned above, the only work that we are aware of where (1.1) with forcing functions in Sobolev spaces with negative exponents has been investigated is [24].

The plan of the paper is the following. In Section2the well-posedness results for the Stokes and the Navier- Stokes equations with f ∈L^q(I;W^−1,p(Ω)) under proper conditions on pand q are presented. Some selected proofs are postponed to Sections 3 and 4. Section 5 presents a sensitivity analysis with respect to the right hand side, and Section 6 an asymptotic stability analysis. The proof of a technical result on the nonlinearity appearing in (1.1) is given in the Appendix.

Notation

In this paper, we denoteL^s(Ω) =Nd

i=1L^s(Ω) andW^1,s₀ (Ω) =Nd

i=1W₀^1,s(Ω) for s∈(1,∞), and we choose the norm in W₀^1,s(Ω) as

kyk_W1,s

0 (Ω)=k∇ykL^s(Ω)= Z

Ω

|∇y|^sdx ¹_s

=



 Z

Ω

[

d

X

j=1

|∇yj|²]^s2dx





1 s

.

We also consider the spaces

H= closure of{φ∈C^∞₀ (Ω) : divφ= 0}inL²(Ω), Ws(Ω) ={y∈W₀^1,s(Ω) : divy= 0}.

Fors= 2 we setH¹₀(Ω) =W^1,2₀ (Ω) andV=W2(Ω). We also define the following spaces W(0, T) ={y∈L²(I;V) : ∂y

∂t ∈L²(I;V⁰)}

Wr,s(0, T) ={y∈L^r(I;Ws(Ω)) :∂y

∂t ∈L^r(I;Ws⁰(Ω)⁰)}

withr, s∈(1,∞), endowed with the norms

kyk_W(0,T)=kyk_L2(I;H¹₀(Ω))+k∂y

∂tk_L2(I;V⁰), kykW_r,s(0,T)=kyk_Lr(I;W^1,s₀ (Ω))+k∂y

∂tk_Lr(I;W_s0(Ω)⁰). Obviously these are reflexive Banach spaces, and W(0, T) =W_r,s(0, T) ifr=s= 2.

Now we consider the interpolation space B_s,r(Ω) = (W_s⁰(Ω)⁰,W_s(Ω))_1−1/r,r. From Chapter III/4.10.2 of [1] we know that W_r,s(0, T)⊂C([0, T];B_s,r(Ω)) and the trace mapping y∈W_r,s(0, T)→y(0)∈B_s,r(Ω) is surjective. If r=s= 2, then it is known that B_2,2(Ω) = (V⁰,V)1

2,2=H. Hence, the embedding W(0, T)⊂ C([0, T];H) holds; see Page 22, Proposition I-2.1 of [22] and Page 143, Remark 3 of [31].

2. Well-posedness results

The aim of this section is to prove the well-posedness of the following Navier-Stokes equations in dimension 2







∂y

∂t −ν∆y+ (y· ∇)y+∇p=f inQ,

divy= 0 inQ, y= 0 on Σ, y(0) =y0in Ω,

(2.1)

where ν > 0 is the kinematic viscosity coefficient, f ∈L^q(I;W^−1,p(Ω)), and y0 ∈Y0 =H+Bp,q(Ω). The parameters pandq are fixed throughout this manuscript, and it is assumed that

4

3 ≤p <2 andq > 2p

p−1 (2.2)

hold; with the exception of Corollary 2.6. Observe that these assumptions imply that q >4. This condition (2.2) is essential for the well-posedness of the bilinear form introduced in Lemma2.1as well as in the proofs of Theorem2.4and Proposition2.7. The low regularity of the force is due to the assumptionp <2. Forp≥2 the solvability of (2.1) is well known. The spaceY0is endowed with the norm

ky0kY₀ = inf

y0=y01+y02

ky01kL²(Ω)+ky02kB_p,q(Ω),

which makes it a Banach space. The assumption Ω ⊂ R² is imposed throughout this paper, except in Theorem 2.5, which addresses the Stokes equation.

Now we introduce the following spaces:

Y= [L²(I;V)∩L^∞(I;H)] +L^q(I;Wp(Ω)), Y =W(0, T) +Wq,p(0, T).

They are Banach spaces with the norms kykY = inf

y=y₁+y₂ky1k_L2(I;H¹₀(Ω))+ky1k_L^∞_(I;L2(Ω))+ky2k_Lq(I;W₀^1,p(Ω)), kyk_Y = inf

y=y₁+y₂ky1k_W(0,T)+ky2k_Wq,p(0,T).

The solution of (2.1) will be found in Y. Before proving the existence of such a solution, let us present the following technical lemma. Its proof is given in the Appendix.

Lemma 2.1. Assume that (2.2)andΩ⊂R² hold. The bilinear operatorB:Y×Y−→L²(I;H⁻¹(Ω))defined by B(y₁,y₂) = (y₁· ∇)y₂ is continuous.

As usual, we can remove the pressure from the equation (2.1) by using divergence free test functions.

Definition 2.2. We say thaty∈ Y is a variational solution of (2.1) if









 hd

dty(t),ψi_W

p0(Ω))⁰,W_p0(Ω)+a(y(t),ψ) +b(y(t),y(t),ψ)

=hf(t),ψi_{W−1,p(Ω),W}1,p0

0 (Ω) in (0, T), ∀ψ∈Wp⁰(Ω), y(0) =y0,

(2.3)

where

a(y(t),ψ) =ν Z

Ω

∇y(x, t) :∇ψ(x) dx=ν

2

X

i=1

Z

Ω

∇yi(x, t)∇ψ_i(x) dx, b(y(t),y(t),ψ) =hB(y(t),y(t)),ψi_H−1(Ω),H¹₀(Ω)=

Z

Ω

[y(t)· ∇]y(t)· ∇ψdx.

A distributionp inQis called an associated pressure if the equation

∂y

∂t −ν∆y+ (y· ∇)y+∇p=f inQ is satisfied in the distribution sense. Then, (y,p) is called a solution of (2.1).

Givenysatisfying (2.3), the pressurepis obtained by using De Rham’s theorem; see Lemma IV-1.4.1 of [27].

The details are obtained in a similar way as in the proof for the case of the Stokes equation, see step (iv) of the proof of Theorem 2.5in Section3.

Remark 2.3. Givens∈(1,∞) andg∈W^−1,s(Ω), we have thatg:W^1,s₀ ⁰(Ω)−→Ris a linear and continuous mapping. We know that Ws⁰(Ω) is a closed subspace ofW^1,s₀ ⁰(Ω). Therefore, for every elementg∈W^−1,s(Ω) we can consider its restriction to Ws⁰(Ω), andkgk_W

s0(Ω)⁰ ≤ kgk_W^−1,s_(Ω) holds. Moreover, from Hahn-Banach Theorem we know that every element ofW_s0(Ω)⁰ is the restriction of an element ofW^−1,s(Ω). It is important to observe that the restriction of an elementg∈W^−1,s(Ω) toW_s⁰(Ω) can be zero even thoughg6= 0. Actually, given an element g∈W_s⁰(Ω)⁰, there are infinitely many elements in W^−1,s(Ω) whose restriction to W_s⁰(Ω) coincide with g. As a consequence, the variational solution y of (2.1), as defined by (2.3), only depends on the restriction of f to W_p⁰(Ω). Thus, different elements f can lead to the same solutiony, but the pressure p changes. The pressure depends on the action off on the whole domainW^1,p₀ ⁰(Ω).

As pointed out in Section 1, the embeddings W(0, T)⊂C([0, T];H) and W_q,p(0, T)⊂C([0, T];B_p,q(Ω)) hold. Hence,Y ⊂C([0, T];Y₀) and, consequently, the initial conditiony(0) =y₀ withy₀∈Y₀makes sense.

The next theorem is the main result of this section.

Theorem 2.4. Suppose that (2.2) and Ω⊂R² hold. Then, system (2.1) has a unique solution (y,p)∈ Y × W^−1,q(I;L^p(Ω)/R). Furthermore, there exists a nondecreasing functionηp,q: [0,∞)−→[0,∞)withηp,q(0) = 0 such that

kyk_Y≤η_p,q

kfk_Lq(I;W_p0(Ω)⁰)+ky₀k_Y0

. (2.4)

For the proof of this result we will use the next two theorems. The first one concerns the associated Stokes equation and holds in arbitrary dimensiond. It is given by







∂y_S

∂t −ν∆yS+∇pS =g inQ,

divy_S = 0 inQ, y_S = 0 on Σ, y_S(0) =y_S0 in Ω.

(2.5)

Given g∈L^r(I;W^−1,s(Ω)) and yS0∈Bs,r(Ω) with 1< r, s <∞, analogously to Definition2.2, we say that yS ∈Wr,s(0, T) is a variational solution of (2.5) if for everyψ∈Ws⁰(Ω)

( hd

dty_S(t),ψi(W_s0(Ω))⁰,W_s0(Ω)+a(y_S(t),ψ) =hg(t),ψi_W−1,s(Ω),W^1,s₀ ⁰(Ω) in (0, T), y_S(0) =y_S0.

(2.6)

A distributionpS in Qis called an associated pressure if the equation

∂yS

∂t −ν∆y_S+∇p_S =g in Q is satisfied in the distribution sense.

Theorem 2.5. Assume thatΩ⊂R^d withd≥2. Giveng∈L^r(I;W^−1,s(Ω))andyS0∈Bs,r(Ω)with1< r, s <

∞, there exists a unique solution (yS,pS)∈Wr,s(0, T)×W^−1,r(I;L^s(Ω)/R)of (2.5). Moreover, there exists a constant Cr,s such that

ky_Sk_Wr,s(0,T)≤C_r,s

kgk_Lr(I;W_s0(Ω)⁰)+ky_S0k_Bs,r(Ω)

. (2.7)

As mentioned at the end of section 1, the embedding Wr,s(0, T)⊂C([0, T];Bs,r(Ω)) holds. Moreover, the trace mappingy∈Wr,s(0, T)→y(0)∈Bs,r(Ω) is continuous and surjective. This motivates our choice for the initial conditionyS0∈Bs,r(Ω).

Though Theorem2.5is expected to hold by experts, it seems that there is no proof available in the literature.

For this reason it is given in the next section. There, in Remark3.1, we shall also assert that Theorem2.5holds for domains which are only Lipschitz, provided that conditions on the ranges ofrandsare met.

As a consequence of Theorem2.5we get the following corollary.

Corollary 2.6. Let assume thatp∈(2,∞)and thatΩ⊂R². Then, given(f,y0)∈L²(I;W^−1,p(Ω))×Bp,2(Ω), the system (2.1)has a unique solution(y,p)∈W(0, T)∩Wp⁰,p(0, T)×W^−1,p0(I;L²(Ω)/R). Furthermore, there exist two constants M2>0 andMp>0 such that

kyk_W

p0,p(0,T)≤M₂

kfk_L2(I;V⁰)+ky₀k_H2

+M_p kfk_Lp0

(I;W_p0(Ω)⁰)+ky₀k_Bp,2(Ω)

. (2.8)

Proof. From our assumptions on p, we have that f ∈L²(I;W^−1,p(Ω))⊂L²(I;H⁻¹(Ω)) and y0 ∈Bp,2(Ω) ⊂ B2,2(Ω) =H. Hence, it is well known that (2.1) has a unique solutiony∈W(0, T). Let us prove that (y· ∇)y∈

L^p0(I;W^−1,p(Ω)). Given an arbitrary element ψ∈W^1,p₀ ⁰(Ω) and using (A.1) withr= 2pwe infer

|h(y(t)· ∇)y(t),ψi

W^−1,p(Ω),W^1,p₀ ⁰(Ω)|=|b(y(t),ψ,y(t))|

≤ ky(t)k²_L2p(Ω)kψk_W1,p0

0 (Ω)≤C2pky(t)k

2 p

L²(Ω)ky(t)k

2 p0

H¹₀(Ω)kψk_W1,p0 0 (Ω)

≤C2pkyk

2 p

L^∞(I;L²(Ω))ky(t)k

2 p0

H¹₀(Ω)kψk_W1,p0 0 (Ω). Using this estimate and Young’s inequality we deduce

k(y· ∇)yk_Lp0

(I;W^−1,p(Ω))≤C2pkyk

2 p

L^∞(I;L²(Ω))kyk

2 p0

L²(I;H¹₀(Ω))≤C⁰

kfkL²(I;V⁰)+ky0kH

² ,

where we have used the standard estimates for the solution of (2.1)y∈L²(I;H¹₀(Ω))∩L^∞(I;H); seee.g.([5], Thm. V.1.4) or ([28], (3.135)).

Sincef ∈L^p0(I;W^−1,p(Ω)), we deduce from Theorem2.5withg=f−(y· ∇)y thaty∈Wp⁰,p(0, T) and kyk_W

p0,p(0,T)≤C_p⁰_,p kgk_Lp0

(I;W_p0(Ω)⁰)+ky₀k_Bp,2(Ω)

≤C_p⁰_,ph

kfk_Lp0(I;W_p0(Ω)⁰)+C⁰

kfk_L2(I;V⁰)+ky₀k_H2

+ky₀k_Bp,2(Ω)i ,

which implies (2.8).

Proposition 2.7. Suppose that (2.2)andΩ⊂R² hold. Given(g,yN0)∈L²(I;H⁻¹(Ω))×H,e1,e2∈Y, and ν0≥0, then the system







∂y_N

∂t −ν∆y_N+ν₀(y_N · ∇)yN + (e₁· ∇)yN + (y_N · ∇)e2+∇pN =g in Q, divyN = 0 in Q, yN = 0 onΣ, yN(0) =yN0 in Ω

(2.9)

has a unique solution (yN,pN)∈W(0, T)×W^−1,∞(I;L²(Ω)/R). Furthermore, there exists a nondecreasing function ηN : [0,∞)−→(0,∞)such that

ky_Nk_L2(I;H¹₀(Ω))+kyk_L∞(I;L²(Ω))≤η_N

ke₂k_Y

kgk_L2(I;V⁰)+ky_N0k_L2(Ω)

, ky_Nk_W(0,T)≤ν₀η_N²(ke₂k_Y)

kgk_L2(I;V⁰)+ky_N0k_L2(Ω)

2

+[(1 +ν+ke1kY+ke2kY)ηN(ke2kY) + 1]

kgk_L2(I;V⁰)+kyN0k_L2(Ω)

.

(2.10)

Similarly to the previous cases, we say thatyN ∈W(0, T) is a variational solution of (2.9) if













 d

dt(y_N(t),ψ)_L2(Ω)+a(y_N(t),ψ) +ν₀b(y_N(t),y_N(t),ψ) +b(e1(t),yN(t),ψ) +b(yN(t),e2(t),ψ)

=hg(t),ψi_H−1(Ω),H¹₀(Ω) in (0, T), ∀ψ∈V, yN(0) =yN0.

(2.11)

Furthermore, a distributionpN inQis called an associated pressure if the equation

∂y_N

∂t −ν∆y_N+ν₀(y_N · ∇)yN+ (e₁· ∇)yN + (y_N · ∇)e2+∇pN =g inQ is satisfied in the distribution sense.

Theorem2.5 and Proposition2.7will be proved in Sections3and4, respectively.

Proof of Theorem 2.4. We are going to prove the existence of a solutiony=yN+yS withyN ∈W(0, T) and yS ∈Wq,p(0, T). To this end we writey0=yN0+yS0 withyN0∈H and yS0∈Bp,q(Ω). Using Theorem2.5 withr=qands=p, we define the functionyS ∈Wq,p(0, T) as the unique solution of the system







∂yS

∂t −ν∆yS+∇pS =f inQ,

divyS = 0 inQ, yS = 0 on Σ, yS(0) =yS0 in Ω.

(2.12)

Now, we takeyN as the solution of







∂y_N

∂t −ν∆yN + (yN · ∇)yN+ (yS· ∇)yN + (yN· ∇)yS+∇pN =−(yS· ∇)yS inQ, divy_N = 0 inQ, y_N = 0 on Σ, y_N(0) =y_N0in Ω

(2.13)

The existence and uniqueness of the solution yN ∈W(0, T) of the above system follows from Proposition2.7 by taking ν0 = 1, e1=e2 =yS ∈Y, and g(t) =−B(yS(t),yS(t)). As a consequence of Lemma 2.1 we have that g∈L²(I;H⁻¹(Ω)). Now, setting y=yN +yS and p=pN +pS, and adding equations (2.12) and (2.13) we obtain that y∈ Y, p∈W^−1,q(I;L^p(Ω)/R)), and (y,p) is a solution of (2.1). Moreover, (2.4) follows from (A.6) to estimateg, (2.7) and (2.10).

It remains to prove the uniqueness. Let y1,y2 ∈ Y and p1,p2∈W^−1,q(I;L^p(Ω)/R) such that (y1,p1) and (y2,p2) are two solutions of (2.1). We take (y,p) = (y2−y1,p2−p1). Subtracting the equations satisfied for both solutions we have







∂y

∂t −ν∆y+∇p=−(y1· ∇)y−(y· ∇)y2 in Q, divy= 0 inQ, y= 0 on Σ, y(0) = 0 in Ω.

(2.14)

The right hand side of the above equation can be written in the formg=−B(y1,y)−B(y,y2), which belongs to L²(I;H⁻¹(Ω)) by Lemma2.1. Let us prove that y∈W(0, T). First, we observe that due to the properties of p and q, in particular p <2 and q > 2, we have that V ⊂Wp(Ω) and Wp⁰(Ω) ⊂V. This implies that V⁰⊂Wp⁰(Ω)⁰ and, hence,W(0, T)⊂W2,p(0, T) andWq,p(0, T)⊂W2,p(0, T). Therefore,y∈ Y ⊂W2,p(0, T) holds. Moreover,W^1,p₀ ⁰(Ω)⊂H¹₀(Ω) yieldsH⁻¹(Ω)⊂W^−1,p(Ω) and, consequently,g∈L²(I;W^−1,p(Ω)). We also have that p ∈W^−1,2(I;L^p(Ω)/R). Now, from Theorem 2.5 we infer that (y,p) is the unique solution of (2.14) inW_2,p(0, T)×W^−1,2(I;L^p(Ω)/R).

On the other hand, (2.14) can be considered as a Stokes system with the right hand side belonging to L²(I;H⁻¹(Ω)). Hence, it is well known that there exists a unique element (ˆy,ˆp)∈W(0, T)×W^−1,∞(I;L²(Ω)/R) solution of (2.14). Using again that W(0, T)×W^−1,∞(I;L²(Ω)/R) ⊂W2,p(0, T)×W^−1,2(I;L^p(Ω)/R), we deduce from Theorem 2.5 that (ˆy,ˆp) = (y,p). Thus, we have y∈ W(0, T). Therefore, we can multiply the equation (2.14) byy and after integration by parts it yields

1 2

d

dtky(t)k²_L2(Ω)+νky(t)k²_H1

0(Ω)=−b(y1,y,y)−b(y,y₂,y)

=−b(y,y2,y)≤ ky2k_H¹

0(Ω)kyk²_L4(Ω)≤Cky2k_H¹

0(Ω)kykL²(Ω)kyk_H¹

0(Ω)

≤ν 2kyk²_H1

0(Ω)+C² 2νky2k²_H1

0(Ω)kyk²_L2(Ω). From this inequality we deduce that

d

dtky(t)k²_L2(Ω)≤ C² ν ky2k²_H1

0(Ω)kyk²_L2(Ω).

Since y(0) = 0, we infer from Gronwall’s inequality that y = 0, and with (2.14) p is the zero element of W^−1,∞(I;L^p(Ω)/R).

Remark 2.8. Let us observe that Y ⊂L⁴(I;L⁴(Ω)). Indeed, given y∈ Y, we can write it in the form y = yN +yS with yN ∈W(0, T) and yS ∈Wq,p(0, T). Using a Gagliardo inequality we obtain for almost every t∈(0, T)

kyN(t)k⁴_L4(Ω)≤C4kyN(t)k²_L2(Ω)kyN(t)k²_H1

0(Ω)≤C4kyNk²_L∞(I;L²(Ω))kyN(t)k²_H1 0(Ω).

The embeddingsW(0, T)⊂L²(I;H¹₀(Ω)) andW(0, T)⊂L^∞(I;L²(Ω)) and the above inequality implyyN ∈ L⁴(I;L⁴(Ω)). On the other hand, sinceWq,p(0, T)⊂L^q(I;Wp(Ω))⊂L^q(I;L⁴(Ω))⊂L⁴(I;L⁴(Ω)), recall (2.2), we infer thatyS ∈L⁴(I;L⁴(Ω)).

The solution of (1.1) enjoys a better regularity than the one established in the previous remark forq≥8 and under additional assumption ony0.

Theorem 2.9. Let us assume that q ≥8 and y0 =yN0+yS0 ∈B2,4(Ω) +Bp,q(Ω). Then the variational solution y of (1.1) belongs to L^q(I;L⁴(Ω)) and depends continuously in this topology on f and y0. Moreover, the estimate

kyk_Lq(I;L⁴(Ω))≤ηq

kfk_Lq(I;W^−1,p(Ω))+kyS0k_Bp,q(Ω)+kyN0k_B2,4(Ω)

(2.15) holds for an increasing monotone functionηq: [0,∞)−→[0,∞)independent of f andy0, withηq(0) = 0.

Proof. As in the proof of Theorem 2.4, we decompose the equation (1.1) in two systems, namely (2.12) and (2.13). The solution yS of (2.12) belongs toWq,p(0, T)⊂L^q(I;L⁴(Ω)) due to the assumption (2.2) onp. We prove thaty_N ∈C([0, T];L⁴(Ω)) ifq= 8. To this end we follow a fixed point approach. Givenz∈L⁸(I;L⁴(Ω)) we consider the equation







∂y

∂t −ν∆y+∇p=gz inQ,

divy= 0 inQ, y= 0 on Σ, y(0) =yN0 in Ω,

(2.16)

where

gz=−(yS· ∇)yS−(z· ∇)z−(yS· ∇)z−(z· ∇)yS. (2.17) It is immediate to check that

kgzkL⁴(I;H⁻¹(Ω))≤(kySkL⁸(I;L⁴(Ω))+kzkL⁸(I;L⁴(Ω)))². (2.18) Then, Theorem 2.5implies that the solutionyz of (2.16) belongs toW4,2(0, T) and satisfies

ky_zk_W4,2(0,T)≤C_4,2

kg_zk_L4(I;H⁻¹(Ω))+ky_N0k_B2,4(Ω)

. (2.19)

We apply Theorem 3 of [2] with

1

8 < s < 1

4 and θ=3 4

to deduce that W4,2(0, T)⊂L^ρ(I;H¹²(Ω))⊂L^ρ(I;L⁴(Ω)) with ρ= _1−4s⁴ . The choice of s implies that ρ >

8. Moreover, the first embedding is compact. Using ([1], Thm. III-4.10.2) we also have that W4,2(0, T) ⊂ C([0, T]; (H⁻¹(Ω),H¹₀(Ω))3

4,4). The embeddingsH¹₀(Ω)⊂W^12,4(Ω) andH⁻¹(Ω)⊂W^−1,4(Ω) imply that (H⁻¹(Ω),H¹₀(Ω))3

4,4⊂(W^−1,4(Ω),W^12,4(Ω))3

4,4=W^18,4(Ω)⊂L⁴(Ω).

Hence, we haveW4,2(0, T)⊂C([0, T];W^18,4(Ω))⊂C([0, T];L⁴(Ω)).

This embedding, (2.19) and H¨older’s inequality imply for 0<¯t≤T kyzk_L8(0,¯t;L⁴(Ω))≤¯t^1/8kyzk_C([0,¯t];L⁴(Ω))≤C1¯t^1/8

kgzk_L4(0,¯t;H⁻¹(Ω))+kyN0k_B2,4(Ω)

≤C1¯t^1/8h

(kySkL⁸(I;L⁴(Ω))+kzk_L8(0,t;L¯ ⁴(Ω)))²+kyN0kB_2,4(Ω)

i .

Let us takeR >0 andzany element of the closed ball ¯BR(0) ofL⁸(0,¯t;L⁴(Ω)). Then for some 0<t¯≤T small enough we obtain from the above inequality

kyzk_Lq(0,¯t;L⁴(Ω))≤C1¯t^1/8h

(kySk_L8(I;L⁴(Ω))+R)²+kyN0k_B2,4(Ω)i

≤R.

Hence, we have a compact mappingz∈B¯R(0)→yz∈B¯R(0). From Schauder’s fixed point theorem we infer the existence of a fixed point. Since the solution of (1.1) is unique, this fixed point must beyN and, consequently, yN belongs to L⁸(0,¯t;L⁴(Ω)). From (2.16) with gz replaced by gy we infer that y∈W4,2(0,¯t) and satisfies (2.19). Therefore, there exists a maximal timeT^∗ ≤T and a solution in the space W4,2. We know there are two possibilities: either T^∗=T and the theorem is proved, orT^∗< T and

¯lim

t→T^∗kyNk_W4,2(0,¯t)=∞ and kyNk_W4,2(0,¯t)<∞ ∀t < T¯ ^∗.

Let us prove that the second option can not occur. Givenε >0, we know from Remark2.8that there exists t_ε>0 close enough toT^∗ so that

kyNk_L4(tε,T^∗;L⁴(Ω))< ε.

From (2.19) and (2.18) with zreplaced by y, and continuous embeddingW_4,2(t_ε,¯t)⊂C([t_ε,¯t];L⁴(Ω)) we get for every ¯t∈(t_ε, T^∗) and anyε >0

ky_Nk_W4,2(tε,t)¯ ≤C_4,2

(ky_Sk_L8(t_ε,¯t;L⁴(Ω))+ky_Nk_L8(t_ε,t;L¯ ⁴(Ω)))²+ky_N(t_ε)k_B2,4(Ω)

≤C4,2

(kySk_L8(0,T;L⁴(Ω))+kyNk^1/2_C([t

ε,¯t];L⁴(Ω))kyNk^1/2_L4(t_ε,¯t;L⁴(Ω)))²+kyN(tε)k_B2,4(Ω)

≤C4,2

(kySk_L8(0,T;L⁴(Ω))+ε^1/2kyNk^1/2_C([t

ε,¯t];L⁴(Ω)))²+kyN(tε)k_B2,4(Ω)

≤C4,2

(kySk_L8(0,T;L⁴(Ω))+C1ε^1/2kyNk^1/2_W

4,2(t_ε,t)¯)²+kyN(tε)k_B2,4(Ω) .

Selecting ε= [4C4,2C1]⁻¹, we deduce from the above inequality kyNk_W4,2(tε,¯t)≤2C4,2

2kySk²_L8(0,T;L⁴(Ω))+kyN(tε)k_B2,4(Ω)

∀t¯∈(tε, T^∗), which proves that the explosion is not possible.

To prove estimate (2.15) we proceed as above to obtain kyNkW_4,2(0,T)

≤C_4,2

(ky_Sk_L8(0,T;L⁴(Ω))+ky_Nk^1/2_C([0,T];L4(Ω))ky_Nk^1/2_L4(0,T;L⁴(Ω)))²+ky_N0k_B2,4(Ω)

≤C4,2

2kySk²_L8(0,T;L⁴(Ω))+ 2kyNk_C([0,T];L4(Ω))kyNk_L4(0,T;L⁴(Ω))+kyN0k_B2,4(Ω)

Applying Corollary 4 of [26] withX=W^18,4(Ω),B=L⁴(Ω),Y =H⁻¹(Ω), andr= 2 we infer the compactness of the embedding W4,2(0, T)⊂C([0, T],L⁴(Ω)). Here we used that W4,2(0, T)⊂C([0, T],L⁴(Ω)), which was established above. Then, using Lions lemma with

ε= 1

4C4,2kyNkL⁴(0,T;L⁴(Ω))

we deduce from the above inequality kyNk_W4,2(0,T)≤2C4,2

kySk_L8(0,T;L⁴(Ω))+ 2CεkyNk_L4(0,T;L⁴(Ω))kyNk_L2(0,T;H⁻¹(Ω))+kyN0k_B2,4(Ω) . Estimate (2.15) follows from (2.7), the above inequality, and the estimate obtained in Remark 2.8 for kyNk_L4(0,T;L⁴(Ω)). Finally, the continuous dependence of y with respect to f and y0 can be proved using Theorem 2.5and (2.15).

For the above proof theL^∞(I;L⁴(Ω)) regularity of the solution of (2.16) is crucial. This is obtained ifq≥8.

3. Proof of Theorem 2.5

We separate the proof in several steps.

(i) Notation and preliminaries.LetL^s_σ(Ω) denote the closure inL^s(Ω) of the space{φ∈C^∞₀ (Ω) : divφ= 0}.

Givenf ∈L^s(Ω) we define the Helmholtz projectionPs:L^s(Ω)→L^s(Ω) byPsf =f− ∇H, where ∆H = divf in Ω, andn·(∇H−f) =0withnequal to the unit outer normal vector to Γ. We have that range(Ps) =L^s_σ(Ω), P_s²=Ps, andP_s⁰=Ps⁰, for the dual operator. The Stokes operator inL^s_σ(Ω) is defined by

As=−νPs∆ :D(As) =Ws(Ω)∩W^2,s(Ω)→L^s_σ(Ω). (3.1) It is a closed bijective operator when considered on the dense domain D(A_s)⊂L^s_σ(Ω). This operator enjoys maximal parabolic regularity [14, 16], ([17], page 147). More precisely, for every (˜g,y˜0) ∈ L^r(I;L^s_σ(Ω))× (L^s_σ(Ω),Ws(Ω)∩W^2,s(Ω))₁₋1

r,r the equation







∂y˜

∂t(t) +Asy(t) = ˜˜ g(t) for a.a.t∈(0, T), y(0) = ˜y₀,

(3.2)

has a unique solution ˜y∈L^r(I;W^2,s(Ω)∩Ws(Ω))∩W^1,r(I;L^s_σ(Ω)). Moreover, the inequality k∂y˜

∂tk_Lr(I;L^s(Ω))+k˜yk_Lr(I;W^2,s(Ω))≤C

k˜gk_Lr(I;L^s(Ω))+ky˜0k

(3.3) holds for some C independent of (˜g,y˜0). Above the norm of ˜y0 is taken in the interpolation space (L^s_σ(Ω),Ws(Ω)∩W^2,s(Ω))₁₋1

r,r.

The fractional power As¹² :D(As¹²)⊂L^s_σ(Ω)→L^s_σ(Ω), is well-defined with D(As¹²) dense in L^s_σ(Ω) and the identity As=A

1

s2A

1

s2 holds. A

1

s2 is an isomorphism when D(As¹²) is endowed with the graph norm of A

1

s2. The graph normkAs¹²yk_Ls(Ω)is equivalent to the normkyk_W1,s

0 (Ω)onD(As¹²). Moreover, the normskAsyk_Ls(Ω)and kykW^2,s(Ω) are equivalent onD(As), seee.g.[10]. We shall use in an essential manner that

D(As¹²) =D((−∆)¹²)∩L^s_σ(Ω) =W_s(Ω), (3.4) and −∆ is understood with homogenous boundary conditions. This was verified in [15], for domains with a

’smooth’ boundary. Using the classical result in [11] on the characterisation ofD((−∆¹²)), we obtain thatD(As¹²) is isomorphic toWs(Ω). The case of aC³boundary can be argued as follows. First we use that [L^s_σ(Ω),D(As)]1

2 = [L^s_σ(Ω),W_s(Ω)∩W^2,s(Ω)]1

2 =D(As¹²), where [·,·]¹

2 denotes complex interpolation. The second equality follows from the fact that the Stokes operator on L^s_σ(Ω) admits an H^∞ calculus [23], see also ([17], pg. 149). Next we note that the Helmhotz projection satisfies P_s=P_sP_s and that the range ofP_s is given by L^s_σ(Ω). Hence L^s_σ(Ω) is a complemented subspace ofL^s(Ω), see ([31], pg. 22). Using standard regularity results for the Stokes equation [3] it follows from the definition of P_s, that it is a bounded linear operator fromL^s(Ω) toL^s_σ(Ω) and from W^2,s(Ω) toW_s(Ω)∩W^2,s(Ω). As a consequence we obtain that [L^s_σ(Ω),W_s(Ω)∩W^2,s(Ω)]1

2 =W_s(Ω), by a general result on interpolation couples involving subspaces, see ([31], pg.118).

(ii) Extending the operator As. In the following we use arguments inspired by §6 of [9] and §11 of [4] where the case of second order elliptic operators is considered. First, note that we can utilize the above arguments for sreplaced by its conjugates⁰. HenceA

1 2

s⁰ :Ws⁰(Ω)→L^s_σ⁰(Ω) is a topological isomorphism, and its adjoint (A_s¹²0)⁰:L^s_σ(Ω)→W_s⁰(Ω)⁰ is an isomorphism too. We also have

(A_s¹²0)⁰y=As¹²y for ally∈ D(As¹²) =W_s(Ω), (3.5) where equality holds inW_s⁰(Ω)⁰. Indeed, for ally∈W_s(Ω) and z∈W_s⁰(Ω) we get

h(A_s¹²0)⁰y,zi_W

s0(Ω)⁰,W_s0(Ω)=hy, A_s¹²0zi_Ls

σ(Ω),L^s_σ⁰(Ω)=hAs¹²y,zi_Ls

σ(Ω),L^s_σ⁰(Ω)=hAs¹²y,zi_W

s0(Ω)⁰,W_s0(Ω), where the identityhy, A_s¹²0zi_Ls

σ(Ω),L^s_σ⁰(Ω)=hAs¹²y,zi_Ls

σ(Ω),L^s_σ⁰(Ω)for ally∈ D(As¹²) =Ws(Ω) and allz∈ D(A_s¹²0) = Ws⁰(Ω) is well known, seee.g.equality (1.23) of [12]. Thus (3.5) holds. SinceWs(Ω) is dense inL^s_σ(Ω) we obtain that (A

1 2

s⁰)⁰ is the extension of A

1

s2 from Ws(Ω) toL^s_σ(Ω). In a similar way we can argue that ((A

1 2

s⁰)⁰)⁻¹ is an extension of (As¹²)⁻¹ fromL^s_σ(Ω) toWs⁰(Ω)⁰.

Now, we define the operator As∈ L(Ws(Ω),Ws⁰(Ω)⁰) by hAsy,zi_W

s0(Ω)⁰,W_s0(Ω)=ν Z

Ω

∇y:∇zdx.

Let us study some properties of this operator. First, we observe that from the definitions ofAsandAsit follows that Asy=Asyfor ally∈ D(As). Hence,As is an extension ofAs.