Control problems for convection-diffusion equations with control localized on manifolds

URL:http://www.emath.fr/cocv/

CONTROL PROBLEMS FOR CONVECTION-DIFFUSION EQUATIONS WITH CONTROL LOCALIZED ON MANIFOLDS

Phuong Anh Nguyen

and Jean-Pierre Raymond

Abstract. We consider optimal control problems for convection-diffusion equations with a pointwise control or a control localized on a smooth manifold. We prove optimality conditions for the control variable and for the position of the control. We do not suppose that the coefficient of the convection term is regular or bounded, we only suppose that it has the regularity of strong solutions of the Navier–Stokes equations. We consider functionals with an observation on the gradient of the state.

To obtain optimality conditions we have to prove that the trace of the adjoint state on the control manifold belongs to the dual of the control space. To study the state equation, which is an equation with measures as data, and the adjoint equation, which involves the divergence ofL^p-vector fields, we first study equations without convection term, and we next use a fixed point method to deal with the complete equations.

Mathematics Subject Classification. 49K20, 49J20, 35K57.

Received May 31, 2000. Revised February 15, 2001.

1. Introduction

We are interested in the following optimal control problem:

Can we control the temperature distribution of a fluid in a three dimensional domain by heating sources localized on a network of wires?

More generally we are interested in heating sources (the control variables) concentrated on thin structures.

For simplicity we consider the case of controls localized on a manifoldγ included in aN-dimensional bounded domain Ω, but the case of a finite union of manifolds (a network of wires) can be considered as well. This optimal control problem clearly refers to a system of equations where the temperature and the fluid velocity are coupled. Such problems have been studied in the case of distributed or boundary controls (see for examples the references in [19]). The case of controls localized on thin structures, which is interesting for technological applications, has not yet been studied in the literature. As it is shown in [20], a fundamental step to tackle the complete Boussinesq system with controls localized on thin structures, first consists in studying a problem in which the fluid velocity is known. Ifydenotes the fluid temperature andV~ the fluid velocity, the temperaturey

Keywords and phrases:Pointwise control, optimal control, convection-diffusion equation, control localized on manifolds.

1 Universit´e Paul Sabatier, UMR CNRS MIP, UFR MIG, 31062 Toulouse Cedex 4, France; e-mail: [email protected] and[email protected]

c EDP Sciences, SMAI 2001

is the solution to the following convection-diffusion equation:

∂y

∂t +Ay+V~ · ∇y=uδγ|^Q in Q, ∂y

∂nA

=uδγ|^Σon Σ, y(0) =y0 in Ω, (1) where Ω is a bounded domain inR^N with a regular boundary Γ, N ≥2,Q= Ω×]0, T[,T >0 is given fixed, Σ = Γ×]0, T[,Ais a second order elliptic operator of the formAy=−PN

i,j=1Di(aij(x)Djy) +a0(x)y,γ⊂Ω is a regular manifold of dimension 0≤D≤N−2,δγ denotes the Dirac distribution onγ, anduis a function fromγ×]0, T[ with values inR.

Convection-diffusion equations are often refered to flow related models, and a computational approach in the case of Neumann boundary control is carried out in [5]. In [4, 8], the case of pointwise controls (which is a particular type of thin structure) is considered for the one-dimensional Burgers’ equation.

In order to next study the complete Boussinesq system we must suppose that V~ is not too regular. In control problems for convection-diffusion equation studied in the literature, it is often supposed that V~ is bounded [5,11,12]. However, it is not reasonable to suppose that the solution (y, ~V) to the Boussinesq system is such thatV~ belongs toL^∞(0, T; (L^∞(Ω))^N). Here we only suppose thatV~ belongs toL^m˜(0, T; (L^m(Ω))^N), for some ˜m >2,m >2 satisfying _m¹_˜ +_2m^N ≤¹2 (forN= 2 orN = 3, the limit cases _m¹_˜ +_2m^N =¹₂, with ˜m=m= 4 ifN = 2, and ˜m= 8 andm= 4 ifN = 3, correspond to the regularity of strong solutions of the Navier–Stokes equations).

These assumptions are sufficient to next study problems where the heat equation is coupled with the Navier–

Stokes equations in the two following cases [20]:

• the Boussinesq system linearized at (z, ~U), where (z, ~U) has the same regularity as strong solutions of the Boussinesq system;

• the two-dimensional nonlinear Boussinesq system.

As it is shown in [13], even in the case when V~ ≡0, studying equation (1) is not completely obvious. (In [13], the domain is supposed to be a 3-dimensional cylindrical domain, and taking advantage of the particular form of the domain, the equation is split into a 2-dimensional elliptic equation with measures as data, and a heat equation with regular source terms.)

Here we shall use new regularity results for parabolic equations with measures as data obtained in [21], where we have studied optimal control problems with controls localized on thin structures for semilinear parabolic equations.

In the present paper, we first study the control problem

(P1) inf{I(y, u)|(y, u)∈L^κ(0, T;W^1,κ(Ω))×KU, (y, u) satisfies (1)}, where

I(y, u) =CQ

Z

Q|∇y−Vd|^κdxdt+CΩ

Z

Ω|y(T)−yd|^θdx+Cγ

Z T 0

Z

γ|u|^σdζ ^q_σ

dt.

(CQ, CΩ, Cγ are nonnegative constants.)

Next, we consider the case whenγ is a pointx0 (that isδγ =δx0. A finite union of points can be considered as well). In this case we are interested in characterizing the best locationx0 which minimizes the distance to an observed profile of temperature. The problem is formulated as follows:

(P2) inf{J(yu,x₀, u, x0)|(yu,x₀, u, x0)∈L¹(0, T;W^1,1(Ω))×KU ×K_Ω, (y, u, x0) satisfies (1)}, where

J(yu,x0, u) = Z

Ω|yu,x0(T)−yd|^θdx, and K_Ω⊂Ω.

This problem can be related to the identification of sources of pollution (see [16]).

Also mention that the techniques we have developed for parabolic equations with measures as data can be adapted to study the corresponding stationnary elliptic equations with measures as data. In both cases (elliptic and parabolic) these new results can be useful to tackle optimal control problems with point observations. In this case the Dirac measures are involved in the adjoint equations [9]. For a review on control problems with pointwise controls we refer to [17] (see also [3]).

Let us briefly present the difficulties encountered in studying (P1) and (P2). Equation (1) is an equation with measures as data which may be studied by the transposition method [6]. However the regularity results in the literature [6, 18] are not sufficient to deal with the control problems (P1) and (P2).

To obtain optimal regularity results for a convection-diffusion equation of the form (1), or for the adjoint equation associated with (P1), throughout the paper the idea consists in studying equations firstly whenV~ ≡0, and next, by using a fixed point method, extending theses results to the general case. The fixed point method is developed in details in the proof of Proposition 2.7, and is next used for different propositions in the paper.

We prove regularity results for the state equation in Section 2. The control problem (P1) is studied in Section 3.

The main difficulty to obtain optimality conditions for (P1) is to prove that the trace of the adjoint state on γ×]0, T[ belongs toL^q0(0, T;L^σ0(γ)). The adjoint equation for (P1) is of the form

−∂p

∂t +Ap−V~ · ∇p=−div~hin Q, ∂p

∂nA

=~h·~non Σ, p(T) =pT in Ω.

Still using the fixed point method described above, we study the minimal regularity required on ~h and pT

to have p|γ×]0,T[ ∈ L^q0(0, T;L^σ0(γ)) (Ths. 3.2, 3.3). Since in the adjoint equation~h is equal to κCQ|∇yu− Vd|^κ−2(∇yu−Vd), whereyu is the solution to (1) corresponding to the optimal controlu, and pT is equal to θCΩ|yu(T)−yd|^θ−2(yu(T)−yd), the conditions on~handpT to havep|γ×]0,T[∈L^q0(0, T;L^σ0(γ)), are satisfied under additional conditions on κand θ (these conditions are stated in assumptions (A9, A10)). Optimality conditions for (P1) are obtained in Theorem 3.5.

The control problem (P2) is studied in Section 4. To obtain optimality conditions for (P2), we prove that the adjoint state belongs toL¹(0, T;C^1,ν(Ω)) (Th. 4.2). Next, we are able to characterize the optimal location of a pointwise control (Th. 4.3).

Numerical experiments for the computation of optimal solutionsufor (P1), and optimal pairs (u, x0) for (P2) are reported in [20] (Chap. 6).

2. State equation

We make the following assumptions on the data.

(A1) The elliptic operatorAis defined byAy=−PN

i,j=1Di(aij(x)Djy) +a0(x)y. The coefficienta0is positive and belongs toC(Ω), the coefficientsaij belong toC^1,ν(Ω) with 0< ν ≤1,aij =aji, and they satisfy

XN i,j=1

aij(x)ξiξj≥m0|ξ|² for everyξ∈R^N and everyx∈Ω, withm0>0.

(A2) Γ is of class C^∞, and γ is a submanifold in Ω of dimension D ≤ N −2, of class C^¯k with ¯k = max 2,[^N_σ⁻₀^D] + 1

.

(A3) V~ belongs toL^m˜(0, T; (L^m(Ω))^N) and satisfies

divV~ = 0 inQ, V~ ·~n= 0 on Σ, 2<m <˜ ∞, 2≤N < m <∞, 1

˜ m + N

2m ≤1 2 , where “div” denotes the divergence operator with respect tox∈Ω.

(A4) ubelongs toL^q(0, T;L^σ(γ)) withq≥2, σ≥N^N−1. (A5) y0 belongs toL^ρ(Ω) withρ=_N−^ND

σ0−q0².

(A6) KU is a closed convex subset of L^q(0, T;L^σ(γ)).EitherKU is bounded inL^q(0, T;L^σ(γ)), orCγ >0.

(A7) The functionVd belongs toL^κ(0, T; (L^κ(Ω))^N), the functionydbelongs toL^θ(Ω), with κ >1,θ >1.

Remark. For simplicity we have supposed that Γ is of classC^∞, but the results of the paper can be extended to less regular domains by using the techniques of [7] (Prop. 5).

Throughout the paper, we denote byTγ (respectivelyTγ×]0,T[,TΣ) the trace mapping onγ(respectively on γ×]0, T[, on Σ). We denote byC, Ci, K, Ki fori∈N, various constants depending on known quantities. The same letter may be used for different constants.

In [21] we have studied equation (1) in the case whenV~ ≡0. By the transposition method, we have proven the following regularity results.

Proposition 2.1. [21] (Prop. 2.3) Suppose that V~ ≡0 and y0 ≡0. Equation (1) admits a unique solution yu in L¹(0, T;W^1,1(Ω)). The mapping u7→yu is continuous fromL^q(0, T;L^σ(γ))intoL^δ1(0, T;W^1,d1(Ω)) for every(δ1, d1) satisfying:

q≤δ1, σ≤d1< N N−D

σ⁰ −1

, N−D

2 + D

2σ +1 q < 1

δ1

+ N 2d1

+1

2, if σ < N−D N −D−1 , q≤δ1, 1< d1< N−D

N−D−1, N−D

2 +1

q < 1 δ1

+N−D 2d1

+1

2, ifσ≥ N−D N−D−1·

(2)

The mapping that associatesyuwithuis continuous fromL^q(0, T;L^σ(γ))intoL^∞(0, T;L^r(Ω))for every1≤r <

inf

N−D

N−D−_q²0,_N−^ND σ0−_q0²

. Moreover,yubelongs toC([0, T];L^r_w(Ω))for every1≤r <inf

N−D

N−D−_q²0,_N−^ND σ0−_q0²

. (C([0, T];L^rw(Ω)) denotes the space of continuous functions from [0, T] into L^r(Ω), endowed with its weak topology.)

Remark. The conditions expressed in (2) can be written in the following shorter form q≤δ1, σ≤d1, N−D

2 + D

2σ+1 q < 1

δ1

+ N 2d1

+1

2· (3)

Indeed if d1≥σ, taking δ1=qin (3), we obtain:

N−D

2 + D

2d1

< N−D

2 + D

2σ < N 2d1

+1 2·

Therefore we have σ ≤ d1 < _N^N₋⁻_D^D₋₁. This means that (3) cannot be used if σ ≥ N^N−⁻D^D−1. Now, if u ∈ L^q(0, T;L^σ(γ)) with σ ≥ N^N−⁻D^D−1, then u ∈ L^q(0, T;L^ˆσ(γ)) for any ˆσ < _N^N₋⁻_D^D₋₁. Therefore y belongs to L^δ1(0, T;W^1,d1(Ω)) for every (δ1, d1) satisfying:

q≤δ1, σˆ=d1< N−D

N−D−1 and N−D

2 + D

2ˆσ+1 q < 1

δ1

+ N 2d1

+1 2,

that is

N−D

2 +1

q < 1 δ1

+N−D 2d1

+1 2,

which is nothing else than the second condition in (2). Finally observe that the condition d1 < _N−^ND σ0−1

follows from (3) by takingδ1 =q. Even if (2) and (3) are equivalent, using (2) avoids forgetting the condition d1< _N^N₋⁻_D^D₋₁.

Proposition 2.2. [21] (Prop. 2.5) Suppose thatV~ ≡0andu≡0. Lety0 belong toL^ρ(Ω)with ρ= _N−^ND σ0−q0² , and lety be the solution to equation (1). The mappingy07→y is continuous fromL^ρ(Ω) intoL^∞(0, T;L^ρ(Ω))∩ L^δ2(0, T;W^1,d2(Ω)), for every(δ2, d2) satisfying

1< δ2<2, ρ≤d2< N ρ

N−ρ, 1 + N 2ρ < 1

δ2

+ N 2d2

+1

2· (4)

In this section we want to extend these results to the case whenV~ satisfies (A3) (Props. 2.7 and 2.8). Due to the weak regularity of V~, we cannot use the transposition method. We obtain existence and regularity results for equation (1) by using a fixed point method. For this, we need some preliminary estimates that are stated below.

2.2. Preliminary estimates

First recall some results for analytic semigroups. We denote by ˜A the operator defined by D( ˜A) =

y∈C²(Ω)| ∂y

∂nA

= 0 on Γ

, Ay˜ =Ay.

For 1 ≤ ` < ∞, we denote by A` the closure of ˜A in L^{`</sup>(Ω). The operator −A` is the generator of a strongly continuous analytic semigroupS`(t)t≥0 in L<sup>`}(Ω) [2]. For 1 < ` <∞ the domain of A` is D(A`) = n

y∈W^2,`(Ω)| ∂n^∂y_A = 0 on Γo

. For any 1≤` <∞, 0 belongs to the resolvent of −A` and there existsδ >0 such that Reσ(A`)≥δ(it is a consequence of (A1) and of the fact thatσ(A`) is independent of `). Therefore, forα >0, there exists a constantK=K(`, α) such that

kA^α<sub>`</sub>S`(t)ϕkL^{`</sup>(Ω)≤Kt<sup>−α</sup>kϕkL<sup>`}(Ω),

for everyt >0 and everyϕ∈L<sup>`</sup>(Ω) (see [14, 22],A<sup>α</sup><sub>`</sub> is theα-power ofA`). Thanks to this result the following lemma can be established.

Lemma 2.1. [2, 24]For every1≤`≤λ≤ ∞with` <∞, there exists a constant K1=K1(λ, `)such that kS`(t)ϕkL^λ(Ω)≤K1t^{−N2(1`−1λ)</sup>kϕkL<sup>`}(Ω) (5) for everyϕ∈L<sup>`</sup>(Ω) and everyt >0. For every 1≤`≤λ≤ ∞with ` <∞, and everyα > 0, there exists a constantK2=K2(λ, `, α) such that

kA^α<sub>`</sub>S`(t)ϕkL^λ(Ω)≤K2t^{−N2(1`−λ1)−α</sup>kϕkL<sup>`}(Ω) (6) for everyϕ∈L^`(Ω) and everyt >0.

Proposition 2.3. [1](Th. 7.58) If assumption (A2) is satisfied, thenTγ is a continuous linear operator from W^r,p(Ω)intoL^q(γ) for all(r, p, q)such that 0≤r≤k,¯ 0< N−rp < D, p≤q < _N^Dp_−rp.

Remark. Theorem 7.58 in [1] is stated with Ω and γ replaced by R^N and R^D, but as it is noticed in [1], just before Theorem 7.58, the statement is also true for domains by using coverings, partitions of unity, and diffeomorphisms of classC^k¯.

Proposition 2.4. [21](Prop. 2.1) Letφ be inD(Ω), and wbe the solution of the Cauchy problem:

∂w

∂t +Aw= 0 inQ, ∂w

∂nA

= 0on Σ, w(0) =A^α_d0φin Ω, (7) where0≤α≤1. The mapping that associates wwith φ is continuous from L^d0(Ω) intoLⁱ(0, T;W^r,j(Ω)) for all(α, q, j, i, r, d)satisfying:

2> r≥0, i≥1, j≥d⁰, α+r 2+ N

2d⁰ <1 i +N

2j · (8)

Proposition 2.5. Letf~be in(D(Q))^N, andz be the solution of the equation:

∂z

∂t +Az= divf~inQ, ∂z

∂nA

= 0onΣ, z(0) = 0 inΩ. (9)

The mapping that associates z with f~is continuous from L^˜η(0, T; (L^η(Ω))^N)intoL^η˜(0, T;W^1,η(Ω)). It is also continuous fromL^η˜(0, T; (L^η(Ω))^N)intoL^δ(0, T;W^r,d(Ω)) for all(r,η, η, δ, d)˜ satisfying:

0≤r <1, 1≤η˜≤δ, 1< η≤d, r 2+1

˜ η +N

2η <1 δ +N

2d+1

2· (10)

The mapping that associateszwithf~is continuous fromL^η˜(0, T; (L^η(Ω))^N)intoL^δ(0, T;L^d(Ω))for all(˜η, η, δ, d) satisfying:

1<η < δ,˜ 1< η≤d, 1

˜ η+ N

2η ≤1 δ +N

2d+1

2· (11)

Remark. Sincef~belongs to (D(Q))^N, equation (9) is defined in a classical sense.

Proof. The first continuity results are already proved in [21] and [27]. Here we only prove the second one. Let wbe the solution of the Cauchy problem (7) whenα= 0, then:

Z

Ω

z(x, t)φ(x) dx= Z t

0

d dτ

Z

Ω

w(x, t−τ)z(x, τ) dx

dτ

= Z t

0

Z

Ω

−∂w

∂t(x, t−τ)z(x, τ) +w(x, t−τ)∂z

∂t(x, τ)

dxdτ

= Z t

0

Z

Ω

n

Aw(x, t−τ)z(x, τ)−w(x, t−τ)Az(x, τ) + divf~(x, τ)w(x, t−τ)o dxdτ

=−Z t 0

Z

Ω∇w(x, t−τ)·f~(x, τ) dxdτ.

Using (6) in Lemma 2.1 with 1≤d⁰≤η⁰≤ ∞, d⁰ <∞,we have:

kz(t)kL^d(Ω)= sup Z

Ω

z(x, t)φ(x) dx

,kφkL^d0(Ω)= 1

= sup Z t

0

Z

Ω

∇w(x, t−τ)·f~(x, τ)dxdτ

,kφkL^d0(Ω)= 1

≤K Z t

0

(t−τ)^{−12−N2(η1−1d)}kf~(τ)k^(Lη(Ω))Ndτ.

The mappingτ 7→ kf~(τ)k(L^η(Ω))^N belongs toL^η˜(0, T). We denote byLⁱ_∗(0, T) the weak-Lⁱ(0, T) space defined as follows ([23], p. 30):

Lⁱ_∗(0, T) :=

g: (0, T)7→R | g is measurable, and sup

ξ>0

(ξⁱL¹{t| |g(t)|> ξ})<∞

,

where L¹ denotes the Lebesgue measure on (0, T). Then the mapping t 7→ t^{−12−N2(1η−d1)} belongs to Lⁱ_∗(0, T) with i >1 defined by ¹₂+^N₂(¹_η −¹d) = ¹_i. Due to (11), 1 + ¹_δ ≥ ¹i +_η¹_˜.From the generalized Young inequality, it follows that the mapping t 7→ Rt

0(t−τ)^{−12−N2(η1−1d)}kf~(τ)k(L^η(Ω))^Ndτ belongs to L^δ(0, T), and the proof is

complete.

Remark. Since (D(Q))^N is dense inL^η˜(0, T; (L^η(Ω))^N), the regularity result of Proposition 2.5 is also true for the solution zto the variational equation

−Z

Q

z∂φ

∂t dxdt+ Z

Q

XN i,j=1

aijDjzDiφdxdt+ Z

Q

a0zφdxdt dxdt=−Z

Q

f~· ∇φdxdt ,

for allφ∈C¹(Q) such thatφ(T) = 0 on Ω, wheref~∈L^η˜(0, T; (L^η(Ω))^N).

Proposition 2.6. Letf be inD(Q), andz be the solution of the equation:

∂z

∂t +Az=f inQ, ∂z

∂nA

= 0on Σ, z(0) = 0in Ω. (12)

The mapping that associateszwithf is continuous fromL^η˜(0, T;L^η(Ω))intoL^δ(0, T;W^r,d(Ω))for all(r,η, η, δ, d)˜ satisfying:

0≤r <2, 1≤η˜≤δ, 1< η≤d, r 2+1

˜ η +N

2η <1 δ + N

2d+ 1. (13)

The mapping that associateszwithf is continuous fromL^η˜(0, T;L^η(Ω))intoL^δ(0, T;W^k,d(Ω))for all(k,η, η, δ, d)˜ satisfying:

k= 0 ork= 1, 1<η < δ,˜ 1< η≤d, k 2+1

˜ η + N

2η ≤ 1 δ+ N

2d+ 1. (14)

Proof. The first continuity result is already proved in [21]. Here we only prove the second one. Let us setα= ^k₂. We haveD(A⁰_d) =L^d(Ω) andD(A

1 2

d) =W^1,d(Ω). Let wbe the solution of the Cauchy problem (7), then:

Z

Ω

A^α_dz(x, t)φ(x) dx= Z t

0

Z

Ω

A^α_d0Sd⁰(t−τ)φ(x)f(x, τ)dxdτ.

Using (6) in Lemma 2.1 with 1≤d⁰≤η⁰≤ ∞, d⁰ <∞,we have:

kz(t)kW^k,d(Ω)≤CkA^αdz(t)kL^d(Ω)≤C sup Z

Ω

A^αdz(x, t)φ(x) dx

,kφkL^d0(Ω)= 1

=C sup Z t

0

Z

Ω

A^α_d0Sd0(t−τ)φf(τ) dxdτ

,kφkL^d0(Ω)= 1

≤C Z t

0

(t−τ)^{−α−N2(η1−1d)}kf(τ)kL^η(Ω)dτ.

The mappingτ 7→ kf(τ)k^Lη(Ω) belongs to L^η˜(0, T). The mappingt 7→t^{−α−N2(η1−1d)} belongs to Lⁱ_∗(0, T) with i > 1 defined by α+ ^N₂(¹_η −¹_d) = ¹_i. Due to (14), 1 +¹_δ ≥ ¹_i + ¹_˜η. From the generalized Young inequality, it follows that the mappingt7→Rt

0(t−τ)^{−α−N2(1η−1d)}kf(τ)kL^η(Ω)dτ belongs toL^δ(0, T).

2.3. State equation

In this section, we prove regularity results for equation (1). We shall say that a functiony∈L^δ(0, T;W^1,d(Ω)) is a weak solution to equation (1) if and only ifδ≥m˜⁰,d≥m⁰ and

− Z

Q

y∂φ

∂t dxdt+ Z

Q

XN i,j=1

aij(x)DjyDiφdxdt+ Z

Q

a0yφdxdt+ Z

Q

V~ · ∇yφdxdt

= Z T

0

Z

γ

uφdζ dt+ Z

Ω

φ(0)y0 dx for allφ∈C¹(Q) such thatφ(T) = 0 on Ω. (15) To simplify the writing, throughout the sequel, we suppose thatγis included in Γ, but the results are true for γ⊂Ω.

Proposition 2.7. We consider the equation

∂y

∂t +Ay+V~ · ∇y= 0 inQ, ∂y

∂nA

=uδγ onΣ, y(0) = 0in Ω. (16)

Equation (16) admits a unique solution y in L^δˆ(0, T;W^1,dˆ(Ω)) for all (ˆδ,d)ˆ obeying m˜⁰ < ˆδ ≤ q, m⁰ < d <ˆ infn

N

N−σ0^D−1,_N^N₋⁻_D^D₋₁o

.Moreover, the mapping that associatesywithuis continuous fromL^q(0, T;L^σ(γ))into L^δ(0, T; W^1,d(Ω)) for every(δ, d)satisfying:

q≤δ, σ≤d < N N−D

σ⁰ −1

, N−D

2 + D

2σ+1 q < 1

δ+ N 2d+1

2, if σ < N−D N−D−1 , q≤δ, m⁰< d < N−D

N−D−1, ^N−₂^D +1 q <1

δ +N−D 2d +1

2, ifσ≥ N−D N−D−1·

(17)

The mapping that associatesywithuis continuous fromL^q(0, T;L^σ(γ))intoL^∞(0, T;L^r(Ω))for every1≤r <

inf

N

N−_N−D^D −_q0² ,_N−^ND σ0−_q0²

. Moreover, y belongs to C([0, T];L^r_w(Ω)) for every 1 ≤ r <

inf

N

N−_N−D^D −q0² ,_N−^ND σ0−q0²

.

Proof. 1 - Existence of a local solution. Let us setQ¯t := Ω×]0,¯t[, Σ¯t := Γ×]0,¯t[. Let (δ, d) be a pair obeying (17). By a fixed point method, we prove that the equation

∂y

∂t +Ay+V~ · ∇y= 0 inQt¯, ∂y

∂nA

=uδγ on Σt¯, y(0) = 0 in Ω, (18) admits a solution for ¯t > 0 small enough. Let ξ belong to L^δ(0,t;¯W^1,d(Ω)), and yξ be the solution to the equation:

∂yξ

∂t +Ayξ =−V~ · ∇ξinQ¯t, ∂yξ

∂nA

=uδγ on Σt¯, yξ(0) = 0 in Ω. (19) Thenyξ = ˆy+ ˜y, where ˆy and ˜y are the solutions to the equations:

∂yˆ

∂t +Aˆy= 0 inQ¯t, ∂ˆy

∂nA

=uδγ on Σ¯t, y(0) = 0 in Ω,ˆ

∂˜y

∂t +Ay˜=−V~ · ∇ξ inQ¯t, ∂˜y

∂nA

= 0 on Σ¯t, y(0) = 0 in Ω.˜

From Proposition 2.1, we know that ˆy ∈L^δ(0,¯t;W^1,d(Ω)). Sinceδ ≥q ≥2 >m˜⁰, and d≥ σ≥ N^N−1 > m⁰, then _m+δ˜^mδ˜ > 1, _m+d^md > 1, and V~ · ∇ξ ∈ L^m+δ˜mδ˜ (0,¯t;L^m+dmd (Ω)). Due to Proposition 2.6, it follows that ˜y ∈ L^δ(0,¯t;W^1,d(Ω)). Thusyξ belongs toL^δ(0,¯t;W^1,d(Ω)).

Letξ1 andξ2 belong toL^δ(0,t;¯W^1,d(Ω)). Still with Proposition 2.6, we have

kyξ₁−yξ₂kL^δ(0,t;W¯ ^1,d(Ω))≤C1kV~kL^m˜(0,¯t;L^m(Ω)^N)kξ1−ξ2kL^δ(0,¯t;W^1,d(Ω)), whereC1can be chosen depending onT, but independent of ¯t.The mappingt7→C₁^m˜ Rt

0kV~(x, τ)k^mL^˜m(Ω)^Ndτ is absolutely continuous, then there exists ¯t > 0 such thatC1(Rmin{t+¯t,T}

t kV~(·, τ)k^mL^˜m(Ω)^Ndτ)^m1˜ ≤C = ¹₂ for all t∈[0, T]. Thus the mappingξ7→yξ is a contraction in the Banach spaceL^δ(0,¯t;W^1,d(Ω)).

2 - Estimate of the local solution. Consider the sequence (ξn)n defined byξ0 = 0 andξn =yξ_n−1. Then (ξn)n converges to the unique solution y of (18). From the definition of yξ₀, and due to Proposition 2.1, we deduce that

kyξ₀kL^δ(0,¯t;W^1,d(Ω)) ≤KkukL^q(0,¯t;L^σ(γ)). Moreover for allnwe have

kyξnkL^δ(0,¯t;W^1,d(Ω)) ≤2kyξ0kL^δ(0,¯t;W^1,d(Ω)) ≤Kkuk^{Lq(0,T;Lσ(γ))}. By lettingngoes to∞, we obtain:

kykL^δ(0,¯t;W^1,d(Ω)) ≤KkukL^q(0,T;L^σ(γ)). (20) 3 -Existence of a global solution. We prove that a solution exists inL^δ(0, T;W^1,d(Ω)), by repeating the above process. Let ˆy be the solution constructed on (0,t) in Step 2. Let ( ˆ¯ ξ1,ξˆ2) belong toL^δ(¯t,2¯t;W^1,d(Ω)).

Define (ξ1, ξ2) belonging toL^δ(0,2¯t;W^1,d(Ω)) byξ1=ξ2= ˆy on (0,¯t), andξ1= ˆξ1,ξ2= ˆξ2 on (¯t,2¯t). We still denote byyξ_i the solution to equation (19) on (0,2¯t) corresponding to ξi fori= 1,2. As in Step 2, we have:

kyξ₁−yξ₂kL^δ(0,2¯t;W^1,d(Ω))=kyξ₁−yξ₂kL^δ(¯t,2¯t;W^1,d(Ω))≤ 1

2kξ1−ξ2kL^δ(0,2¯t;W^1,d(Ω)).

Thus the mapping ξ 7→ yξ admits a unique fixed point in the metric space {π ∈ L^δ(0,2¯t;W^1,d(Ω)) | π = ˆ

yon ]0,¯t[}. We want to estimate the solution inL^δ(0,2¯t;W^1,d(Ω)).Let (ξn)n be the sequence defined by:

ξ0= ˆy on (0,¯t), ξ0= 0 on (¯t,2¯t), ξn=yξ_n−1 on (0,2¯t).

Then (ξn)n converges toy in L^δ(0,2¯t;W^1,d(Ω)). From the properties of the fixed point, we have kyξ_n−yξ₀kL^δ(0,2¯t;W^1,d(Ω)) ≤ kyξ₀−ξ0kL^δ(0,2¯t;W^1,d(Ω)).

Since yξ0 is the solution to the equation

∂y

∂t +Ay=−V~ · ∇ξ0 inQ2¯t, ∂y

∂nA

=uδγ on Σ2¯t, y(0) = 0 in Ω, then

kyξ0kL^δ(0,2¯t;W^1,d(Ω))≤K

1 +kV~kL^m˜(0,T;(L^m(Ω))^N)

kukL^q(0,T;L^σ(γ)). Therefore, we have:

kykL^δ(0,2¯t;W^1,d(Ω)) ≤K

1 +kV~kL^m˜(0,T;(L^m(Ω))^N)

kukL^q(0,T;L^σ(γ)). (21)

4 -Estimate of the global solution inL^δ(0, T;W^1,d(Ω)). By induction, it is easy to prove that kykL^δ(0,T;W^1,d(Ω))≤Kn

1 +kV~kL^m˜(0,T;(L^m(Ω))^N)+...+kV~kⁿL^−m˜1(0,T;(L^m(Ω))^N)

kukL^q(0,T;L^σ(γ)), (22)

where n= [^Tt¯] + 1, and whereKndepends onn(observe thatndepends on ¯t, and ¯tdepends onV~). Therefore, there exists a constant ˜C depending onV~ andT,such that

kykL^δ(0,T;W^1,d(Ω)) ≤C˜kukL^q(0,T;L^σ(γ)).

5 -Estimate inL^∞(0, T;L^r(Ω)). Observe thaty=y1+y2, wherey1andy2are the solutions to the equations:

∂y1

∂t +Ay1= 0 inQ, ∂y1

∂nA

=uδγ on Σ, y1(0) = 0 in Ω,

∂y2

∂t +Ay2=−V~ · ∇y in Q, ∂y2

∂nA

= 0 on Σ, y2(0) = 0 in Ω.

Due to Proposition 2.1, y1 belongs to L^∞(0, T;L^r(Ω)) for every 1 ≤ r < inf

N

N−N^D−D−q0² ,_N−^ND σ0−q0²

≤ inf

N−D

N−D−q0²,_N−^ND σ0−q0²

. Ifσ < _N^N₋⁻_D^D₋₁, then inf

N

N−_N−D^D −q0² ,_N−^ND σ0−q0²

=_N−^ND

σ0−q0² , andσ < _N−^ND

σ0−1. Letrsatisfyσ < _N−^ND σ0−1 <

r <_N−^ND

σ0−_q0². Observe that¹_r <_m¹+_σ¹, ¹_q+_2r^N−2m^N < ^N_2r+¹₂, and¹_q+¹₂(N−σ^D0−1) = ¹₂(N−σ^D0−q²⁰)+¹₂ <_2r^N+¹₂. Therefore, there existsdsatisfying

sup N

2r− N 2m, 1

2

N−D σ⁰ −1

< N 2d<inf

N 2σ, N

2r+1 2−1

q

·