Local exact controllability for the 2 and 3-d compressible Navier-Stokes equations
LLocal exact controllability for the and -d compressibleNavier-Stokes equations ∗ Sylvain Ervedoza † Olivier Glass ‡ Sergio Guerrero § November 13, 2018
Abstract
The goal of this article is to present a local exact controllability result for the and -dimensionalcompressible Navier-Stokes equations on a constant target trajectory when the controls act on thewhole boundary. Our study is then based on the observability of the adjoint system of some linearizedversion of the system, which is analyzed thanks to a subsystem for which the coupling terms aresomewhat weaker. In this step, we strongly use Carleman estimates in negative Sobolev spaces. We consider the isentropic compressible Navier-Stokes equation in dimension two or three in space, in asmooth bounded domain Ω ⊂ R d , d = 2 or d = 3 : (cid:26) ∂ t ρ S + div( ρ S u S ) = 0 in (0 , T ) × Ω ,ρ S ( ∂ t u S + u S · ∇ u S ) − µ ∆ u S − ( λ + µ ) ∇ div( u S ) + ∇ p ( ρ S ) = 0 in (0 , T ) × Ω . (1.1)Here ρ S is the density, u S the velocity and p is the pressure, which follows the standard polytropic law: p ( ρ S ) = κρ γ S , (1.2)for some γ ≥ and κ > . (Actually, our proof will only require p to be C locally around the targetdensity.)The parameters µ and λ correspond to constant viscosity parameters and are assumed to satisfy µ > and dλ + 2 µ ≥ (the only condition required for our result is µ > and λ + 2 µ > ).In this work, we intend to consider the local exact controllability around constant trajectories ( ρ, u ) ∈ R ∗ + × R d \{ } . Here, the controls do not appear explicitly in (1.1) as we are controlling the whole externalboundary (0 , T ) × ∂ Ω for the equation of the velocity and the incoming part u S · (cid:126)n < of the boundaryfor the equation of the density, (cid:126)n being the unit outward normal on ∂ Ω , see e.g. [24, Chapter 5].Given e a direction in the unit sphere S d − of R d , we define the thickness of some nonempty open set A ⊂ R d in the direction e as the following nonnegative number: sup { (cid:96) ≥ / ∃ x ∈ A, x + (cid:96)e ∈ A } . Our main result is the following.
Theorem 1.1.
Let d ∈ { , } , ρ > and u ∈ R d \ { } . Let L > be larger than the thickness of Ω inthe direction u/ | u | , and assume T > L / | u | . (1.3) ∗ The authors were partially supported by the ANR-project CISIFS 09-BLAN- 0213-03 and by the project ECOSEAfrom FNRAE. † Institut de Mathématiques de Toulouse ; UMR5219; Université de Toulouse ; CNRS ; UPS IMT, F-31062 ToulouseCedex 9, France, [email protected] ‡ Ceremade (UMR CNRS no. 7534), Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 ParisCedex 16, France, [email protected] § Université Pierre et Marie Curie - Paris 6, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005 France, [email protected] a r X i v : . [ m a t h . A P ] D ec here exists δ > such that for all ( ρ , u ) ∈ H (Ω) × H (Ω) satisfying (cid:107) ( ρ , u ) (cid:107) H (Ω) × H (Ω) ≤ δ, (1.4) there exists a solution ( ρ S , u S ) of (1.1) with initial data ρ S (0 , x ) = ρ + ρ ( x ) , u S (0 , x ) = u + u ( x ) in Ω , (1.5) and satisfying the control requirement ρ S ( T, x ) = ρ, u S ( T, x ) = u in Ω . (1.6) Besides, the controlled trajectory ( ρ S , u S ) has the following regularity: ( ρ S , u S ) ∈ C ([0 , T ]; H (Ω)) × ( L (0 , T ; H (Ω)) ∩ C ([0 , T ]; H (Ω))) . (1.7)Theorem 1.1 extends the results in [8] to the multi-dimensional case. As in the one dimensionalcase, our result proves local controllability to constant states having non-zero velocity. This restrictionappears explicitly in the condition (1.3). As expected, this condition is remanent from the transportequation satisfied by the density which allows the information to travel at a velocity (close to) u .This transport phenomenon and its consequences on the controllability of compressible Navier-Stokesequations have been also developed and explained in the articles [25, 5, 21] focusing on the linearizedequations in the case of zero-velocity. Using then moving controls, [22, 3] managed to show that control-lability for a system of linear viscoelasticity can be reestablished if the control set travels in the wholedomain (among some other geometric conditions, see [3] for further details). Let us also mention thework [4] where the compressible Navier-Stokes equations in d linearized around a constant state withnon-zero velocity are studied thoroughly using a spectral approach and suitable Ingham-type inequalities.In order to prove Theorem 1.1, we will deal with system (1.1)–(1.2) thinking to it as a couplingof parabolic and transport equations, and we shall therefore borrow some ideas from previous worksstudying controllability of systems coupling parabolic and hyperbolic effects, in particular the works [1]focusing on a system of linear thermoelasticity, [8] for the d compressible Navier-Stokes equation arounda constant state with non-zero velocity, [3] for a system of viscoelasticity with moving controls, or [2]for non-homogeneous incompressible Navier-Stokes equations. All these works are all based on suitableCarleman estimates designed simultaneously for the control of a parabolic equation following the ideasin [12] and for the control of the hyperbolic equation.Our approach will follow this path and use Carleman estimates with weight functions traveling atvelocity u similarly as in [8, 3, 2]. But we will also need to construct smooth trajectories in order toguarantee that the velocity field belongs to L (0 , T ; H (Ω)) . This space is natural as it is included in thespace L (0 , T ; Lip (Ω)) ensuring the existence and continuity of the flow. Therefore, in order to obtainvelocity fields in L (0 , T ; H (Ω)) , we will use duality and develop observability estimates in negativeSobolev spaces in the spirit of the work [18].We will not deal with the Cauchy problem for system (1.1)–(1.2), as our strategy directly constructsa solution of (1.1)–(1.2). We refer the interested reader to the pioneering works by P.-L. Lions [20] andE. Feireisl et al. in [10]. Nevertheless, we emphasize that our approach will use on the adjoint equationsa new variable which is similar to the so-called viscous effective flux introduced by P.-L. Lions in [20] inorder to gain compactness properties.Let us briefly mention other related works in the literature. In particular, we shall quote the works onthe controllability of compressible Euler equations, namely the ones obtained in [19] in the -dimensionalsetting in the context of classical C solutions, and the ones developed by the second author in the contextof weak entropy solutions obtained in [14] for isentropic -d Euler equations and [15] for non-isentropic dEuler equations. We also refer to the work [23] for a global approximate controllability result for the -dEuler equations with controls spanned by a finite number of modes. When considering incompressibleflows, the literature is large. We refer for instance to the works [16, 11, 17] for several results on the localexact controllability to trajectories for the (homogeneous) incompressible Navier-Stokes equations, andto the works [6, 13] for global exact controllability results for incompressible perfect fluids.2 utline. The article is organized as follows. Section 2 presents the general strategy of the proof ofTheorem 1.1. Section 3 shows the controllability of a suitable system of one parabolic and one transportequation. Section 4 deduces from it a controllability result for the linearized Navier-Stokes equations.Section 5 then explains how to perform a fixed point argument using the controllability results developedbeforehand, thus proving Theorem 1.1. Section 6 provides some open problems.
Since we are controlling the whole external boundary, Ω can be embedded into some torus T L , where T L is identified with [0 , L ] d with periodic conditions. The length L is large enough (for instance L = diam (Ω) + 5 | u | T ) and we may consider the control problem in the cube [0 , L ] d completed with periodicboundary conditions with controls appearing as source terms supported in T L \ Ω . Our control systemthen reads as follows: (cid:26) ∂ t ρ S + div( ρ S u S ) = ˇ v ρ , in (0 , T ) × T L ,ρ S ( ∂ t u S + u S · ∇ u S ) − µ ∆ u S − ( λ + µ ) ∇ div( u S ) + ∇ p ( ρ S ) = ˇ v u , in (0 , T ) × T L , (2.1)where ˇ v ρ and ˇ v u are control functions supported in [0 , T ] × ( T L \ Ω) . Then we set ˇ ρ := ρ S − ρ, ˇ u := u S − u. We also extend the initial data ( ρ , u ) to T L such that (cid:107) (ˇ ρ , ˇ u ) (cid:107) H ( T L ) × H ( T L ) ≤ C L (cid:107) ( ρ , u ) (cid:107) H (Ω) × H (Ω) ≤ C L δ. (2.2)With these notations, one needs to solve the following control problem: Given (ˇ ρ , ˇ u ) small in H ( T L ) × H ( T L ) , find control functions ˇ v ρ and ˇ v u supported in [0 , T ] × ( T L \ Ω) such that the solution of (cid:26) ∂ t ˇ ρ + ( u + ˇ u ) · ∇ ˇ ρ + ρ div(ˇ u ) = ˇ v ρ + ˇ f ρ (ˇ ρ, ˇ u ) , in (0 , T ) × T L ,ρ ( ∂ t ˇ u + ( u + ˇ u ) · ∇ ˇ u ) − µ ∆ˇ u − ( λ + µ ) ∇ div(ˇ u ) + p (cid:48) ( ρ ) ∇ ˇ ρ = ˇ v u + ˇ f u (ˇ ρ, ˇ u ) , in (0 , T ) × T L , (2.3)with initial data ˇ ρ (0 , x ) = ˇ ρ ( x ) , ˇ u (0 , x ) = ˇ u ( x ) in T L , (2.4)and source terms ˇ f ρ (ˇ ρ, ˇ u ) = − ˇ ρ div(ˇ u ) , (2.5) ˇ f u (ˇ ρ, ˇ u ) = − ˇ ρ ( ∂ t ˇ u + ( u + ˇ u ) · ∇ ˇ u ) − ∇ ( p ( ρ + ˇ ρ ) − p (cid:48) ( ρ )ˇ ρ ) , (2.6)satisfies ˇ ρ ( T ) = 0 , ˇ u ( T ) = 0 in T L . (2.7)To take the support of the control functions ˇ v ρ and ˇ v u into account, we introduce a smooth cut-offfunction χ ∈ C ∞ ( T L ; [0 , satisfying (cid:26) χ ( x ) = 0 for all x such that d ( x, Ω) ≤ ε,χ ( x ) = 1 for all x such that d ( x, Ω) ≥ ε, (2.8)and we will look for ˇ v ρ and ˇ v u in the form ˇ v ρ = v ρ χ and ˇ v u = v u χ. Now in order to solve the controllability problem (2.3)–(2.7), we will use a fixed point argument. Adifficulty arising when building this argument is that the term ˇ u · ∇ ˇ ρ in (2.3) (1) is very singular. Hencewe start by removing this term via a diffeomorphism close to the identity. To be more precise, we definethe flow X ˇ u = X ˇ u ( t, τ, x ) corresponding to ˇ u and defined for ( t, τ, x ) ∈ [0 , T ] × [0 , T ] × T L by the equation dX ˇ u dt ( t, τ, x ) = u + ˇ u ( t, X ˇ u ( t, τ, x )) , t ∈ [0 , T ] , X ˇ u ( τ, τ, x ) = x. (2.9)3n order to give a sense to (2.9), we require ˇ u ∈ L (0 , T ; W , ( T L )) and div(ˇ u ) ∈ L (0 , T ; W , ∞ ( T L )) (see [7]). But as we will work in Hilbert spaces with integer indexes, we will rather assume the strongerassumption ˇ u ∈ L (0 , T ; H ( T L )) , in which case, the flow X ˇ u is well-defined classically by Cauchy-Lipschitz’s theorem. We then set, for ( t, x ) ∈ [0 , T ] × T L , Y ˇ u ( t, x ) = X ˇ u ( t, T, X ( T, t, x )) , Z ˇ u ( t, x ) = X ( t, T, X ˇ u ( T, t, x )) , (2.10)which are inverse one from another, i.e. Y ˇ u ( t, Z ˇ u ( t, x )) = Z ˇ u ( t, Y ˇ u ( t, x )) = x for all ( t, x ) ∈ [0 , T ] × T L . For ˇ u suitably small, both transformations Y ˇ u ( t, · ) and Z ˇ u ( t, · ) , t ∈ [0 , T ] , are diffeomorphism of T L whichare close to the identity map on the torus. This change of variable is reminiscent of the Lagrangiancoordinates and allow to straighten the characteristics.We thus set, for ( t, x ) ∈ [0 , T ] × T L , ρ ( t, x ) = ˇ ρ ( t, Y ˇ u ( t, x )) , u ( t, x ) = ˇ u ( t, Y ˇ u ( t, x )) , (2.11)After tedious computations developed in Appendix A, our problem can now be reduced to find controlledsolutions ( ρ, u ) of (cid:26) ∂ t ρ + u · ∇ ρ + ρ div( u ) = v ρ χ + f ρ ( ρ, u ) , in (0 , T ) × T L ,ρ ( ∂ t u + u · ∇ u ) − µ ∆ u − ( λ + µ ) ∇ div( u ) + p (cid:48) ( ρ ) ∇ ρ = v u χ + f u ( ρ, u ) , in (0 , T ) × T L , (2.12)for some ε > small enough, with initial data given by ρ (0 , x ) = ˇ ρ ( Y ˇ u (0 , x )) , u (0 , x ) = ˇ u ( Y ˇ u (0 , x )) in T L , (2.13)and source terms f ρ ( ρ, u ) given by f ρ ( ρ, u ) = − ρDZ t ˇ u ( t, Y ˇ u ( t, x )) : Du − ρ ( DZ t ˇ u ( t, Y ˇ u ( t, x )) − I ) : Du, and f u ( ρ, u ) by f i,u ( ρ, u ) = − ρ ( ∂ t u i + u · ∇ u i ) + d (cid:88) j =1 ∂ i Z j, ˇ u ( t, Y ˇ u ( t, x )) ∂ j ( p ( ρ + ρ ) − p (cid:48) ( ρ ) ρ )+ µ d (cid:88) j,k,(cid:96) =1 ∂ k,(cid:96) u i ( ∂ j Z k, ˇ u ( t, Y ˇ u ( t, x )) − δ j,k ) ( ∂ j Z (cid:96), ˇ u ( t, Y ˇ u ( t, x )) − δ j,(cid:96) ) + d (cid:88) k =1 ∂ k u i ∆ Z k, ˇ u ( t, Y ˇ u ( t, x )) + ( λ + µ ) d (cid:88) j,k,(cid:96) =1 ∂ k,(cid:96) u j ( ∂ j Z k, ˇ u ( t, Y ˇ u ( t, x )) − δ j,k )( ∂ i Z (cid:96), ˇ u ( t, Y ˇ u ( t, x )) − δ i,(cid:96) ) + ( λ + µ ) d (cid:88) j,k =1 ∂ i,j Z k, ˇ u ( t, Y ˇ u ( t, x )) ∂ k u j − p (cid:48) ( ρ ) d (cid:88) j =1 ( ∂ i Z j, ˇ u ( t, Y ˇ u ( t, x )) − δ i,j ) ∂ j ρ , where δ j,k is the Kronecker symbol ( δ j,k = 1 if j = k , δ j,k = 0 if j (cid:54) = k ), and satisfying the controllabilityrequirement ρ ( T ) = 0 , u ( T ) = 0 in T L . (2.14)The corresponding control functions in (2.12) will then be given for ( t, x ) ∈ [0 , T ] × T L by ˇ v ρ ( t, x ) = χ ( Z ˇ u ( t, x )) v ρ ( t, Z ˇ u ( t, x )) , ˇ v u ( t, x ) = χ ( Z ˇ u ( t, x )) v u ( t, Z ˇ u ( t, x )) , (2.15)which are supported in [0 , T ] × ( T L \ Ω) provided that χ ( Z ˇ u ( t, x )) = 0 for all ( t, x ) ∈ [0 , T ] × Ω . (2.16)Let us then remark that the map Y ˇ u can be computed starting from u . Indeed, we have that Y ˇ u ( t, X ( t, T, x )) = X ˇ u ( t, T, x ) so that by differentiation with respect to the time variable, we get, for all ( t, x ) ∈ [0 , T ] × T L , ∂ t Y ˇ u ( t, X ( t, T, x )) + u · ∇ Y ˇ u ( t, X ( t, T, x )) = u + ˇ u ( t, X ˇ u ( t, T, x )) .
4n particular, using this equation at the point X ( T, t, x ) , we obtain ∂ t Y ˇ u ( t, x ) + u · ∇ Y ˇ u ( t, x ) = u + ˇ u ( t, X ˇ u ( t, T, X ( T, t, x ))) = u + ˇ u ( t, Y ˜ u ( t, x )) = u + u ( t, x ) . Next we shall introduce a map F : ( (cid:98) ρ, (cid:98) u ) (cid:55)→ ( ρ, u ) defined on a convex subset of some weightedSobolev spaces, corresponding to some Carleman estimate described later. This fixed point map isconstructed as follows. Given ( (cid:98) ρ, (cid:98) u ) small in a suitable norm, we first define (cid:98) Y = (cid:98) Y ( t, x ) as the solutionof ∂ t (cid:98) Y + u · ∇ (cid:98) Y = u + (cid:98) u, in (0 , T ) × T L , (cid:98) Y ( T, x ) = x, in T L , (2.17)Then we define (cid:98) Z = (cid:98) Z ( t, x ) as follows: for all t ∈ [0 , T ] , (cid:98) Z ( t, · ) is the inverse of (cid:98) Y ( t, · ) on T L . In otherwords, for all ( t, x ) ∈ [0 , T ] × T L , (cid:98) Z ( t, (cid:98) Y ( t, x )) = x, (cid:98) Y ( t, (cid:98) Z ( t, x )) = x. (2.18)We will see that for suitably small (cid:98) u , (cid:98) Y ( t, · ) is invertible for all t ∈ [0 , T ] , see Proposition 5.1.Corresponding to the initial data, we introduce (cid:98) ρ ( x ) = ˇ ρ ( (cid:98) Y (0 , x )) , (cid:98) u ( x ) = ˇ u ( (cid:98) Y (0 , x )) , in T L , (2.19)and, corresponding to the source terms, f ρ ( (cid:98) ρ, (cid:98) u ) = − (cid:98) ρD (cid:98) Z t ( t, (cid:98) Y ( t, x )) : D (cid:98) u − ρ ( D (cid:98) Z t ( t, (cid:98) Y ( t, x )) − I ) : D (cid:98) u, (2.20)and f i,u ( (cid:98) ρ, (cid:98) u ) = − (cid:98) ρ ( ∂ t (cid:98) u i + u · ∇ (cid:98) u i ) + d (cid:88) j =1 ∂ i (cid:98) Z j ( t, (cid:98) Y ( t, x )) ∂ j ( p ( ρ + (cid:98) ρ ) − p (cid:48) ( ρ ) (cid:98) ρ ) . (2.21) + µ d (cid:88) j,k,(cid:96) =1 ∂ k,(cid:96) (cid:98) u i (cid:16) ∂ j (cid:98) Z k ( t, (cid:98) Y ( t, x )) − δ j,k (cid:17) (cid:16) ∂ j (cid:98) Z (cid:96) ( t, (cid:98) Y ( t, x )) − δ j,(cid:96) (cid:17) + d (cid:88) k =1 ∂ k (cid:98) u i ∆ (cid:98) Z k ( t, (cid:98) Y ( t, x )) + ( λ + µ ) d (cid:88) j,k,(cid:96) =1 ∂ k,(cid:96) (cid:98) u j ( ∂ j (cid:98) Z k ( t, (cid:98) Y ( t, x )) − δ j,k )( ∂ i (cid:98) Z (cid:96) ( t, (cid:98) Y ( t, x )) − δ i,(cid:96) ) + ( λ + µ ) d (cid:88) j,k =1 ∂ i,j (cid:98) Z k ( t, (cid:98) Y ( t, x )) ∂ k (cid:98) u j − p (cid:48) ( ρ ) d (cid:88) j =1 ( ∂ i (cid:98) Z j ( t, (cid:98) Y ( t, x )) − δ i,j ) ∂ j (cid:98) ρ . We then look for ( ρ, u ) solving the controllability problem (cid:26) ∂ t ρ + u · ∇ ρ + ρ div( u ) = v ρ χ + f ρ ( (cid:98) ρ, (cid:98) u ) , in (0 , T ) × T L ,ρ ( ∂ t u + u · ∇ u ) − µ ∆ u − ( λ + µ ) ∇ div( u ) + p (cid:48) ( ρ ) ∇ ρ = v u χ + f u ( (cid:98) ρ, (cid:98) u ) , in (0 , T ) × T L , (2.22)with initial data ρ (0 , x ) = (cid:98) ρ ( x ) , u (0 , x ) = (cid:98) u ( x ) in T L , (2.23)with source terms f ρ ( (cid:98) ρ, (cid:98) u ) , f u ( (cid:98) ρ, (cid:98) u ) as in (2.20)–(2.21), and satisfying the controllability objective (2.14).We are therefore reduced to study the controllability of the linear system (cid:40) ∂ t ρ + u · ∇ ρ + ρ div( u ) = v ρ χ + (cid:98) f ρ , in (0 , T ) × T L ,ρ ( ∂ t u + u · ∇ u ) − µ ∆ u − ( λ + µ ) ∇ div( u ) + p (cid:48) ( ρ ) ∇ ρ = v u χ + (cid:98) f u , in (0 , T ) × T L . (2.24)Since this is a linear system, the controllability of (2.24) is equivalent to the observability property forthe adjoint equation (cid:26) − ∂ t σ − u · ∇ σ − p (cid:48) ( ρ ) div( z ) = g σ , in (0 , T ) × T L , − ρ ( ∂ t z + u · ∇ z ) − µ ∆ z − ( λ + µ ) ∇ div( z ) − ρ ∇ σ = g z , in (0 , T ) × T L . (2.25)5he main idea to get an observability inequality for (2.25) is to remark that, taking the divergence ofthe equation of z , the equations of σ and div( z ) form a closed coupled system: (cid:40) − ∂ t σ − u · ∇ σ − p (cid:48) ( ρ ) div( z ) = g σ , in (0 , T ) × T L , − ρ ( ∂ t div( z ) + u · ∇ div( z )) − ν ∆ div( z ) − ρ ∆ σ = div( g z ) , in (0 , T ) × T L , (2.26)where ν := λ + 2 µ > . Now we are led to introduce a new variable q as follows: q := ν div( z ) + ρσ. (2.27)System (2.26) can then be rewritten with the unknown ( σ, q ) as − ∂ t σ − u · ∇ σ + p (cid:48) ( ρ ) ρν σ = g σ + p (cid:48) ( ρ ) ν q, in (0 , T ) × T L , − ρν ( ∂ t q + u · ∇ q ) − ∆ q − p (cid:48) ( ρ ) ρ ν q = div( g z ) + ρ ν g σ − p (cid:48) ( ρ ) ρ ν σ, in (0 , T ) × T L . (2.28)The advantage of considering system (2.28) rather than (2.26) directly is that now the coupling betweenthe two equations is of lower order. In particular, the observability can now be obtained directly byconsidering independently the observability inequality for the equation of σ , which is of transport type,and for the equation of z , which is of parabolic type, considering in both cases the coupling term as asource term. Let us emphasize that the quantity q in (2.27) can be seen as a version of the so-calledeffective viscous flux ν div( u ) − p ( ρ ) , which has been used for the analysis of the Cauchy problem forcompressible fluids [10, 20], but for the dual operator.Now, let us again remark that as system (2.28) is linear, its observability is equivalent to a controlla-bility statement for the adjoint equation written in the dual variables ( r, y ) , where the adjoint is takenwith respect to the variables ( σ, q ) . This leads to the controllability problem: ∂ t r + u · ∇ r + p (cid:48) ( ρ ) ρν r = f r − p (cid:48) ( ρ ) ρ ν y + v r χ , in (0 , T ) × T L ,ρν ( ∂ t y + u · ∇ y ) − ∆ y − p (cid:48) ( ρ ) ρ ν y = f y + p (cid:48) ( ρ ) ν r + v y χ , in (0 , T ) × T L , ( r (0 , · ) , y (0 , · )) = ( r , y ) in T L , ( r ( T, · ) , y ( T, · )) = (0 , in T L , (2.29)where in order to add a margin on the control zone we introduce χ is a smooth cut-off function satisfyingSupp χ (cid:98) { χ = 1 } and χ ( x ) = 1 for all x ∈ T L such that d ( x, Ω) ≥ ε. (2.30)Now in order to solve the controllability problem (2.29), we use again another fixed point argument, andbegin by considering the following decoupled controllability problem: ∂ t r + u · ∇ r + p (cid:48) ( ρ ) ρν r = ˜ f r + v r χ , in (0 , T ) × T L ,ρν ∂ t y − ∆ y = ˜ f y + v y χ , in (0 , T ) × T L , ( r (0 , · ) , y (0 , · )) = ( r , y ) in T L , ( r ( T, · ) , y ( T, · )) = (0 , in T L . (2.31)Getting suitable estimates on the controllability problem (2.31) will allow us to solve the controllabilityproblem (2.29) by a fixed point argument. Note that the control problem for (2.31) simply consists ofthe control of two decoupled equations, the one in r of transport type, the other one in y of parabolictype. We are then reduced to these two classical problems.It turns out that our main difficulty then will be to show the existence of smooth controlled tra-jectory for smooth source terms. Indeed, this is needed as we would like to consider velocity fields u ∈ L (0 , T ; Lip ( T L )) . As Carleman estimates are the basic tools to establish the controllability of6arabolic equations and to estimate the regularity of controlled trajectories and since they are based onthe Hilbert structures of the underlying functional spaces, it is therefore natural to try getting velocityfields u ∈ L (0 , T ; H ( T L )) ∩ H (0 , T ; H ( T L )) ∩ C ([0 , T ]; H ( T L )) . This regularity corresponds to the following regularity properties on the other functions: • g z ∈ L (0 , T ; H − ( T L )) , z ∈ L (0 , T ; H − ( T L )) , • q ∈ L (0 , T ; H − ( T L )) , σ ∈ L (0 , T ; H − ( T L )) , g σ ∈ L (0 , T ; H − ( T L )) , • f r , f y , ˜ f r , ˜ f y ∈ L (0 , T ; H ( T L )) , r ∈ L (0 , T ; H ( T L )) , y ∈ L (0 , T ; H ( T L )) . The controllability and observability properties of the systems described in Section 2.1 will be studiedby using Carleman estimates. These require the introduction of several notations, in particular to definethe weight function. We first construct a function ˜ ψ = ˜ ψ ( t, x ) ∈ C ([0 , T ] × T L ) satisfying the followingproperties. First, it is assumed that ∀ ( t, x ) ∈ [0 , T ] × T L , ˜ ψ ( t, x ) ∈ [0 , . (2.32) We assume that ˜ ψ is constant along the characteristics of the target flow, i.e. ˜ ψ solves ∂ t ˜ ψ + u · ∇ ˜ ψ = 0 in (0 , T ) × T L , (2.33)or equivalently ˜ ψ ( t, x ) = Ψ( x − ut ) with Ψ( · ) := ˜ ψ (0 , · ) . (2.34) We finally assume the existence of a subset ω (cid:98) { χ = 1 } such that inf (cid:110) |∇ ˜ ψ ( t, x ) | , ( t, x ) ∈ [0 , T ] × ( T L \ ω ) (cid:111) > . (2.35)The existence of a function ˜ ψ satisfying those assumptions is easily obtained for L large enough, e.g. L = diam (Ω) + 5 | u | T : it suffices to choose Ψ taking values in [0 , and having its critical points in some ω such that dist ( ω , T L \ Supp χ ) ≥ | u | T and then to propagate ˜ ψ with (2.34). This leaves room todefine ω .Now once ˜ ψ is set, we define ψ ( t, x ) := ˜ ψ ( t, x ) + 6 . (2.36)Next we pick T > , T > and ε > small enough so that T + 2 T < T − L + 12 ε | u | . (2.37)Now for any α ≥ , we introduce the weight function in time θ ( t ) defined by θ = θ ( t ) such that ∀ t ∈ [0 , T ] , θ ( t ) = 1 + (cid:18) − tT (cid:19) α , ∀ t ∈ [ T , T − T ] , θ ( t ) = 1 , ∀ t ∈ [ T − T , T ) , θ ( t ) = 1 T − t ,θ is increasing on [ T − T , T − T ] ,θ ∈ C ([0 , T )) . (2.38)Then we consider the following weight function ϕ = ϕ ( t, x ) : ϕ ( t, x ) = θ ( t ) (cid:0) λ e λ − exp( λ ψ ( t, x )) (cid:1) , (2.39)7here s, λ are positive parameters with s ≥ , λ ≥ and α is chosen as α = sλ e λ , (2.40)which is always larger than , thus being compatible with the condition θ ∈ C ([0 , T )) . Actually, in thesequel we will use that s can be chosen large enough, but for what concerns λ , it can be fixed from thebeginning as equal to the constant λ obtained in Theorem 3.2 below.Also note that, due to the definition of ψ in (2.36), to the condition (2.32) and to λ ≥ , we havefor all ( t, x ) ∈ [0 , T ) × T L , t ) ≤ ϕ ( t, x ) ≤ Φ( t ) , (2.41)where Φ( t ) := θ ( t ) λ e λ . (2.42)We emphasize that the weight functions θ and ϕ depend on the parameters s and λ but we will omitthese dependences in the following for simplicity of notations. Notations.
In the following, we will consider functional spaces depending on the time and spacevariables. This introduces heavy notations, that we will keep in the statements of the theorems, but thatwe shall simplify in the proof by omitting the time interval (0 , T ) and the spatial domain T L as soon asno confusion can occur. Thus, we will use the notations: (cid:107)·(cid:107) L ( L ) = (cid:107)·(cid:107) L (0 ,T ; L ( T L )) , (cid:107)·(cid:107) L ( H ) = (cid:107)·(cid:107) L (0 ,T ; H ( T L )) , and so on for the other functional spaces. Similarly, we will often denote by (cid:107)·(cid:107) H and (cid:107)·(cid:107) H the norms (cid:107)·(cid:107) H ( T L ) , (cid:107)·(cid:107) H ( T L ) . The goal of this section is to solve the controllability problem (2.29):
Theorem 3.1.
Let ( u, T, ε ) be as in (2.37) .Let ( r , y ) ∈ L ( T L ) × L ( T L ) . There exist C > and s ≥ large enough such that for all s ≥ s , if f r and f y satisfy the integrability conditions (cid:13)(cid:13)(cid:13) θ − / f r e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + (cid:13)(cid:13)(cid:13) θ − / f y e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) < ∞ , (3.1) there exists a controlled trajectory ( r, y ) solving (2.29) and satisfying the following estimate: (cid:13)(cid:13)(cid:13) θ − / re sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + s (cid:107) ye sϕ (cid:107) L (0 ,T ; L ( T L )) + (cid:13)(cid:13) θ − ∇ ye sϕ (cid:13)(cid:13) L (0 ,T ; L ( T L )) + (cid:13)(cid:13)(cid:13) θ − / χ v r e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + s − / (cid:13)(cid:13)(cid:13) θ − / χ v y e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) θ − / f r e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + s − / (cid:13)(cid:13)(cid:13) θ − / f y e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + (cid:13)(cid:13)(cid:13) r e sϕ (0) (cid:13)(cid:13)(cid:13) L ( T L ) + (cid:13)(cid:13)(cid:13) y e sϕ (0) (cid:13)(cid:13)(cid:13) L ( T L ) (cid:19) . (3.2) Besides, if ( r , y ) ∈ H ( T L ) × H ( T L ) , and f r and f y satisfy f r e s Φ ∈ L (0 , T ; H ( T L )) , f y e s Φ ∈ L (0 , T ; H ( T L )) , (3.3) we furthermore have the following estimate: (cid:13)(cid:13)(cid:13) re s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) + (cid:13)(cid:13)(cid:13) ye s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) + (cid:13)(cid:13)(cid:13) χ v r e s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) + (cid:13)(cid:13)(cid:13) χ v y e s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) ≤ C (cid:18)(cid:13)(cid:13) f r e s Φ (cid:13)(cid:13) L (0 ,T ; H ( T L )) + (cid:13)(cid:13) f y e s Φ (cid:13)(cid:13) L (0 ,T ; H ( T L )) + (cid:13)(cid:13)(cid:13) r e s Φ(0) (cid:13)(cid:13)(cid:13) H ( T L ) + (cid:13)(cid:13)(cid:13) y e s Φ(0) (cid:13)(cid:13)(cid:13) H ( T L ) (cid:19) , (3.4) for some constant C independent of s ≥ s .
8s explained in Section 2, Theorem 3.1 will be proved by a fixed point theorem based on the un-derstanding of the controllability problem (2.31), which amounts to understand two independent con-trollability problems, one for the heat equation satisfied by y , the other one for the transport equationsatisfied by r .The section is then organized as follows. Firstly, we recall the controllability properties of the heatequation. Secondly, we explain how to exhibit a null-controlled trajectory for the transport equation.Thirdly, we explain how these constructions can be combined in order to get Theorem 3.1. In this paragraph we deal with the following controllability problem: given y and ˜ f y , find a controlfunction v y such that the solution y of (cid:40) ρν ∂ t y − ∆ y = ˜ f y + v y χ , in (0 , T ) × T L ,y (0 , · ) = y , in T L , (3.5)satisfies y ( T, · ) = 0 , in T L . (3.6) As it is done classically, the study of the controllability properties of (3.5) is based on the observabilityof the adjoint system, which is obtained with the following Carleman estimate:
Theorem 3.2 (Theorem 2.5 in [2]) . There exist constants C > and s ≥ and λ ≥ large enoughsuch that for all smooth functions w on [0 , T ] × T L and for all s ≥ s , s / λ (cid:13)(cid:13)(cid:13) ξ / we − sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + s / λ (cid:13)(cid:13)(cid:13) ξ / ∇ we − sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + sλ / e λ (cid:13)(cid:13)(cid:13) w (0) e − sϕ (0) (cid:13)(cid:13)(cid:13) L ( T L ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) − ρν ∂ t − ∆ (cid:19) we − sϕ (cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + C s / λ (cid:13)(cid:13)(cid:13) ξ / χ we − sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) . (3.7) where we have set ξ ( t, x ) = θ ( t ) exp( λ ψ ( t, x )) . (3.8)Using Theorem 3.2 and the remark that for some constant C ≥ independent of s , θ ( t ) C ≤ ξ ( t, x ) ≤ Cθ ( t ) , for all ( t, x ) ∈ [0 , T ) × T L , we obtain the following controllability result: Theorem 3.3 (Inspired by Theorem 2.6 in [2]) . There exist positive constants
C > and s ≥ suchthat for all s ≥ s , for all ˜ f y satisfying (cid:13)(cid:13)(cid:13) θ − / ˜ f y e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) < ∞ (3.9) and y ∈ L ( T L ) , there exists a solution ( y, v y ) of the control problem (3.5) – (3.6) which furthermoresatisfies the following estimate: s / (cid:107) ye sϕ (cid:107) L (0 ,T ; L ( T L )) + (cid:13)(cid:13)(cid:13) θ − / χ v y e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + s / (cid:13)(cid:13) θ − ∇ ye sϕ (cid:13)(cid:13) L (0 ,T ; L ( T L )) ≤ C (cid:13)(cid:13)(cid:13) θ − / ˜ f y e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + Cs / (cid:13)(cid:13)(cid:13) y e sϕ (0) (cid:13)(cid:13)(cid:13) L ( T L ) . (3.10) Besides, this solution ( y, v y ) can be obtained through a linear operator in ( y , ˜ f y ) .If y ∈ H ( T L ) , we also have s − / (cid:13)(cid:13) θ − ∇ ye sϕ (cid:13)(cid:13) L (0 ,T ; L ( T L )) ≤ C (cid:13)(cid:13)(cid:13) θ − / ˜ f y e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + Cs / (cid:13)(cid:13)(cid:13) y e sϕ (0) (cid:13)(cid:13)(cid:13) L ( T L ) + Cs − / (cid:13)(cid:13)(cid:13) ∇ y e sϕ (0) (cid:13)(cid:13)(cid:13) L ( T L ) . (3.11)9he proof of Theorem 3.3 is done in [2] for an initial data y = 0 . We shall therefore not provideextensive details for its proof, but only explain how it should be adapted to the case y (cid:54) = 0 , see theproof in Section 3.1.2.We now explain what can be done when the source term ˜ f y is more regular and lies in L (0 , T ; H ( T L )) or in L (0 , T ; H ( T L )) . Proposition 3.4.
Consider the controlled trajectory ( y, v y ) constructed in Theorem 3.3. Then for someconstant C > independent of s , we have the following properties:1. v y ∈ L (0 , T ; H ( T L )) and (cid:13)(cid:13)(cid:13) χ v y e s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) θ − / ˜ f y e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + (cid:13)(cid:13)(cid:13) y e s Φ(0) (cid:13)(cid:13)(cid:13) L ( T L ) (cid:19) .
2. If y ∈ H ( T L ) and ˜ f y e s Φ / ∈ L (0 , T ; H ( T L )) , θ − / ˜ f y e sϕ ∈ L (0 , T ; L ( T L )) , (cid:13)(cid:13)(cid:13) ye s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ˜ f y e s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) + (cid:13)(cid:13)(cid:13) θ − / ˜ f y e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + (cid:13)(cid:13)(cid:13) y e s Φ(0) (cid:13)(cid:13)(cid:13) H ( T L ) (cid:19) .
3. If y ∈ H ( T L ) and ˜ f y e s Φ / ∈ L (0 , T ; H ( T L )) , θ − / ˜ f y e sϕ ∈ L (0 , T ; L ( T L )) , (cid:13)(cid:13)(cid:13) ye s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ˜ f y e s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) + (cid:13)(cid:13)(cid:13) θ − / ˜ f y e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + (cid:13)(cid:13)(cid:13) y e s Φ(0) (cid:13)(cid:13)(cid:13) H ( T L ) (cid:19) . The proof is done below in Section 3.1.2 and is mainly based on regularity results.
Sketch of the proof of Theorem 3.3.
For later purpose, let us briefly present how the proof of Theorem 3.3works. It mainly consists in introducing the functional J ( w ) = 12 (cid:90) T (cid:90) T L | ( − ρν ∂ t − ∆) w | e − sϕ + s (cid:90) T (cid:90) T L χ θ | w | e − sϕ − (cid:90) T (cid:90) T L ˜ f y w + (cid:90) T L w (0 , · ) y ( · ) , (3.12)considered on the set H obs = { w ∈ C ∞ ([0 , T ] × T L ) } (cid:107)·(cid:107) obs . (3.13)Here the overline refers to the completion with respect to the Hilbert norm (cid:107)·(cid:107) obs defined by (cid:107) w (cid:107) obs = (cid:90) T (cid:90) T L | ( − ρν ∂ t − ∆) w | e − sϕ + s (cid:90) T (cid:90) T L χ θ | w | e − sϕ . (3.14)Thanks to the Carleman estimate (3.7), (cid:107)·(cid:107) obs is a norm. The assumptions y ∈ L and (3.9) implythat J is well-defined, convex and coercive on H obs . Therefore it has a unique minimizer W in H obs andthe couple ( y, v y ) given by y = e − sϕ ( − ρν ∂ t − ∆) W, v y = − s θ χ W e − sϕ (3.15)10olves the controllability problem (3.5)–(3.6). Using the coercivity of J immediately yields L ( L ) estimates on y and v y and on (cid:107) W (cid:107) obs by using J ( W ) ≤ J (0) = 0 : s (cid:90) T (cid:90) T L | y | e sϕ + (cid:90) T (cid:90) T L θ − | v y | e sϕ = s (cid:107) W (cid:107) obs ≤ Cs (cid:32) s (cid:90) T (cid:90) T L θ − | ˜ f y | e sϕ + 1 s (cid:90) T L | y | e sϕ (0) (cid:33) . (3.16)The other estimates on y are derived by weighted energy estimates similar to the ones developed in [2,Theorem 2.6], the only difference coming from the integrations by parts in time leading to new termsinvolving y . Details of the proof are left to the reader. Proof of Proposition 3.4. Item 1.
The control v y is given by (3.15) with W ∈ X obs with an estimate on (cid:107) W (cid:107) obs given by (3.16). Therefore, v y e s Φ / = s θ χ W e s Φ / − sϕ . We look at the equation satisfiedby W ∗ = e − s Φ / W : ( − ρν ∂ t − ∆) W ∗ = e − s Φ / ( − ρν ∂ t − ∆) W + 106105 s ρν ∂ t Φ e − s Φ / W, and W ∗ ( T ) = 0 (in D (cid:48) ( T L ) ). Using (3.16) and (3.7), we get an L ( L ) bound on the right hand-side since ϕ ≤ Φ . Maximal regularity estimates then yield W ∗ = W e − s Φ / ∈ L ( H ) . From / − ϕ ≤ − / , see (2.41), we thus get the claimed estimates. Items 2 and 3.
Let us give some partial details on the proof of item 2. We set y ∗ = ye s Φ / . It solves: ρν ∂ t y ∗ − ∆ y ∗ = e s Φ / ( ˜ f y + v y ) + 67 s ρν ∂ t Φ e s Φ / y. We then simply use classical parabolic regularity estimates for y ∗ , item 1 and (3.10)–(3.11). We study the following control problem: Given ˜ f r and r , find a control function v r such that the solution r of (cid:40) ∂ t r + u · ∇ r + p (cid:48) ( ρ ) ρν r = ˜ f r + v r χ , in (0 , T ) × T L ,r (0 , · ) = r , in T L , (3.17)satisfies the controllability requirement r ( T, · ) = 0 in T L . (3.18)We show the following existence result: Theorem 3.5.
Let ( u, T, ε ) be as in (2.37) .For all ˜ f r with (cid:13)(cid:13)(cid:13) θ − / ˜ f r e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) < ∞ (3.19) and r ∈ L ( T L ) , there exists a function v r ∈ L (0 , T ; L ( T L )) such that the solution r of (3.17) satisfiesthe control requirement (3.18) . Besides, the controlled trajectory r and the control function v r satisfy (cid:13)(cid:13)(cid:13) θ − / re sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + (cid:13)(cid:13)(cid:13) θ − / v r e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) θ − / ˜ f r e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + (cid:13)(cid:13)(cid:13) r e sϕ (0) (cid:13)(cid:13)(cid:13) L ( T L ) (cid:19) . (3.20) If r ∈ H ( T L ) and ˜ f r satisfies (3.17) and ˜ f r e s Φ / ∈ L (0 , T ; H ( T L )) , then r furthermore belongs to L (0 , T ; H ( T L )) and satisfies (cid:13)(cid:13)(cid:13) re s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) + (cid:13)(cid:13)(cid:13) v r e s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ˜ f r e s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) + (cid:13)(cid:13)(cid:13) θ − / ˜ f r e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + (cid:13)(cid:13)(cid:13) r e s Φ(0) (cid:13)(cid:13)(cid:13) H ( T L ) (cid:19) . (3.21)11 f r ∈ H ( T L ) and ˜ f r satisfies (3.17) and ˜ f r e s Φ / ∈ L (0 , T ; H ( T L )) , then r belongs to L (0 , T ; H ( T L )) and satisfies (cid:13)(cid:13)(cid:13) re s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) + (cid:13)(cid:13)(cid:13) v r e s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ˜ f r e s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) + (cid:13)(cid:13)(cid:13) θ − / ˜ f r e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ( T L )) + (cid:13)(cid:13)(cid:13) r e s Φ(0) (cid:13)(cid:13)(cid:13) H ( T L ) (cid:19) . (3.22) Besides, this solution ( r, v r ) can be obtained through a linear operator in ( r , ˜ f r ) .Proof. The proof of Theorem 3.5 consists in an explicit construction solving the control problem (3.17)–(3.18) and then on suitable estimates on it.
An explicit construction.
Let η be a smooth cut-off function taking value on { x ∈ T L , with d ( x, Ω) ≤ ε + | u | T } and vanishing on { x ∈ T L , d ( x, Ω) ≥ ε + | u | T } . We then introduce η the solution of (cid:26) ∂ t η + u · ∇ η = 0 , in (0 , T ) × T L ,η (0 , · ) = η in T L , (3.23)and the solutions r f and r b (here ‘ f ’ stands for forward, ‘ b ’ for backward) of (cid:26) ∂ t r f + u · ∇ r f + ar f = ˜ f r , in (0 , T ) × T L ,r f (0 , · ) = r in T L , (3.24)and (cid:26) ∂ t r b + u · ∇ r b + ar b = ˜ f r , in (0 , T ) × T L ,r b ( T, · ) = 0 in T L , (3.25)where a denotes the constant a = p (cid:48) ( ρ ) ρν . We then set r = η ( x ) ( ηr f + (1 − η ) r b ) + (1 − η ( x )) η ( t ) r f , in (0 , T ) × T L , (3.26)where η ( t ) is a smooth cut-off function taking value on [0 , T / and vanishing for t ≥ T and η = η ( x ) is a smooth cut-off function taking value for x with d ( x, Ω) ≤ ε and vanishing for x with d ( x, Ω) ≥ ε .One easily checks that r solves ∂ t r + u · ∇ r + ar = η ˜ f r + (1 − η ) η ˜ f r + u · ∇ η ( ηr f + (1 − η ) r b ) − η u · ∇ η r f + (1 − η ) ∂ t η r f in (0 , T ) × T L , (3.27)thus corresponding to a control function v r = ( η −
1) ˜ f r + (1 − η ) η ˜ f r + u · ∇ η ( ηr f + (1 − η ) r b ) − η u · ∇ η r f + (1 − η ) ∂ t η r f , (3.28)localized in the support of χ due to the condition on the support of η . Besides, r given by (3.26)satisfies r (0 , · ) = r in T L , r ( T, · ) = 0 in T L due to the conditions on the support of η , η , η and the condition (2.37) on the flow corresponding to u .Actually, thanks to the choice of ε > , T > and T > in (2.37) we have η (1 − η ) = 0 for all ( t, x ) ∈ [0 , T ] × T L , and η η = 0 for all ( t, x ) ∈ [ T − T , T ] × T L . (3.29) Estimates on r . Let us start with estimates on r f . To get estimates on r f , we perform weighted energyestimates on (3.24) on the time interval (0 , T − T ) . Multiplying (3.24) by θ − r f e sϕ , we obtain dd (cid:18) (cid:90) T L θ − | r f | e sϕ (cid:19) ≤ (cid:90) T L | r f | (cid:0) − aθ − e sϕ + ( ∂ t + u · ∇ )( θ − e sϕ ) (cid:1) + (cid:18)(cid:90) T L θ − | r f | e sϕ (cid:19) / (cid:18)(cid:90) T L θ − | ˜ f r | e sϕ (cid:19) / . (3.30)12ut, for all t ∈ (0 , T − T ) and x ∈ T L , ( ∂ t + u · ∇ )( θ − e sϕ ) ≤ . We thus conclude (cid:13)(cid:13)(cid:13) θ − / r f e sϕ (cid:13)(cid:13)(cid:13) L ∞ (0 ,T − T ; L ) ≤ C (cid:13)(cid:13)(cid:13) θ − / ˜ f r e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T − T ; L ) + C (cid:13)(cid:13)(cid:13) r e sϕ (0) (cid:13)(cid:13)(cid:13) L . Similarly, one can show that r b satisfies (cid:13)(cid:13)(cid:13) θ − / r b e sϕ (cid:13)(cid:13)(cid:13) L ∞ ( T ,T ; L ) ≤ C (cid:13)(cid:13)(cid:13) θ − / ˜ f r e sϕ (cid:13)(cid:13)(cid:13) L ( T ,T ; L ) . To conclude that (cid:13)(cid:13)(cid:13) θ − / re sϕ (cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; L ) ≤ C (cid:13)(cid:13)(cid:13) θ − / ˜ f r e sϕ (cid:13)(cid:13)(cid:13) L (0 ,T ; L ) + C (cid:13)(cid:13)(cid:13) r e sϕ (0) (cid:13)(cid:13)(cid:13) L , we use the explicit definition of r in (3.26) and identity (3.29), and notice that η , η and η belong to L ∞ , and ( η, (cid:98) u ) to L ∞ ( L ∞ ) .The estimate on v r in (3.20) is also a simple consequence of its explicit value in (3.28) and the factthat η ∈ W , ∞ , η ∈ W , ∞ , η ∈ L ∞ ( L ∞ ) , η ∈ L ∞ . Regularity results.
To obtain regularity results on r and v r , it is then sufficient to get regularity estimateson r f solution of (3.24) on the time interval (0 , T − T ) and on r b solution of (3.25) on the time interval ( T , T ) . Of course, these estimates will be of the same nature, so we only focus on r f , the other casebeing completely similar.To get weighted estimates in higher norms, we do higher order energy estimates on (3.24). Forinstance, ∇ r f satisfies the equation (cid:26) ∂ t ∇ r f + ( u · ∇ ) ∇ r f + a ∇ r f = ∇ ˜ f r , in (0 , T ) × T L , ∇ r f (0 , · ) = ∇ r in T L , (3.31)Hence, using that ∂ t Φ ≤ on (0 , T − T ) , energy estimates directly provide (cid:13)(cid:13)(cid:13) ∇ r f e s Φ / (cid:13)(cid:13)(cid:13) L ∞ (0 ,T − T ; L ) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ∇ ˜ f r e s Φ / (cid:13)(cid:13)(cid:13) L ( L ) + (cid:13)(cid:13)(cid:13) ∇ r e s Φ / (cid:13)(cid:13)(cid:13) L (cid:19) . (3.32)This implies (3.21).The equation of ∇ r f has the same form. For all ( i, j ) ∈ { , · · · , d } , (cid:26) ∂ t ∂ i,j r f + ( u · ∇ ) ∂ i,j r f + a∂ i,j r f = ∂ i,j ˜ f r , in (0 , T ) × T L ,∂ i,j r f (0 , · ) = ∂ i,j r in T L . (3.33)An energy estimate for ∇ r f on (0 , T − T ) directly yields (cid:13)(cid:13)(cid:13) ∇ r f e s Φ / (cid:13)(cid:13)(cid:13) L ∞ (0 ,T − T ; L ) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ∇ ˜ f r e s Φ / (cid:13)(cid:13)(cid:13) L ( L ) + (cid:13)(cid:13)(cid:13) ∇ r e s Φ / (cid:13)(cid:13)(cid:13) L (cid:19) , thus concluding the proof of Theorem 3.5. Existence of a solution to the control problem.
We construct the controlled trajectory using a fixed pointargument.We introduce the sets C rs = { r ∈ L (0 , T ; L ( T L )) such that θ − / re sϕ ∈ L (0 , T ; L ( T L )) } , C ys = { y ∈ L (0 , T ; H ( T L )) such that ye sϕ , θ − ∇ ye sϕ ∈ L (0 , T ; L ( T L )) } . (3.34)13or ˜ r ∈ C rs and ˜ y ∈ C ys , we introduce ˜ f r := ˜ f r (˜ y ) = f r − p (cid:48) ( ρ ) ρ ν ˜ y, ˜ f y := ˜ f y (˜ r, ˜ y ) = f y + p (cid:48) ( ρ ) ν ˜ r − ρuν · ∇ ˜ y + p (cid:48) ( ρ ) ρ ν ˜ y. As f r and f y satisfy (3.1), for (˜ r, ˜ y ) ∈ C rs × C ys , ˜ f r satisfies (3.19) and ˜ f y satisfies (3.9).Therefore, one can define a map Λ s on C rs × C ys which to a data (˜ r, ˜ y ) ∈ C rs × C ys associates ( r, y ) ,where r is the solution of the controlled problem (cid:40) ∂ t r + u · ∇ r + p (cid:48) ( ρ ) ρν r = ˜ f r + v r χ , in (0 , T ) × T L ,r (0 , · ) = r ( · ) , r ( T, · ) = 0 , in T L , (3.35)given by Theorem 3.5, and y is the solution of the controlled problem (cid:40) ρν ∂ t y − ∆ y = ˜ f y + v y χ , in (0 , T ) × T L ,y (0 , · ) = y ( · ) , y ( T, · ) = 0 , in T L , (3.36)given by Theorem 3.3.Then we remark that Theorems 3.3 and 3.5 both yield a linear construction, respectively for ( y , ˜ f y ) (cid:55)→ ( y, v y ) and for ( r , ˜ f r ) (cid:55)→ ( r, v r ) . In order to apply Banach’s fixed point theorem, let us show that themap Λ s is a contractive mapping for s large enough.Let (˜ r a , ˜ y a ) and (˜ r b , ˜ y b ) be elements of C rs × C ys , and call their respective images ( r a , y a ) = Λ s (˜ r a , ˜ y a ) ,and ( r b , y b ) = Λ s (˜ r b , ˜ y b ) . Setting R = r a − r b , Y = y a − y b , ˜ R = ˜ r a − ˜ r b , ˜ Y = ˜ y a − ˜ y b , ˜ F r = ˜ f r (˜ y a ) − ˜ f r (˜ y b ) and ˜ F y = ˜ f y (˜ r a , ˜ y a ) − ˜ f y (˜ r b , ˜ y b ) , by Theorem 3.5 we have (cid:13)(cid:13)(cid:13) θ − / R e sϕ (cid:13)(cid:13)(cid:13) L ( L ) ≤ C (cid:13)(cid:13)(cid:13) θ − / ˜ F r e sϕ (cid:13)(cid:13)(cid:13) L ( L ) ≤ C (cid:13)(cid:13)(cid:13) θ − / ˜ Y e sϕ (cid:13)(cid:13)(cid:13) L ( L ) , while Theorem 3.3 implies s / (cid:107)Y e sϕ (cid:107) L ( L ) + s / (cid:13)(cid:13) θ − ∇Y e sϕ (cid:13)(cid:13) L ( L ) ≤ C (cid:13)(cid:13)(cid:13) θ − / ˜ F y e sϕ (cid:13)(cid:13)(cid:13) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) θ − / ˜ R e sϕ (cid:13)(cid:13)(cid:13) L ( L ) + (cid:13)(cid:13)(cid:13) θ − / ˜ Y e sϕ (cid:13)(cid:13)(cid:13) L ( L ) + (cid:13)(cid:13)(cid:13) θ − / ∇ ˜ Y e sϕ (cid:13)(cid:13)(cid:13) L ( L ) (cid:19) . In particular, we have (cid:13)(cid:13)(cid:13) θ − / R e sϕ (cid:13)(cid:13)(cid:13) L ( L ) + s (cid:107)Y e sϕ (cid:107) L ( L ) + (cid:13)(cid:13) θ − ∇Y e sϕ (cid:13)(cid:13) L ( L ) ≤ Cs − / (cid:18)(cid:13)(cid:13)(cid:13) θ − / ˜ R e sϕ (cid:13)(cid:13)(cid:13) L ( L ) + s (cid:13)(cid:13)(cid:13) ˜ Y e sϕ (cid:13)(cid:13)(cid:13) L ( L ) + (cid:13)(cid:13)(cid:13) θ − ∇ ˜ Y e sϕ (cid:13)(cid:13)(cid:13) L ( L ) (cid:19) . Thus the quantity (cid:107) ( r, y ) (cid:107) C rs × C ys = (cid:13)(cid:13)(cid:13) θ − / re sϕ (cid:13)(cid:13)(cid:13) L ( L ) + s (cid:107) ye sϕ (cid:107) L ( L ) + (cid:13)(cid:13) θ − ∇ ye sϕ (cid:13)(cid:13) L ( L ) defines a norm on C rs × C ys for which the map Λ s satisfies (cid:107) Λ s (˜ r a , ˜ y a ) − Λ s (˜ r b , ˜ y b ) (cid:107) C rs × C ys ≤ Cs − / (cid:107) ( r a , y a ) − ( r b , y b ) (cid:107) C rs × C ys . (3.37)Consequently, if s is chosen large enough, the map Λ s is a contractive mapping and by Banach’s fixedpoint theorem, Λ s has a unique fixed point ( r, y ) in C rs × C ys . By construction, this fixed point ( r, y ) solves the controllability problem (2.29). Besides, estimating ˜ f r ( y ) and ˜ f y ( r ) by (cid:13)(cid:13)(cid:13) θ − / ˜ f r ( y ) e sϕ (cid:13)(cid:13)(cid:13) L ( L ) ≤ C (cid:107) ye sϕ (cid:107) L ( L ) + C (cid:13)(cid:13)(cid:13) θ − / f r e sϕ (cid:13)(cid:13)(cid:13) L ( L ) (3.38)14nd (cid:13)(cid:13)(cid:13) θ − / ˜ f y ( r, y ) e sϕ (cid:13)(cid:13)(cid:13) L ( L ) ≤ C (cid:13)(cid:13)(cid:13) θ − / f y e sϕ (cid:13)(cid:13)(cid:13) L ( L ) + C (cid:18)(cid:13)(cid:13)(cid:13) θ − / re sϕ (cid:13)(cid:13)(cid:13) L ( L ) + (cid:13)(cid:13)(cid:13) θ − / ye sϕ (cid:13)(cid:13)(cid:13) L ( L ) + (cid:13)(cid:13)(cid:13) θ − / ∇ ye sϕ (cid:13)(cid:13)(cid:13) L ( L ) (cid:19) , (3.39)one gets with Theorems 3.3 and 3.5 that ( r, y ) solution of (2.28) satisfies (cid:13)(cid:13)(cid:13) θ − / re sϕ (cid:13)(cid:13)(cid:13) L ( L ) + s (cid:107) ye sϕ (cid:107) L ( L ) + (cid:13)(cid:13) θ − ∇ ye sϕ (cid:13)(cid:13) L ( L ) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) θ − / f r e sϕ (cid:13)(cid:13)(cid:13) L ( L ) + s − / (cid:13)(cid:13)(cid:13) θ − / f y e sϕ (cid:13)(cid:13)(cid:13) L ( L ) (cid:19) + C (cid:16)(cid:13)(cid:13)(cid:13) r e sϕ (0) (cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13) y e sϕ (0) (cid:13)(cid:13)(cid:13) L (cid:17) , that is, the estimate (3.2). Regularity estimates on the solution of the control problem.
We now further assume that ( r , y ) ∈ H ( T L ) × H ( T L ) and f r , f y satisfy (3.3). We use a bootstrap argument. For preciseness, we set f r ( y ) = f r − p (cid:48) ( ρ ) ρ ν y,f y ( r, y ) = f y + p (cid:48) ( ρ ) ν r − ρuν · ∇ y + p (cid:48) ( ρ ) ρ ν y. First, thanks to (3.11), we have θ − ye sϕ ∈ L (0 , T ; H ( T L )) , so that f r ( y ) e s Φ / ∈ L (0 , T ; H ( T L )) (recall condition (3.3)). Using (3.22), we deduce re s Φ / ∈ L (0 , T ; H ( T L )) . Hence we obtain that f y ( r, y ) e s Φ / ∈ L (0 , T ; H ( T L )) . By Proposition 3.4, ye s Φ / ∈ L (0 , T ; H ( T L )) . Therefore, we have f y ( r, y ) e s Φ / ∈ L (0 , T ; H ( T L )) , and using again Proposition 3.4, we get ye s Φ / ∈ L (0 , T ; H ( T L )) .The above regularity results come with estimates. Tracking them yields estimate (3.4). In this section, we study the controllability of (2.24). This is given in the following statement.
Theorem 4.1.
There exists s ≥ , such that for all s ≥ s , for all ( (cid:98) ρ , (cid:98) u ) ∈ H ( T L ) × H ( T L ) , (cid:98) f ρ , (cid:98) f u such that (cid:98) f ρ e s Φ ∈ L (0 , T ; H ( T L )) , (cid:98) f u e s Φ / ∈ L (0 , T ; H ( T L )) , there exist control functions v ρ , v u and a corresponding controlled trajectory ( ρ, u ) solving (2.24) with initial data ( (cid:98) ρ , (cid:98) u ) , satisfying thecontrollability requirement (2.14) , and depending linearly on the data ( (cid:98) ρ , (cid:98) u , (cid:98) f ρ , (cid:98) f u ) . Besides, we havethe estimate: (cid:13)(cid:13)(cid:13) ( ρe s Φ / , ue s Φ / ) (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) × L (0 ,T ; H ( T L )) + (cid:13)(cid:13)(cid:13) ( χv ρ e s Φ / , χv u e s Φ / ) (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) × L (0 ,T ; H ( T L )) ≤ C (cid:13)(cid:13)(cid:13) ( (cid:98) f ρ e s Φ , (cid:98) f u e s Φ / ) (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) × L (0 ,T ; H ( T L )) + C (cid:13)(cid:13)(cid:13) ( (cid:98) ρ e s Φ(0) , (cid:98) u e s Φ(0) / ) (cid:13)(cid:13)(cid:13) H ( T L ) × H ( T L ) . (4.1) In particular, this implies (cid:13)(cid:13)(cid:13) ( ρe s Φ / , ue s Φ / ) (cid:13)(cid:13)(cid:13) ( C ([0 ,T ]; H ( T L )) ∩ H (0 ,T ; L ( T L ))) × ( L (0 ,T ; H ( T L )) ∩ C ([0 ,T ]; H ( T L )) ∩ H (0 ,T ; H ( T L ))) ≤ C (cid:13)(cid:13)(cid:13) ( (cid:98) f ρ e s Φ , (cid:98) f u e s Φ / ) (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) × L (0 ,T ; H ( T L )) + C (cid:13)(cid:13)(cid:13) ( (cid:98) ρ e s Φ(0) , (cid:98) u e s Φ(0) / ) (cid:13)(cid:13)(cid:13) H ( T L ) × H ( T L ) . (4.2) This allows to define a linear operator G defined on the set (cid:110) ( (cid:98) ρ , (cid:98) u , (cid:98) f ρ , (cid:98) f u ) ∈ H ( T L ) × H ( T L ) × L (0 , T ; H ( T L )) × L (0 , T ; H ( T L )) with (cid:98) f ρ e s Φ ∈ L (0 , T ; H ( T L )) and (cid:98) f u e s Φ / ∈ L (0 , T ; H ( T L )) (cid:111) by G ( (cid:98) ρ , (cid:98) u , (cid:98) f ρ , (cid:98) f u ) = ( ρ, u ) , where ( ρ, u ) denotes a controlled trajectory solving (2.24) with initial condi-tion ( (cid:98) ρ , (cid:98) u ) , satisfying the control requirement (2.14) and estimates (4.1) – (4.2) . roof. We first recover observability estimates for (2.28). We write down (cid:13)(cid:13) ( σe − s Φ , qe − s Φ ) (cid:13)(cid:13) L ( H − ) × L ( H − ) + (cid:13)(cid:13)(cid:13) ( σ (0) e − s Φ(0) , q (0) e − s Φ(0) ) (cid:13)(cid:13)(cid:13) H − × H − = sup (cid:107) ( f r e s Φ ,f y e s Φ ) (cid:107) L H × L H ≤ (cid:107) ( r e s Φ(0) ,y e s Φ(0) ) (cid:107) H × H ≤ (cid:104) ( f r , f y ) , ( σ, q ) (cid:105) L ( H ) ,L ( H − ) + (cid:104) ( r , y ) , ( σ (0) , q (0)) (cid:105) H × H ,H − × H − But by construction, if we associate to ( r , y ) ∈ H ( T L ) × H ( T L ) and f r , f y satisfying f r e s Φ , f y e s Φ ∈ L (0 , T ; H ( T L )) the controlled trajectory of (2.29) in Theorem 3.1, then we get: (cid:104) ( f r , f y ) , ( σ, q ) (cid:105) L ( H ) ,L ( H − ) + (cid:104) ( r , y ) , ( σ (0) , q (0)) (cid:105) H × H ,H − × H − = (cid:104) ( g σ , div g z + ρ ν g σ ) , ( r, y ) (cid:105) L ( H − ) × L ( H − ) ,L ( H ) × L ( H ) + (cid:104) ( σ, q ) , χ ( v r , v y ) (cid:105) L ( H − ) ,L ( H ) . Using (3.4), we obtain: (cid:13)(cid:13) ( σe − s Φ , qe − s Φ ) (cid:13)(cid:13) L ( H − ) × L ( H − ) + (cid:13)(cid:13)(cid:13) ( σ (0) e − s Φ(0) , q (0) e − s Φ(0) ) (cid:13)(cid:13)(cid:13) H − × H − ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ( g σ e − s Φ / , g z e − s Φ / ) (cid:13)(cid:13)(cid:13) L ( H − ) × L ( H − ) + (cid:13)(cid:13)(cid:13) χ ( σe − s Φ / , qe − s Φ / ) (cid:13)(cid:13)(cid:13) L ( H − ) × L ( H − ) (cid:19) . (4.3)Then we use the equation of z in (2.25) to recover estimates on z , that we rewrite as follows: − ρ ( ∂ t z + u · ∇ z ) − µ ∆ z = g z + ρ (cid:18) − λ + µν (cid:19) ∇ σ + λ + µν ∇ q, in (0 , T ) × T L . Then we use the duality with the following controllability problem for the heat equation: (cid:26) ρ ( ∂ t y + u · ∇ y ) − ∆ y = ˜ f y + v y χ , in (0 , T ) × T L ,y (0 , · ) = y , y ( T, · ) = 0 in T L . (4.4)Replacing s by s/ in Proposition 3.4 items 1 & 2, we get that if y ∈ H ( T L ) and ˜ f y e s Φ / ∈ L (0 , T ; H ( T L )) , then there exists a controlled trajectory y satisfying (4.4) with control v y with (cid:13)(cid:13) ye s Φ (cid:13)(cid:13) L ( H ) + (cid:13)(cid:13) v y e s Φ (cid:13)(cid:13) L ( H ) ≤ C (cid:13)(cid:13)(cid:13) ˜ f y e s Φ / (cid:13)(cid:13)(cid:13) L ( H ) + C (cid:13)(cid:13)(cid:13) y e s Φ(0) / (cid:13)(cid:13)(cid:13) H . Arguing by duality, we thus obtain (cid:13)(cid:13)(cid:13) ze − s Φ / (cid:13)(cid:13)(cid:13) L ( H − ) + (cid:13)(cid:13)(cid:13) z (0) e − s Φ(0) / (cid:13)(cid:13)(cid:13) H − ≤ C (cid:16)(cid:13)(cid:13) χ ze − s Φ (cid:13)(cid:13) L ( H − ) + (cid:13)(cid:13) ( σ, q ) e − s Φ (cid:13)(cid:13) L ( H − ) + (cid:13)(cid:13) g z e − s Φ (cid:13)(cid:13) L ( H − ) (cid:17) ≤ C (cid:18)(cid:13)(cid:13)(cid:13) ( g σ e − s Φ / , g z e − s Φ / ) (cid:13)(cid:13)(cid:13) L ( H − ) × L ( H − ) + (cid:13)(cid:13)(cid:13) χ ( σe − s Φ / , qe − s Φ / ) (cid:13)(cid:13)(cid:13) L ( H − ) × L ( H − ) + (cid:13)(cid:13) χ ze − s Φ (cid:13)(cid:13) L ( H − ) (cid:17) . As χ = 1 in Supp χ (recall (2.30)), we have χ χ = χ and and χ div z = χ div( χz ) . Now using that χ is a multiplier on H − we get (cid:13)(cid:13)(cid:13) χ qe − s Φ / (cid:13)(cid:13)(cid:13) L ( H − ) ≤ C (cid:13)(cid:13)(cid:13) χ div ze − s Φ / (cid:13)(cid:13)(cid:13) L ( H − ) + C (cid:13)(cid:13)(cid:13) χ σe − s Φ / (cid:13)(cid:13)(cid:13) L ( H − ) ≤ C (cid:13)(cid:13)(cid:13) χze − s Φ / (cid:13)(cid:13)(cid:13) L ( H − ) + C (cid:13)(cid:13)(cid:13) χσe − s Φ / (cid:13)(cid:13)(cid:13) L ( H − ) , (cid:13)(cid:13) σe − s Φ (cid:13)(cid:13) L ( H − ) + (cid:13)(cid:13)(cid:13) σ (0) e − s Φ(0) (cid:13)(cid:13)(cid:13) H − + (cid:13)(cid:13)(cid:13) ze − s Φ / (cid:13)(cid:13)(cid:13) L ( H − ) + (cid:13)(cid:13)(cid:13) z (0) e − s Φ(0) / (cid:13)(cid:13)(cid:13) H − ≤ C (cid:13)(cid:13)(cid:13) ( g σ e − s Φ / , g z e − s Φ / ) (cid:13)(cid:13)(cid:13) L ( H − ) × L ( H − ) + C (cid:13)(cid:13)(cid:13) χ ( σ, z ) e − s Φ / (cid:13)(cid:13)(cid:13) L ( H − ) × L ( H − ) . (4.5)Using that ( σ, z ) satisfies Equation (2.25), we again argue by duality to deduce that System (2.24) iscontrollable and the estimate (4.1) follows immediately.To conclude (4.2), we look at the equations satisfied by ρe s Φ / and ue s Φ / and perform regularityestimates on each equation. To estimate the regularity of ρe s Φ / , as Φ does not satisfy the transportequation anymore ( Φ is independent of the space variable x ), this induces a small loss in the parameter s ,which is reflected by the fact that we estimate ρe s Φ / instead of ρe s Φ / . This is similar for the estimateon the velocity field u . In this section, we fix the parameter s = s so that Theorem 4.1 applies. We introduce the set on whichthe fixed point argument will take place: C R = { ( ρ, u ) with ρ ∈ L ∞ (0 , T ; H ( T L )) ∩ H (0 , T ; L ( T L )) u ∈ L (0 , T ; H ( T L )) ∩ L ∞ (0 , T ; H ( T L )) ∩ H (0 , T ; H ( T L )) , (cid:13)(cid:13)(cid:13) ( ρe s Φ / , ue s Φ / ) (cid:13)(cid:13)(cid:13) ( L ∞ ( H ) ∩ H ( L )) × ( L ( H ) ∩ L ∞ ( H ) ∩ H ( H )) ≤ R } . The precise definition of our fixed point map is then given as follows: F ( (cid:98) ρ, (cid:98) u ) = G ( (cid:98) ρ , (cid:98) u , f ρ ( (cid:98) ρ, (cid:98) u ) , f u ( (cid:98) ρ, (cid:98) u )) , (5.1)where G is defined in Theorem 4.1, ( (cid:98) ρ , (cid:98) u ) is defined in (2.19) and f ρ ( (cid:98) ρ, (cid:98) u ) , f u ( (cid:98) ρ, (cid:98) u ) are defined in(2.20)–(2.21). Therefore, our first goal is to check that F is well-defined on C R , and for that purpose,we shall in particular show that, for ( (cid:98) ρ, (cid:98) u ) ∈ C R one has (cid:98) ρ e s Φ(0) ∈ H ( T L ) , (cid:98) u e s Φ(0) / ∈ H ( T L ) , f ρ ( (cid:98) ρ, (cid:98) u ) e s Φ ∈ L (0 , T ; H ( T L )) and f u ( (cid:98) ρ, (cid:98) u ) e s Φ / ∈ L (0 , T ; H ( T L )) . F in (5.1) is well-defined on C R In order to show that F is well-defined, we first study the maps (cid:98) Y and (cid:98) Z defined in (2.17)–(2.18) andprove some of their properties, in particular that they are close to the identity map. We can then definethe source term f ρ ( (cid:98) ρ, (cid:98) u ) , f u ( (cid:98) ρ, (cid:98) u ) and the initial data ( (cid:98) ρ , (cid:98) u ) . Accordingly, we will deduce that the map F in (5.1) is well-defined on C R for R > small enough. (cid:98) Y and (cid:98) Z in (2.17)–(2.18) We start with the following result:
Proposition 5.1.
Let (cid:98) u ∈ L (0 , T ; H ( T L )) ∩ H (0 , T ; H ( T L )) with (cid:13)(cid:13)(cid:13)(cid:98) ue s Φ / (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) ∩ H (0 ,T ; H ( T L )) ≤ R. (5.2) Then the map (cid:98) Y defined in (2.17) satisfies, for some constant C independent of R > : (cid:13)(cid:13)(cid:13) ( (cid:98) Y ( t, x ) − x ) e s Φ( t ) / (cid:13)(cid:13)(cid:13) C ([0 ,T ]; H ( T L )) + (cid:13)(cid:13)(cid:13) ( (cid:98) Y ( t, x ) − x ) e s Φ( t ) / (cid:13)(cid:13)(cid:13) C ([0 ,T ]; H ( T L )) ≤ CR. (5.3)17 herefore, there exists R ∈ (0 , such that for all R ∈ (0 , R ) the map (cid:98) Z defined in (2.18) is well-definedand satisfies (cid:13)(cid:13)(cid:13) ( D (cid:98) Z ( t, (cid:98) Y ( t, x )) − I ) e s Φ( t ) / (cid:13)(cid:13)(cid:13) C ([0 ,T ]; H ( T L )) ≤ CR, (5.4) (cid:13)(cid:13)(cid:13) ( D (cid:98) Z ( t, (cid:98) Y ( t, x )) − I ) e s Φ( t ) / (cid:13)(cid:13)(cid:13) W / , (0 ,T ; H / ( T L )) ≤ CR, (5.5) (cid:13)(cid:13)(cid:13) D (cid:98) Z ( t, (cid:98) Y ( t, x )) e s Φ( t ) / (cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H ( T L )) ≤ CR, (5.6) and χ ( (cid:98) Z ( t, x )) = 0 for all ( t, x ) ∈ [0 , T ] × Ω , (5.7) where χ is defined by (2.8) .Proof. Let us consider the equation satisfied by the map (cid:98) δ ( t, x ) = (cid:98) Y ( t, x ) − x. Using (2.17), direct computations show that (cid:98) δ satisfies ∂ t (cid:98) δ + u · ∇ (cid:98) δ = (cid:98) u, in (0 , T ) × T L , (cid:98) δ ( T, x ) = 0 , in T L . (5.8)Therefore, we immediately get that (cid:98) δ ( t, x ) = − (cid:90) Tt (cid:98) u ( τ, x + ( τ − t ) u ) dτ. As for all t ∈ [0 , T ] , Φ( t ) ≤ τ ∈ [ t,T ] { Φ( τ ) } , we deduce from the above formula and (5.2) that (cid:13)(cid:13)(cid:13)(cid:98) δe s Φ / (cid:13)(cid:13)(cid:13) C ( H ) ≤ CR.
Using Equation (5.8) and the bound (5.2), we also derive (cid:13)(cid:13)(cid:13) ∂ t (cid:98) δe s Φ / (cid:13)(cid:13)(cid:13) C ( H ) ≤ CR, from which, together with | s∂ t Φ | ≤ Ce s Φ / independently of s , we immediately deduce (5.3).In particular, (cid:13)(cid:13)(cid:13) D (cid:98) δ (cid:13)(cid:13)(cid:13) C ( H ) ≤ CR, so that for R small enough, for all ( t, x ) ∈ [0 , T ] × T L , D (cid:98) Y ( t, x ) = I + D (cid:98) δ ( t, x ) is invertible. Consequently, (cid:98) Z defined by (2.18) is well-defined by the inverse function theorem (note that C ( H ) ⊂ C ( C ) indimension d ≤ ), and (cid:98) Z ∈ C ( C ) with (cid:13)(cid:13)(cid:13) (cid:98) Z (cid:13)(cid:13)(cid:13) C ( C ) ≤ CR.
In order to get estimates on (cid:98) Z in weighted norms, we start from the formula (cid:98) Z ( t, (cid:98) Y ( t, x )) = x anddifferentiate it with respect to x : We obtain, for all t ∈ [0 , T ] and x ∈ T L , D (cid:98) Z ( t, (cid:98) Y ( t, x )) D (cid:98) Y ( t, x ) = I, (5.9)i.e. D (cid:98) Z ( t, (cid:98) Y ( t, x ))( I + D (cid:98) δ ( t, x )) = I. D (cid:98) Z ( t, (cid:98) Y ( t, x )) = I + ∞ (cid:88) n =1 ( − n ( D (cid:98) δ ( t, x )) n . Using then that C ([0 , T ]; H ( T L )) is an algebra (as d ≤ ) and that e s Φ / ≥ , (cid:13)(cid:13)(cid:13) ( D (cid:98) Z ( t, (cid:98) Y ( t, x )) − I ) e s Φ / (cid:13)(cid:13)(cid:13) C ( H ) ≤ ∞ (cid:88) n =1 C n − (cid:13)(cid:13)(cid:13) D (cid:98) δe s Φ / (cid:13)(cid:13)(cid:13) nC ( H ) ≤ ∞ (cid:88) n =1 C n − ( CR ) n ≤ CR − C R ≤ CR, for R small enough ( R ≤ / C ), i.e. (5.4).By interpolation, we have (cid:13)(cid:13)(cid:13) D (cid:98) δe s Φ / (cid:13)(cid:13)(cid:13) W / , ( H / ) ≤ C (cid:13)(cid:13)(cid:13) D (cid:98) δe s Φ / (cid:13)(cid:13)(cid:13) / C ( H ) (cid:13)(cid:13)(cid:13) D (cid:98) δe s Φ / (cid:13)(cid:13)(cid:13) / C ( H ) ≤ CR.
We now remark that W / , (0 , T ; H / ( T L )) is an algebra ( × / > and / > d/ ), and we cantherefore derive, similarly as above, that (cid:13)(cid:13)(cid:13) ( D (cid:98) Z ( t, (cid:98) Y ( t, x )) − I ) e s Φ / (cid:13)(cid:13)(cid:13) W / , ( H / ) ≤ ∞ (cid:88) n =1 C n − (cid:13)(cid:13)(cid:13) D (cid:98) δe s Φ / (cid:13)(cid:13)(cid:13) nW / , ( H / ) (cid:13)(cid:13)(cid:13) e − s Φ / (cid:13)(cid:13)(cid:13) n − W / , ( H / ) ≤ ∞ (cid:88) n =1 C n − ( CR ) n ≤ CR − C R ≤ CR, which concludes the proof of (5.5).The proof of (5.6) consists in writing D (cid:98) Z ( t, (cid:98) Y ( t, x )) = D ( D (cid:98) Z ( t, (cid:98) Y ( t, x )))( D (cid:98) Y ( t, x )) − = D ( D (cid:98) Z ( t, (cid:98) Y ( t, x ))) D (cid:98) Z ( t, (cid:98) Y ( t, x )) , where the last identity comes from (5.9). From (5.4), we have D ( D (cid:98) Z ( t, (cid:98) Y ( t, x ))) e s Φ / ∈ L ∞ ( H ) and D (cid:98) Z ( t, (cid:98) Y ( t, x )) e s Φ / ∈ L ∞ ( H ) , hence (5.6) easily follows as the product of a function in H ( T L ) by afunction in H ( T L ) belongs to H ( T L ) ( d ≤ ).We finally focus on the proof of (5.7). As (cid:98) Y ( t, x ) − x belongs to L ∞ ((0 , T ) × T L ) by (5.3) and (cid:98) Y ( t, · ) is a C diffeomorphism for all t ∈ [0 , T ] , (cid:13)(cid:13)(cid:13) (cid:98) Z ( t, x ) − x (cid:13)(cid:13)(cid:13) L ∞ ( L ∞ ) = (cid:13)(cid:13)(cid:13) (cid:98) Z ( t, (cid:98) Y ( t, x )) − (cid:98) Y ( t, x ) (cid:13)(cid:13)(cid:13) L ∞ ( L ∞ ) = (cid:13)(cid:13)(cid:13) x − (cid:98) Y ( t, x ) (cid:13)(cid:13)(cid:13) L ∞ ( L ∞ ) ≤ CR.
In particular, taking R small enough ( R < ε/C ), condition (5.7) is obviously satisfied for χ defined in(2.8). f ρ ( (cid:98) ρ, (cid:98) u ) , f u ( (cid:98) ρ, (cid:98) u ) Lemma 5.2.
Let ( (cid:98) ρ, (cid:98) u ) ∈ C R for some s ≥ s and R ∈ (0 , R ) , where R is given by Proposition 5.1.Then we have the following estimate (cid:13)(cid:13)(cid:13) ( f ρ ( (cid:98) ρ, (cid:98) u ) e s Φ , f u ( (cid:98) ρ, (cid:98) u ) e s Φ / ) (cid:13)(cid:13)(cid:13) L (0 ,T ; H ( T L )) × L (0 ,T ; H ( T L )) ≤ CR , (5.10) where f ρ ( (cid:98) ρ, (cid:98) u ) , f u ( (cid:98) ρ, (cid:98) u ) are defined in (2.20) – (2.21) .Proof. In the following, we will repeatedly use the estimates derived in Proposition 5.1 and the crucialremark that / / > / . 19 Concerning f ρ ( (cid:98) ρ, (cid:98) u ) . We perform the following estimates: as H ( T L ) is an algebra in dimension d ≤ , (cid:13)(cid:13)(cid:13) ( (cid:98) ρD (cid:98) Z t ( t, (cid:98) Y ( t, x )) : D (cid:98) u ) e s Φ (cid:13)(cid:13)(cid:13) L ( H ) ≤ C (cid:13)(cid:13)(cid:13)(cid:98) ρe s Φ / (cid:13)(cid:13)(cid:13) L ∞ ( H ) (cid:13)(cid:13)(cid:13) D (cid:98) Z ( t, (cid:98) Y ( t, x )) (cid:13)(cid:13)(cid:13) L ∞ ( H ) (cid:13)(cid:13)(cid:13)(cid:98) ue s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ CR , and (cid:13)(cid:13)(cid:13) ρ ( D (cid:98) Z t ( t, (cid:98) Y ( t, x )) − I ) : D (cid:98) ue s Φ (cid:13)(cid:13)(cid:13) L ( H ) ≤ C (cid:13)(cid:13)(cid:13) ( D (cid:98) Z ( t, (cid:98) Y ( t, x )) − I ) e s Φ / (cid:13)(cid:13)(cid:13) L ∞ ( H ) (cid:13)(cid:13)(cid:13) D (cid:98) ue s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ CR . • Concerning f u ( (cid:98) ρ, (cid:98) u ) . Using that the product is continuous from H ( T L ) × H ( T L ) into H ( T L ) , (cid:13)(cid:13)(cid:13)(cid:98) ρ∂ t (cid:98) ue s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ (cid:13)(cid:13)(cid:13)(cid:98) ρe s Φ / (cid:13)(cid:13)(cid:13) L ∞ ( H ) (cid:13)(cid:13)(cid:13) ∂ t (cid:98) ue s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ CR . Using again that H ( T L ) is an algebra, (cid:13)(cid:13)(cid:13)(cid:98) ρu · ∇ (cid:98) ue s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ C (cid:13)(cid:13)(cid:13)(cid:98) ρe s Φ / (cid:13)(cid:13)(cid:13) L ∞ ( H ) (cid:13)(cid:13)(cid:13)(cid:98) ue s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ CR . For i, j, k, (cid:96) ∈ { , · · · , d } , we get (cid:13)(cid:13)(cid:13) ∂ k,(cid:96) (cid:98) u j ( ∂ j (cid:98) Z k ( t, (cid:98) Y ( t, x )) − δ j,k )( ∂ i (cid:98) Z (cid:96) ( t, (cid:98) Y ( t, x )) − δ i,(cid:96) ) e s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ C (cid:13)(cid:13)(cid:13) D (cid:98) ue s Φ / (cid:13)(cid:13)(cid:13) L ( H ) (cid:13)(cid:13)(cid:13) ( D (cid:98) Z ( t, Y ( t, x )) − I ) e s Φ / (cid:13)(cid:13)(cid:13) L ∞ ( H ) ≤ CR . Similarly, for i, j, k ∈ { , · · · , d } , (cid:13)(cid:13)(cid:13) ∂ i,j (cid:98) Z k ( t, (cid:98) Y ( t, x )) ∂ k (cid:98) u j e s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ C (cid:13)(cid:13)(cid:13) D (cid:98) Z ( t, Y ( t, x )) e s Φ / (cid:13)(cid:13)(cid:13) L ∞ ( H ) (cid:13)(cid:13)(cid:13) Due s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ CR . In order to estimate the terms coming from the pressure, we write p ( ρ + (cid:98) ρ ) − p ( ρ ) − p (cid:48) ( ρ ) (cid:98) ρ = (cid:98) ρ h ( (cid:98) ρ ) where h is a C function depending on the pressure law (here we use that the pressure law p belongsto C locally around ρ ), so we have ∇ ( p ( ρ + (cid:98) ρ ) − p (cid:48) ( ρ ) (cid:98) ρ ) = (cid:0) (cid:98) ρh ( (cid:98) ρ ) + (cid:98) ρ h (cid:48) ( (cid:98) ρ ) (cid:1) ∇ (cid:98) ρ. As (cid:107) (cid:98) ρ (cid:107) L ∞ ( L ∞ ) ≤ CR ≤ C , we thus obtain, for i, j ∈ { , · · · , d } , (cid:13)(cid:13)(cid:13) ∂ i (cid:98) Z j ( t, (cid:98) Y ( t, x )) ∂ j ( p ( ρ + (cid:98) ρ ) − p (cid:48) ( ρ ) (cid:98) ρ ) e s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ C (cid:13)(cid:13)(cid:13) D (cid:98) Z ( t, (cid:98) Y ) (cid:13)(cid:13)(cid:13) L ∞ ( H ) (cid:13)(cid:13)(cid:13)(cid:98) ρe s Φ / (cid:13)(cid:13)(cid:13) L ( H ) (cid:13)(cid:13)(cid:13) ∇ (cid:98) ρe s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ CR . For i, j ∈ { , · · · , d } , (cid:13)(cid:13)(cid:13) p (cid:48) ( ρ )( ∂ i (cid:98) Z j ( t, (cid:98) Y ( t, x ) − δ i,j ) ∂ j (cid:98) ρe s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ C (cid:13)(cid:13)(cid:13) ( D (cid:98) Z ( t, (cid:98) Y ( t, x )) − I ) e s Φ / (cid:13)(cid:13)(cid:13) L ∞ ( H ) (cid:13)(cid:13)(cid:13)(cid:98) ρe s Φ / (cid:13)(cid:13)(cid:13) L ( H ) ≤ CR . Combining all the above estimates yields Lemma 5.2.20 .1.3 Estimates on ( (cid:98) ρ , (cid:98) u ) We finally state the following estimates on ( (cid:98) ρ , (cid:98) u ) defined in (2.19): Lemma 5.3.
Let (cid:98) u ∈ L (0 , T ; H ( T L )) ∩ H (0 , T ; H ( T L )) satisfying (5.2) for some R ≤ R given byProposition 5.1. Let δ ∈ (0 , and (ˇ ρ , ˇ u ) ∈ H ( T L ) × H ( T L ) with (cid:107) (ˇ ρ , ˇ u ) (cid:107) H ( T L ) × H ( T L ) ≤ C L δ. (5.11) Define ( (cid:98) ρ , (cid:98) u ) as in (2.19) . Then there exists a constant C > independent of R such that (cid:107) ( (cid:98) ρ , (cid:98) u ) (cid:107) H ( T L ) × H ( T L ) ≤ Cδ. (5.12)
Proof.
It is a straightforward consequence of the estimate (5.3) derived in Proposition 5.1.
Putting together the estimates obtained in Proposition 5.1, Lemmas 5.2 and 5.3 and using Theorem 4.1,we get the following result:
Proposition 5.4.
Let (ˇ ρ , ˇ u ) in H ( T L ) × H ( T L ) satisfying (5.11) for some δ > , ( (cid:98) ρ, (cid:98) u ) ∈ C R forsome R ∈ (0 , R ) with R given by Proposition 5.1. Then the map F in (5.1) is well-defined, and thereexist a constant C such that ( ρ, u ) = F ( (cid:98) ρ, (cid:98) u ) satisfies (cid:13)(cid:13)(cid:13) ( ρe s Φ / , ue s Φ / ) (cid:13)(cid:13)(cid:13) ( L ∞ ( H ) ∩ H ( L )) × ( L ( H ) ∩ L ∞ ( H ) ∩ H ( H )) ≤ CR + Cδ. (5.13)
Besides, the condition (5.7) is satisfied for χ defined in (2.8) . Proposition 5.4 is the core of the fixed point argument developed below.
Let ( ρ , u ) ∈ H (Ω) × H (Ω) satisfying (1.4) with δ > . Choosing an extension (ˇ ρ , ˇ u ) of ( ρ , u ) satisfying (2.2), (ˇ ρ , ˇ u ) satisfies (5.11). Therefore, from Proposition 5.4, the map F in (5.1) is well-defined for ( (cid:98) ρ, (cid:98) u ) ∈ C R for R ∈ (0 , R ) , with R > given by Proposition 5.1, and ( ρ, u ) = F ( (cid:98) ρ, (cid:98) u ) satisfies (5.13) with some constant C > . We now choose R ∈ (0 , R ) such that CR < / and δ = R/ (2 C ) , so that as a consequence of Proposition 5.4, F maps C R into itself. We are then in asuitable position to use a Schauder fixed point argument, and we now fix the parameters R > , δ > such that C R is stable by the map F .Let us then notice that the set C R is convex and compact when endowed with the ( L (0 , T ; L ( T L ))) -topology, as a simple consequence of Aubin-Lions’ Lemma, see e.g. [26].We then focus on the continuity of the map F on C R endowed with the ( L (0 , T ; L ( T L ))) topology.Let us consider a sequence ( (cid:98) ρ n , (cid:98) u n ) in C R converging strongly in ( L (0 , T ; L ( T L ))) to some element ( (cid:98) ρ, (cid:98) u ) . The set C R is closed under the topology of ( L (0 , T ; L ( T L ))) and therefore ( (cid:98) ρ, (cid:98) u ) ∈ C R and wehave the weak- ∗ convergence of ( (cid:98) ρ n , (cid:98) u n ) towards ( (cid:98) ρ, (cid:98) u ) in ( L ∞ ( H ) ∩ H ( L )) × ( L ( H ) ∩ H ( H )) .Using the bounds defining C R , by Aubin-Lions’ lemma we also have the following convergences to ( (cid:98) ρ, (cid:98) u ) : (cid:98) ρ n → n →∞ (cid:98) ρ strongly in L ∞ (0 , T ; L ∞ ( T L )) , (cid:98) ρ n → n →∞ (cid:98) ρ strongly in L (0 , T ; H ( T L )) , (cid:98) u n → n →∞ (cid:98) u strongly in L (0 , T ; H ( T L )) . (5.14)To each (cid:98) u n , we associate the corresponding flow (cid:98) Y n defined by ∂ t (cid:98) Y n + u · ∇ (cid:98) Y n = u + (cid:98) u n in (0 , T ) × T L , (cid:98) Y n ( T, x ) = x in T L , (5.15)and the respective inverse (cid:98) Z n as in (2.18) (which is well-defined thanks to Proposition 5.1). The sequence (cid:98) Y n is bounded in L ∞ ( H ) ∩ W , ∞ ( H ) according to (5.3) and then converge weakly- ∗ in L ∞ ( H ) ∩ W , ∞ ( H ) . Passing to the limit in Equation (5.15), we easily get that the weak limit of the sequence (cid:98) Y n is21 Y defined by (2.17). Besides, by Aubin-Lions’ lemma and the weak convergence of (cid:98) Y n towards (cid:98) Y , we alsohave the strong convergence of (cid:98) Y n to (cid:98) Y in W / , (0 , T ; H / ( T L )) and therefore in C ([0 , T ]; C ( T L )) .Consequently, the sequence (cid:98) Z n strongly converges to (cid:98) Z in C ([0 , T ]; C ( T L )) . These strong convergencesallow to show that D (cid:98) Z n ( t, (cid:98) Y n ( t, x )) (cid:42) n →∞ D (cid:98) Z ( t, (cid:98) Y ( t, x )) in D (cid:48) ((0 , T ) × T L ) , ( (cid:98) ρ ,n , (cid:98) u ,n ) (cid:42) n →∞ ( (cid:98) ρ , (cid:98) u ) in ( D (cid:48) ((0 , T ) × T L )) , (5.16)where ( (cid:98) ρ ,n , (cid:98) u ,n ) = (ˇ ρ ( (cid:98) Y n (0 , x )) , ˇ u ( (cid:98) Y n (0 , x ))) . From the uniform bounds (5.4)–(5.5) on the quantity D (cid:98) Z n ( t, (cid:98) Y n ( t, x )) − I and Aubin-Lions’ Lemma, we also deduce that D (cid:98) Z n ( t, (cid:98) Y n ( t, x )) − I → n →∞ D (cid:98) Z ( t, (cid:98) Y ( t, x )) − I strongly in L (0 , T ; H ( T L )) . (5.17)Using then the uniform bound (5.6), the identity D (cid:98) Z n ( t, (cid:98) Y n ( t, x )) = D ( D (cid:98) Z n ( t, (cid:98) Y n ( t, x ))) D (cid:98) Z n ( t, (cid:98) Y n ( t, x )) , and the convergence (5.17), we also conclude that D (cid:98) Z n ( t, (cid:98) Y n ( t, x )) (cid:42) n →∞ D (cid:98) Z ( t, (cid:98) Y ( t, x )) weakly in L (0 , T ; H ( T L )) . (5.18)Combining the above convergences, we easily obtain that the functions f ρ ( (cid:98) ρ n , (cid:98) u n ) and f u ( (cid:98) ρ n , (cid:98) u n ) weaklyconverge to f ρ ( (cid:98) ρ, (cid:98) u ) and f u ( (cid:98) ρ, (cid:98) u ) in L (0 , T ; H ( T L )) , and with Lemma 5.2, weakly in the weightedSobolev space described by (5.10). As the control process G in Theorem 4.1 is linear continuous in ( (cid:98) ρ , (cid:98) u , (cid:98) f ρ , (cid:98) f u ) , it is weakly continuous. Hence ( ρ n , u n ) = F ( (cid:98) ρ n , (cid:98) u n ) = G ( (cid:98) ρ ,n , (cid:98) u ,n , f ρ ( (cid:98) ρ n , (cid:98) u n ) , f u ( (cid:98) ρ n , (cid:98) u n )) weakly converges to ( ρ, u ) = F ( (cid:98) ρ, (cid:98) u ) = G ( (cid:98) ρ , (cid:98) u , f ρ ( (cid:98) ρ, (cid:98) u ) , f u ( (cid:98) ρ, (cid:98) u )) in the sense of distributions. But weknow that C R is stable by the map F , so that ( ρ n , u n ) all belong to the set C R , which is compact for the ( L (0 , T ; L ( T L ))) topology. Therefore, ( ρ n , u n ) = F ( (cid:98) ρ n , (cid:98) u n ) strongly converges to ( ρ, u ) = F ( (cid:98) ρ, (cid:98) u ) in ( L (0 , T ; L ( T L ))) .We conclude by applying Schauder’s fixed point theorem to the map F on C R . This yields a fixedpoint ( ρ, u ) = F ( ρ, u ) which by construction solves the control problem (2.12)–(2.13)–(2.14).To go back to the original system (2.3), we define Y as the solution of ∂ t Y + u · ∇ Y = u + u in (0 , T ) × T L , Y ( T, x ) = x in T L , and Z = Z ( t, x ) such that for all t ∈ [0 , T ] , Z ( t, · ) is the inverse of Y ( t, · ) , which is well-defined accordingto Proposition 5.1. We then simply set, for all ( t, x ) ∈ [0 , T ] × T L , ˇ ρ ( t, x ) = ρ ( t, Z ( t, x )) , ˇ u ( t, x ) = u ( t, Z ( t, x )) . (5.19)By construction, (ˇ ρ, ˇ u ) solves (2.3)–(2.4) and the controllability requirement (2.7) with control functions (ˇ v ρ , ˇ v u ) defined for ( t, x ) ∈ [0 , T ] × T L by ˇ v ρ ( t, x ) = χ ( Z ( t, x )) v ρ ( t, Z ( t, x )) , ˇ v u ( t, x ) = χ ( Z ( t, x )) v u ( t, Z ( t, x )) . These control functions are supported in [0 , T ] × ( T L \ Ω) thanks to (5.7), so that by restriction on Ω ,we get a solution ( ρ S , u S ) = ( ρ, u ) + (ˇ ρ, ˇ u ) of (1.1) satisfying (1.5)–(1.6).To get the regularity estimate in (1.7), we first show that the fixed point ( ρ, u ) of F satisfies ( ρ, u ) ∈ C ([0 , T ]; H ( T L )) × ( L (0 , T ; H ( T L )) ∩ C ([0 , T ]; H ( T L ))) , which is a consequence of (4.2). From these regularity results on ( ρ, u ) , (5.19) and the regularity estimatesobtained on Z in Proposition 5.1, we deduce that (ˇ ρ, ˇ u ) ∈ C ([0 , T ]; H ( T L )) × ( L (0 , T ; H ( T L )) ∩ C ([0 , T ]; H ( T L ))) , and, consequently, (1.7). 22 Further comments
The case of non-constant trajectories.
Our result only considers the local exact controllabilityaround a constant state. The next question concerns the case of local exact controllability aroundnon-constant target trajectories, similarly as what has been done in the context of non-homogeneousincompressible Navier-Stokes equations in [2] and in the context of compressible Navier-Stokes equationin one space dimension in the recent preprint [9]. We expect such results to be true provided the targettrajectory is sufficiently smooth and assuming some geometric condition on the flow of the target velocityfield u . Namely, if we denote by X the flow corresponding to u , i.e. given by ddt X ( t, τ, x ) = u ( t, X ( t, τ, x )) , X ( τ, τ, x ) = x, (6.1)it is natural to expect a geometric condition of the form ∀ x ∈ Ω , ∃ t ∈ (0 , T ) , s.t. X ( t, , x ) / ∈ Ω , (6.2)corresponding to the time condition (1.3) in the case of a constant velocity field. But considering the caseof a non-constant velocity field would introduce many new terms in the proof and make it considerablymore intricate, including for instance the difficulty to control the density when recirculation appearsclose to the boundary of Ω . This issue needs to be carefully analyzed and discussed. Controllability from a subset of the boundary.
From a practical point of view, it seems morereasonable to control the velocity and the density from some part of the boundary in which the targetvelocity field u enters in the domain. But in this case, one needs to make precise what are the boundaryconditions on the velocity field u S .One could think for instance to Dirichlet boundary conditions of the form u S = u on the outflowboundary Γ out . But this would mean that one should find a solution ( ρ, u ) of the control problem (2.12)–(2.23)–(2.14) with boundary conditions u = 0 on the outflow boundary Γ out . This would introduce alot of additional technicalities as the dual variable z would also satisfy Dirichlet homogeneous boundaryconditions on the outflow boundary Γ out . The variable q in (2.27) would therefore have non-homogeneousDirichlet boundary conditions on (0 , T ) × Γ out and careful estimates should be done to recover estimateson z , for instance based on the delicate Carleman estimates proved in [17]. A Computations of the equations satisfied by ( ρ, u ) According to (2.11), we have, for all ( t, x ) ∈ [0 , T ] × T L , ρ ( t, X ( t, T, x )) = ˇ ρ ( t, X ˇ u ( t, T, x )) , u ( t, X ( t, x )) = ˇ u ( t, X ˇ u ( t, T, x )) , so that differentiating in t , we easily derive ( ∂ t ρ + u · ∇ ρ )( t, Z ˇ u ( t, x )) = ( ∂ t ˇ ρ + ( u + ˇ u ) · ∇ ˇ ρ )( t, x )( ∂ t u + u · ∇ u )( t, Z ˇ u ( t, x )) = ( ∂ t ˇ u + ( u + ˇ u ) · ∇ ˇ u )( t, x ) . We then write ˇ ρ ( t, x ) = ρ ( t, Z ˇ u ( t, x )) , ˇ u ( t, x ) = u ( t, Z ˇ u ( t, x )) , which allows us to obtain div(ˇ u )( t, x ) = d (cid:88) i,j =1 ∂ i Z j, ˇ u ( t, x ) ∂ j u i ( t, Z ˇ u ( t, x )) , (cid:16) div(ˇ u )( t, x ) = DZ t ˇ u ( t, x ) : Du ( t, Z ˇ u ( t, x )) (cid:17) ∆ˇ u i ( t, x ) = d (cid:88) j,k,(cid:96) =1 ∂ k,(cid:96) u i ( t, Z ˇ u ( t, x )) ∂ j Z k, ˇ u ( t, x ) ∂ j Z (cid:96), ˇ u ( t, x ) + d (cid:88) k =1 ∂ k u i ( t, Z ˇ u ( t, x ))∆ Z k, ˇ u ( t, x ) ,∂ i div ˇ u ( t, x ) = d (cid:88) j,k,(cid:96) =1 ∂ k,(cid:96) u j ( t, Z ˇ u ( t, x )) ∂ j Z k, ˇ u ( t, x ) ∂ i Z (cid:96), ˇ u ( t, x ) + d (cid:88) j,k =1 ∂ i,j Z k, ˇ u ( t, x ) ∂ k u j ( t, Z ˇ u ( t, x )) ,∂ i ˇ ρ ( t, x ) = ∂ j ρ ( t, Z ˇ u ( t, x )) ∂ i Z j, ˇ u ( t, x ) , (cid:16) ∇ ˇ ρ ( t, x ) = DZ ˇ u ( t, x ) t ∇ ρ ( t, Z ˇ u ( t, x )) (cid:17) . ( t, x ) ∈ [0 , T ] × T L , ( ∂ t ˇ ρ + ( u + ˇ u ) · ∇ ˇ ρ + ρ div ˇ u )( t, x ) = ( ∂ t ρ + u · ∇ ρ + ρDZ t ˇ u ( t, Y ˇ u ( t, · )) : Du )( t, Z ˇ u ( t, x ))= ( ∂ t ρ + u · ∇ ρ + ρ div u + ρ ( DZ t ˇ u ( t, Y ˇ u ( t, · )) − I ) : Du )( t, Z ˇ u ( t, x )) . Then we simply check that ˇ f (ˇ ρ, ˇ u )( t, x ) = − (cid:0) ρDZ t ˇ u ( t, Y ( t, · )) : Du (cid:1) ( t, Z ˇ u ( t, x )) . We shall therefore look for ρ satisfying the equation ∂ t ρ + u · ∇ ρ + ρ div u = χv ρ + f ρ ( ρ, u ) , where f ρ ( ρ, u ) is defined by f ρ ( ρ, u ) = − ρDZ t ˇ u ( t, Y ( t, x )) : Du − ρ ( DZ t ˇ u ( t, Y ˇ u ( t, x )) − I ) : Du ) . Similarly, for all ( t, x ) ∈ [0 , T ] × T L , ( ρ ( ∂ t ˇ u i + ( u + ˇ u ) · ∇ ˇ u i ) − µ ∆ˇ u i − ( λ + µ ) ∂ i div(ˇ u ) + p (cid:48) ( ρ ) ∂ i ˇ ρ ) ( t, Y ˇ u ( t, x ))= ( ρ ( ∂ t u i + u · ∇ u i ) − µ ∆ u i − ( λ + µ ) ∂ i div( u ) + p (cid:48) ( ρ ) ∂ i ρ ) ( t, x ) − µ d (cid:88) j,k,(cid:96) =1 ∂ k,(cid:96) u i ( ∂ j Z k, ˇ u ( t, Y ˇ u ( t, x )) − δ j,k ) ( ∂ j Z (cid:96), ˇ u ( t, Y ˇ u ( t, x )) − δ j,(cid:96) ) + d (cid:88) k =1 ∂ k u i ∆ Z k, ˇ u ( t, Y ˇ u ( t, x )) − ( λ + µ ) d (cid:88) j,k,(cid:96) =1 ∂ k,(cid:96) u j ( ∂ j Z k, ˇ u ( t, Y ˇ u ( t, x )) − δ j,k )( ∂ i Z (cid:96), ˇ u ( t, Y ˇ u ( t, x )) − δ i,(cid:96) ) − ( λ + µ ) d (cid:88) j,k =1 ∂ i,j Z k, ˇ u ( t, Y ˇ u ( t, x )) ∂ k u j + p (cid:48) ( ρ ) (cid:0) ( DZ t ˇ u ( t, Y ˇ u ( t, x )) − I ) ∇ ρ (cid:1) , where as before δ j,k is the Kronecker symbol. We then compute ˇ f u (ˇ ρ, ˇ u )( t, Y ˇ u ( t, x )) = − ρ ( ∂ t u + u · ∇ u ) + DZ ˇ u ( t, Y ˇ u ( t, x )) t ∇ ( p ( ρ + ρ ) − p (cid:48) ( ρ ) ρ ) . We are therefore led to look for u as a solution of the equation ρ ( ∂ t u + u · ∇ u ) − µ ∆ u − ( λ + µ ) ∇ div( u ) + p (cid:48) ( ρ ) ∇ ρ = χv u + f u ( ρ, u ) , where f u ( ρ, u ) is given componentwise by f i,u ( ρ, u ) = − ρ ( ∂ t u i + u · ∇ u i ) + d (cid:88) j =1 ∂ i Z j, ˇ u ( t, Y ˇ u ( t, x )) ∂ j ( p ( ρ + ρ ) − p (cid:48) ( ρ ) ρ )+ µ d (cid:88) j,k,(cid:96) =1 ∂ k,(cid:96) u i ( ∂ j Z k, ˇ u ( t, Y ˇ u ( t, x )) − δ j,k ) ( ∂ j Z (cid:96), ˇ u ( t, Y ˇ u ( t, x )) − δ j,(cid:96) ) + d (cid:88) k =1 ∂ k u i ∆ Z k, ˇ u ( t, Y ˇ u ( t, x )) + ( λ + µ ) d (cid:88) j,k,(cid:96) =1 ∂ k,(cid:96) u j ( ∂ j Z k, ˇ u ( t, Y ˇ u ( t, x )) − δ j,k )( ∂ i Z (cid:96), ˇ u ( t, Y ˇ u ( t, x )) − δ i,(cid:96) ) + ( λ + µ ) d (cid:88) j,k =1 ∂ i,j Z k, ˇ u ( t, Y ˇ u ( t, x )) ∂ k u j − p (cid:48) ( ρ ) d (cid:88) j =1 ( ∂ i Z j, ˇ u ( t, Y ˇ u ( t, x )) − δ i,j ) ∂ j ρ . eferences [1] P. Albano and D. Tataru. 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