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1 Technische Universität Berlin Institut für Mathematik Regularity of the adjoint state for the instationary Navier-Stokes equations Arnd Rösch, Daniel Wachsmuth Preprint 1-4 Preprint-Reihe des Instituts für Mathematik Technische Universität Berlin Report 1-4 April 4

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3 Adjoint state regularity 1 Regularity of the adjoint state for the instationary Navier-Stokes equations 1 Arnd Rösch, Daniel Wachsmuth 3 Abstract. In this article, we are considering imbeddings of abstract functions in spaces of functions being continuous in time. A family of functions depending on certain parameters is discussed in detail. In particular, this example shows that such functions do not belong to the space C([, T ], H). In the second part, we investigate an optimal control problem for the instationary Navier-Stokes equation. We will answer the question, in which sense the initial value problem for the adjoint equation can be solved. Key words. Vector-valued functions, imbeddings, optimal control, Navier-Stokes equations, regularity of adjoints AMS subject classifications. Primary 46E4, Secondary 49N6 1. Introduction. In this paper, we will study the regularity of abstract functions. The discussed properties are heavily connected to the optimal control of instationary Navier-Stokes equations. Here, the gradient of a given objective functional is evaluated by means of an adjoint state. The adjoint state is itself the solution of an evolution equation. The discussion of abstract functions in the first part of the paper will reflect important properties of the adjoint state. The aim of the present article is two-folded. At first, we want to shed light on imbeddings of abstract functions in spaces of continuous functions. We have to refer to the mostly classical results due to Lions, [4]. Given a Gelfand triple V H V, the space { W (, T ) = y L d (, T ; V ) : dt y L (, T ; V ) is continuously imbedded in C([, T ], H). In a recent research paper, the question of compact imbeddings is considered, []. However, to the knowledge of the authors there are no further results in the literature generalizing the result of Lions substantially except the following one in the book of Dautray and Lions [4, XVIII.3.5, p. 51]. They wrote, that it suffices to require d dt y L1 (, T ; V ) to get the continuity y C([, T ], H). Since the adjoint state of the Navier-Stokes equation does not belong to W (, T ) in general, this result in [4] is used in different papers concerning the optimal control of the instationary Navier-Stokes equation, see [8, 1, 11, 14]. In this paper, we will show that this more general imbedding result cannot be true. To this aim we discuss a family of functions depending on certain parameters in detail. Nevertheless, the authors want to point out that in our opinion the incorrectness of the imbedding result in [4] does not influence the main results in the mentioned papers [8, 1, 11, 14]. 1 This work was partially supported by DFG SFB 557 Control of complex turbulent shear flows at TU Berlin. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, A-44 Linz, Austria. 3 Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, D-163 Berlin, Germany.

4 Arnd Rösch, Daniel Wachsmuth These examples shows also that the result of Amann [] is really sharp. The second part of the article deals with the adjoint state connected with an optimal control problem for the instationary Navier-Stokes equations. For certain regularity of the data it belongs to the space W (, 4/3; V, V ) := { y L (, T ; V ) : d dt y L4/3 (, T ; V ). As already mentioned, one cannot expect that this space is imbedded in C([, T ], H). Naturally, there arises the question: is there an imbedding of W (, 4/3; V, V ) in C([, T ], X), where X is a space of weaker topology than H? The article is organized as follows: In Section, we construct families of functions and study regularity properties. In the second part, Section 3, we give a brief overview of the theory of optimal control for instationary Navier-Stokes equations. Finally, we present a regularity result of the adjoint state. A last example shows that this regularity cannot be improved by imbedding arguments.. Counterexamples. Here, we will deal with imbeddings of abstract functions in spaces of continuous functions. At first, we state the most classical result in this field. Let V H V be a Gelfand triple. Theorem.1. The space W (p, q; V, V ) := { y L p (, T ; V ) : is continuously imbedded in C([, T ], H) if 1/p + 1/q 1. d dt y Lq (, T ; V ) For the proof in the case p = q = we refer to [4]. It can be easily adapted to the case 1/p + 1/q = 1, cf. [5]. In the sequel, we will construct several functions which are in W (p, q; V, V ), where p, q do not meet the assumptions of the previous theorem. We prove that in the case 1/p+1/q > 1 there is no imbedding W (p, q; V, V ) C([, T ], H). We are also looking for a positive result of the kind: for given p, q the space W (p, q; V, V ) is continuously imbedded in C([, T ], X), where X is a space with weaker topology than H. Consider the following example: Let Ω = [, 1] and T >. Set V := H 1 (Ω), H = L (Ω), and V induced by the H-scalar product such that V H V forms a Gelfand-triple. Define a function f α,k over Ω [, T ] by f α,k (x, t) = n 1/ e ( nαt) sin n k πx, (.1) where k is a natural number. Lemma.. The function f α,k given by (.1) has the following properties: (i) f α,k C([, T ]; H) for α >, (ii) f α,k L p (, T ; V ) for p < α k+1/, (iii) d dt f α,k L q (, T ; V ) for q < α α+1/ k. Proof. Set v n (x) := sin n k πx. At first, observe that the functions v n are orthogonal with respect to the H as well as to the V -scalar product. It holds v n H = 1 and v n V = 1 n k π.

5 Adjoint state regularity 3 Now, we want to derive the V -norm of v n. Let φ V be a test function. After partial integration, we find using the Cauchy-Schwarz inequality v n, φ V,V = 1 sin(n k πx) φ(x) dx = 1 n k π 1 cos(n k πx) φ (x) dx 1 nk π φ V. This allows us to conclude v n V 1 nk π. Setting φ(x) := v n (x), we obtain v n V = 1 π n k. (i) Let f N α,k be the function defined by the finite series fα,k(x, N t) = n 1/ e ( nαt) sin n k πx. For t =, we obtain f N α,k(x, ) = Consequently, we find for the L -norm f N α,k(, ) H = n 1/ sin n k πx. 1 T sin n k πx dx = n 1 n. This series grows unboundedly for N. Therefore, f α,k () cannot belong to H, which implies that f α,k is not in C([, T ]; H). (ii) Again, we consider the finite series fα,k N. We want to estimate the Lp (, T ; V )- norm of fα,k N. We derive first using Hölders inequality ( ( T N ) p fα,k N L p (V ) = n 1/ e ( nαt) v n V dt )1/p ( ) T ) 1/p p (n 1/ e ( nαt) v n V dt. The integral on the right-hand side can be computed by ) p ( ) p π (n 1/ e ( nαt) v n V dt = n p(k 1/) e ( pnαt) dt 1 ( ) p π n p(k 1/) α. p

6 4 Arnd Rösch, Daniel Wachsmuth Hence, we arrive at the estimate f N α,k Lp (V ) This series will be finite for N, if π p p n k 1/ α/p. k 1/ α/p < 1 or equivalently p < α k + 1/ holds. (iii) Similarly, the L q (, T ; V )-estimate can be proven. We shall begin with ( ( d dt f T N q )1/q α,k N = n α 1/ e ( nαt) v n V ) dt Lq (V ) ( ) T ( 1/q q n α 1/ e ( nαt) v n V ) dt. We find for the time integral ( ) ( ) q q 1 T n α 1/ e ( nαt) v n V dt = n q(α k 1/) π 1 q ( 1 π ) q n q(α k 1/) α. e ( qnα t) dt This implies d dt f α,k N 1 Lq (V ) q q π n α k 1/ α/q. The series on the right hand side is uniformly bounded for α 1/ k α/q < 1, which is equivalent to q < α α + 1/ k, and completes the proof. Remark.3. For α = p(k + 1/) + ε with some fixed ε >, we find that (i) and (ii) are automatically fulfilled. Moreover, we obtain from (iii) q < pk + p/ + ε (p 1)k + p/ + ε + 1/.

7 Adjoint state regularity 5 If k is sufficiently large, the value of q is arbitrary close to p = p p 1 = p This shows, that the proposition of Theorem.1 is sharp. Further, we can conclude that there is no imbedding of W (, 1; V, V ) in C([, T ], H) as stated in [4, p. 51]. In the following, we will denote by H s (Ω) for 1 s the Sobolev-Slobodeckij spaces of fractional order. We have H 1 = V and H = H with the notation already introduced. Corollary.4. Let us consider the function This function satisfies f α,k,l (x, t) = n l e ( nαt) sin n k πx. (i) f α,k,l C([, T ]; H) for α > and l 1, (ii) f α,k,l L p (, T ; V ) for p < α k l+1, (iii) d dt f α,k,l L q (, T ; V α ) for q < α k l+1, (iv) f α,k,l C([, T ]; H s ) for s > 1 l k and l < 1, (v) f α,k,l C([, T ]; H s ) for s < 1/ l k and l < 1. Proof. The points (i) (iii) can be shown similarly to Lemma.. To prove (iv) and (v) we use interpolation theory. Given v H, we have v H s with θ 1 = s. For a function v V, we obtain c 1 v θ1 V v 1 θ1 H with θ = 1 1+s. Hence, for v n(x) = sin n k πx we find v H c v θ H s v 1 θ V (.) v n H s c 1 v n s V v n 1 s H = c 1 π s n sk =: c 3 n sk. (.3) Let us denote by fα,k,l N the finite series f N α,k,l(x, t) = n l e ( nαt) sin n k πx. Now, we are ready to prove (iv). We want to show that f α,k,l is in C([, T ], H s ). To this aim, let t [, T ] be given. Let the pre-requisite s > 1 l k be fulfilled. We derive using (.3) f α,k,l (t) H s = n l e ( nαt) v n H s n l v n H s c 3 n l sk.

8 6 Arnd Rösch, Daniel Wachsmuth By assumption, we have l sk < 1 for the exponent. Therefore, we obtain uniform convergence of f α,k,l (t). This uniform convergence and the fact n l e ( nα t) v n C([, T ]; H s ) for each n and fixed s ensures the continuity of the abstract function. (v) Starting from (.), we get fα,k,l N H s c s 1 fα,k,l N s+1 H f N α,k,l s V, since obviously fα,k,l N holds. For sufficiently large N and l < 1/, we estimate the H-Norm of fα,k,l N by f N α,k,l H = 1 n l 1 N 1 x l dx = l (N 1 l 1) l N 1 l. Note, that the estimate is also correct for negative values of l. Similarly, we have to derive a bound of the V -norm. obtain ( fα,k,l N V = π n l n k π 1 + = π ( 1 + N For sufficiently large N, we (x + 1) k l dx 1 ( (N + 1) 1+k l 1 )) 1 + k l π 1 + k l N 1+k l. Since by assumption l < 1/, it holds 1 + k l > for k >. Altogether, we found f N α,k,l H s cn s+1 (1 l) s (1+k l) = cn 1/ sk l, which tends to infinity if s < 1/ l k. Hence, f α,k,l cannot be a function continuous in time with values in such H s spaces. This examples shows that under certain conditions it may happens that abstract functions are continuous with value in some space H s but discontinuous with values in spaces of integrable functions. 3. Application to an optimal control problem. In this section, we will consider optimal control of the instationary Navier-Stokes equations. As model problem serves the minimization of the quadratic objective functional min J(y, u) = α T Ω y(x, T ) y T (x) dx + α Q y(x, t) y Q (x, t) dxdt Q + α R curl y(x, t) dxdt + γ u(x, t) dxdt (3.1) subject to the instationary Navier-Stokes equations Q Q ) y t ν y + (y )y + p = u + f in Q, div y = in Q, y = on Γ, y() = y in Ω. (3.)

9 Adjoint state regularity 7 The control has to be in a set of admissible controls, u U ad, given by U ad = {u L (Q) : u a,i (x, t) u i (x, t) u b,i (x, t) a.e. on Q, i = 1,. Here, Ω is an open bounded subset of R with C -boundary Γ, such that Ω is locally on one side of Γ, and Q is defined by Q = (, T ) Ω. Further, functions y T L (Ω), y Q L (Q), and y H L (Ω) are given. The source term f is required to belong to L (, T ; V ). The parameters γ and ν are positive real numbers. The constraints u a, u b are required to be in L (Q) with u a,i (x, t) u b,i (x, t) a.e. on Q, i = 1, Notations and preliminary results. First, we introduce some notations and provide some results that we need later on. To begin with, we define the solenoidal spaces H := {v L (Ω) : div v =, V := {v H 1 (Ω) : div v =. These spaces are Hilbert spaces with their norms denoted by H respectively V and scalar products (, ) H respectively (, ) V. The dual of V with respect to the scalar product of H we denote by V with the duality pairing, V,V. We shall work in the standard space of abstract functions from [, T ] to a real Banach space X, L p (, T ; X), endowed with its natural norm, ( ) 1/p T y Lp (X) := y Lp (,T ;X) = y(t) p X dt y L (X) := vrai max y(t) X. t (,T ) 1 p <, In the sequel, we will identify the spaces L p (, T ; L p (Ω) ) and L p (Q) for 1 < p <, and denote their norm by u p := u L p (Q). The usual L (Q) -scalar product we denote by (, ) Q to avoid ambiguity. In all what follows, stands for norms of abstract functions, while denotes norms of stationary spaces like H and V. To deal with the time derivative in (3.), we introduce the common spaces of functions y whose time derivatives y t exist as abstract functions, W α (, T ; V ) := {y L (, T ; V ) : y t L α (, T ; V ), W (, T ) := W (, T ; V ), where 1 α <. Endowed with the norm y W α := y W α (,T ;V ) = y L (V ) + y t L α (V ), these spaces are Banach spaces, respectively Hilbert spaces in the case of W (, T ). Every function of W (, T ) is, up to changes on sets of zero measure, equivalent to a function of C([, T ], H), and the imbedding W (, T ) C([, T ], H) is continuous, cf. [1, 1]. As we saw above, there is no imbedding W α (, T ; V ) in C([, T ], H) for α <. However, the space W (, T ) enjoys the following imbedding property: Lemma 3.1. The space W (, T ) is continuously imbedded in L 4 (Q).

10 8 Arnd Rösch, Daniel Wachsmuth Proof. For v V, the interpolation inequality v 4 c v 1/ H v W (, T ) be given. Then, we can readily estimate v 1/ V holds, cf. [13]. Let v 4 4 v(t) 4 4 dt c v H v V dt c v L (H) v L (V ) c v 4 W, which proves the claim. For convenience, we define the trilinear form b : V V V R by together with b(u, v, w) = ((u )v, w) = b Q (u, v, w) = Ω i,j=1 u i v j x i w j dx b(u(t), v(t), w(t)) dt. An important property of b is that for u V and sufficient regular v, w it holds b(u, v, w) = b(u, w, v). (3.3) There are several estimates of b respectively b Q available. We mention only the following one which we will need in the sequel. For detailed discussions consult [3, 13, 15]. Lemma 3.. Let u, w L 4 (Q) and v L (, T ; V ) be given. Then there is a constant c > independently of u, v, w such that holds. b Q (u, v, w) c u 4 v L (,T ;V ) w 4 To specify the problem setting, we introduce a linear operator A : L (, T ; V ) L (, T ; V ) by and a nonlinear operator B by (Ay)(t), v(t) V,V dt := (y(t), v(t)) V dt, ( ) T B(y) (t), v(t) V dt := b(y(t), y(t), v(t))dt.,v As a conclusion of Lemma 3.1, eq. (3.3), and Lemma 3., we find that B is continuous as operator from W (, T ) to L (, T ; V ). Testing system (3.) by divergence-free functions, one obtains the solenoidal form of the Navier-Stokes equations y t + νay + B(y) = u + f y() = y, where the first equation has to be understood in the sense of L (, T ; V ). It is wellknown that for all initial values y H and source terms u, f L (, T ; V ) there exists a unique weak solution y W (, T ) of (3.), cf. [3, 13].

11 Adjoint state regularity 9 We introduce the linearized equation by y t + νay + B (ȳ)y = u y() = y. (3.4) Here, ȳ is a given state y W (, T ). This equation is solvable for all u L (, T ; V ) and y H. Its unique solution y is in W (, T ). 3.. Optimality condition. Now, we return to the optimization problem (3.1). We will call a control u U ad locally optimal, if there exists ρ > such that J(ȳ, ū) J(y, u) for all u U ad with u ū ρ. Here, ȳ and y denote the states associated with ū and u, respectively. The first-order necessary condition for local optimality is stated in the next theorem. Theorem 3.3. Let ū be a locally optimal control with associated state ȳ = y(ū). Then there exists a unique solution λ W 4/3 (, T ; V ) of the adjoint equation λ t + νa λ + B (ȳ) λ = αq (ȳ y Q ) + α R curl curl ȳ λ(t ) = α T (ȳ(t ) y T ). (3.5) Moreover, the variational inequality is satisfied. (γū + λ, u ū) L (Q) u U ad (3.6) Proofs can be found in [6, 7, 14]. The regularity of λ is proven in [1] for homogeneous initial conditions, λ(t ) =. The adjoint state λ is the solution of a linearized adjoint equation backward in time. So it is natural, to look for its dependance of the given data. For convenience, we denote by g the right-hand side of (3.5), and by λ T the initial value α T (ȳ(t ) y T ). Theorem 3.4. Let λ T H, g L (, T ; V ), and ȳ L (, T ; V ) L (, T ; H) be given. Then there exists a unique weak solution λ of (3.5) satisfying λ W 4/3 (, T ). The mapping (g, λ T ) λ is continuous in the mentioned spaces. Proof. At first, denote by w the weak solution of w t + νaw = g w(t ) = λ T. Its existence and regularity w W (, T ) follows from solvability of the instationary Stokes-equation, cf. [13]. Moreover, we get the continuity estimate Further, let z be the weak solution of w W c { g L (V ) + λ T H. (3.7) z t + νaz + B (ȳ) z = B (ȳ) w z(t ) =.

12 1 Arnd Rösch, Daniel Wachsmuth Since ȳ and w are in W (, T ), we get B (ȳ) w L 4/3 (, T ; V ) W (, T ) as follows. We write for v W (, T ) [B (ȳ) w]v = b(ȳ, v, w) + b(v, ȳ, w) dt c { ȳ 4 v L (V ) w 4 + v 4 ȳ L (V ) w 4. By Lemma 3., we conclude [B (ȳ) w]v c ȳ W w W { v L (V ) + v 4. Since v 4 c v W, we get B (ȳ) w W (, T ). The space V is continuously imbedded in L 4 (Ω), which allows us to conclude B (ȳ) w L 4/3 (, T ; V ). Therefore, we arrive at B (ȳ) w W + B (ȳ) w L 4/3 (V ) c ȳ W w W. (3.8) Now, Proposition..1 in [9] respectively Proposition.4 in [1] imply the existence of z together with the regularity z W 4/3 (, T ) and the estimate z W 4/3 c { B (ȳ) w L 4/3 (V ) + B (ȳ) w W c ȳ W w W. (3.9) We construct a solution of the inhomogeneous adjoint equation (3.5) by λ = z + w. Using (3.7) and (3.9), λ W 4/3 z W 4/3 + w W c (1 + ȳ W ) { g L (V ) + λ T H is found, and the claim is proven. Observe, that the conditions of the previous theorem requires the initial value to be in H, whereas the regularity λ W 4/3 (, T ) does not guarantee λ(t) λ T in H for t T. If the data is more regular then things are much easier. If for instance f L (Q) and y V is given together with y T V, then the state y and the adjoint λ admits the same regularity: it holds that λ belongs to a space H,1 which is continuously imbedded in C([, T ], V ), confer [9, 13] Example. In this last section, we will answering the question: can the adjoint state be represented by a continuous abstract function? Clearly, if λ t L 4/3 (, T ; V ) together with λ(t ) V hold, then it is obvious that λ is a continuous function with values in V. Nevertheless, we are looking for a sharper imbedding result. Let ε > and integer k > 3/ + 3ε be given. Set l = 1 k/3 + ε and α = 8/3k. Notice that by the definition of k and l we have l < 1/. Then the function f := f α,k,l introduced in Section, fulfills: ε 4/3k ε (i) f L p (, T ; V ) for p < 8/3k 4/3k ε = + (ii) d dt f Lq (, T ; V ) for q < 8/3k k ε = /3ε k ε (iii) f C([, T ], H s ) for s > ε k (iv) f C([, T ], H s ) for s < 1 3 1/+ε k.

13 Adjoint state regularity 11 Here, we observe that f W (, 4/3; V, V ) for all possible k and ε. Thus, it has the same regularity as the adjoint state λ. And we can say that the space W (, 4/3; V, V ) is not continuous imbedded in C([, T ], H s ) for s < 1/3. However, there is a positive result available. Theorem 3.5. The space W (p, q; V, V ) is compactly imbedded in C([, T ], H s ) for s > 1 p + 1 q p 1 q. Proof. The notation is adapted to the one used in the present article. For the proof and detailed discussions, we refer to Amann []. We combine these conclusions to Corollary 3.6. The space W (, 4/3; V, V ) is continuously imbedded in the space C([, T ], H s ) for s > 1/3. If s < 1/3 holds, then W (, 4/3; V, V ) can not be imbedded in C([, T ], H s ). REFERENCES [1] R. A. Adams. Sobolev spaces. Academic Press, San Diego, [] H. Amann. Compact embeddings of vector-valued Sobolev and Besov spaces. Glasnik Matematicki, 35: ,. [3] P. Constantin and C. Foias. Navier-Stokes equations. The University of Chicago Press, Chicago, [4] R. Dautray and J. L. Lions. Evolution problems I, volume 5 of Mathematical analysis and numerical methods for science and technology. Springer, Berlin, 199. [5] H. Gajewski, K. Gröger, and K. Zacharias. Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin, [6] M. D. Gunzburger and S. Manservisi. The velocity tracking problem for Navier-Stokes flows with bounded distributed controls. SIAM J. Contr. Optim., 37: , [7] M. D. Gunzburger and S. Manservisi. Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. SIAM J. Numer. Anal., 37: ,. [8] M. Hintermüller and M. Hinze. A SQP-semi-smooth Newton-type algorithm applied to control of the instationary Navier-Stokes system subject to control constraints. Technical Report TR3-11, Department of Computational and Applied Mathematics, Rice University, submitted. [9] M. Hinze. Optimal and instantaneous control of the instationary Navier-Stokes equations. Habilitation, TU Berlin, revised version,. [1] M. Hinze and K. Kunisch. Second order methods for optimal control of time-dependent fluid flow. SIAM J. Contr. Optim., 4:95 946, 1. [11] M. Hinze and K. Kunisch. Second order methods for boundary control of the instationary Navier-Stokes system. ZAMM, 84: , 4. [1] J. L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications, volume I. Springer, Berlin, 197. [13] R. Temam. Navier-Stokes equations. North Holland, Amsterdam, [14] F. Tröltzsch and D. Wachsmuth. Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations. Preprint 3-3, Institut für Mathematik, TU Berlin, submitted. [15] W. von Wahl. The equations of Navier-Stokes and abstract parabolic equations. Vieweg, Braunschweig, 1985.