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Universität des Saarlandes U N I V E R S I T A S S A R A V I E N I S S Fachrichtung 6.1 Mathematik Preprint Nr. 311 An approximation theorem for vector fields in BD div Michael Bildhauer, Joachim Naumann

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Universität des Saarlandes U N I V E R S I T A S S A R A V I E N I S S Fachrichtung 6.1 Mathematik Preprint Nr. 311 An approximation theorem for vector fields in BD div Michael Bildhauer, Joachim Naumann and Jörg Wolf Saarbrücken 2012 Fachrichtung 6.1 Mathematik Preprint No. 311 Universität des Saarlandes submitted: An approximation theorem for vector fields in BD div Michael Bildhauer Saarland University Department of Mathematics P.O. Box Saarbrücken Germany Joachim Naumann Humboldt University Berlin Department of Mathematics Rudower Chaussee Berlin Germany Jörg Wolf University of Magdeburg Faculty of Mathematics Universitätsplatz Magdeburg Germany Edited by FR 6.1 Mathematik Universität des Saarlandes Postfach Saarbrücken Germany Fax: WWW: Contents 1. Introduction 1 2. Minimization problems 4 3. The space BD Basic notions The space BD div An approximation theorem for BD div The safe load condition in BD div 17 Appendix. Inhomogeneous boundary data 19 References 22 Abstract. We consider the equations of slow stationary motion of a perfectly plastic fluid in a bounded domain R n n = 2 or n = 3. The proof of the existence of a weak solution to these equations leads to the minimization of a functional of linear growth on the space { } BD div = u BD : u φ dx = 0 φ W 1,n. The elements of this space are divergence free BD-vector fields with vanishing normal component of the trace in suitable sense. The main result of our paper is an approximation theorem for these vector fields by smooth, compactly supported and divergence free vector fields. This approximation theorem implies the equality between the infimum of the above mentioned functional on its natural energy space and the infimum of the extension of this functional on BD div. 1. Introduction Let R n n = 2 or n = 3 be an open bounded set with Lipschitz boundary. The slow motion of a homogeneous incompressible fluid in is modeled by the following 2010 AMS Subject Classification: 35J50, 76A05, 76D99 Keywords: power law fluids; perfectly plastic fluids; minimization problems with linear growth; bounded deformations; space BD div ; relaxation 1 system of PDEs div u = 0 in, S + P = f in, where u = u 1,..., u n velocity, S = {S ij } i,j=1,...,n deviatoric stress, P = pressure, f = external force. The full stress tensor of the fluid is then given by = S + P I. We complete 1.1, 1.2 by the assumption that the fluid adheres to the boundary of, i.e. 1.3 u = 0 on. To proceed, for vector fields u = u 1,..., u n we introduce the notation Du = 1 2 u + u rate of strain. We then consider constitutive laws of the form S = ρ Du Du 1, where ρ is a nonnegative real function defined on [0, +. For notational simplicity, in this section we write D in place of Du. I Power law model. Let ν = const 0 be fixed. For 1 p + we define S = S p by i ii 2 p + : S p := ν D p 2 D, { 0 if D = 0, 1 p 2 : S p := ν D p 2 D if D 0, and the fluid is called dilatant if 2 p +, Newtonian if p = 2, pseudo-plastic if 1 p 2. The above power law model is widely used in chemical engineering to express an elementary non-newtonian behavior of an incompressible fluid see, e.g., [10] for a 1 For A = {A ij }, B = {B ij } i, j = 1,..., n we define A : B = A ij B ij, A = A ij A ij 1/2 repeated indices imply summation over 1,..., n. Notice that D = Du = {D ij u}, D ij u = 1 2 iu j + j u i i, j = 1,..., n. 2 discussion of these fluids from the chemical engineering point of view. II Bingham plastics. Bingham, Green [9]; Bingham [8] Let g = const 0 be fixed. For ν 0, we consider the constitutive law S ν,g g if D = 0, S ν,g = ν + g D D if D 0. This constitutive law characterizes an incompressible, visco-plastic fluid, where g is its yield value and ν its viscosity. The properties of such a fluid can be possibly easier understood by considering the following equivalent formulation of 1.4: if S ν,g g, then D = 0, if S ν,g g, then D = 1 ν 1 g S ν,g S ν,g cf. also [18], [11], Chap. VI, 1. Thus, inside of the region where S ν,g g only rigid motions of the continuum are possible i.e. the continuum moves as a plug of solid. On the other hand, if S ν,g g, then the continuum moves as a Newtonian fluid with S ν,g = ν D + g i.e. g is the value of activation of the viscous flow. The formal passage to the limit p 1 in ii gives while ν 0 in 1.4 leads to S 1 = 0 if D = 0, S 1 = g D D if D 0, S 0,g g if D = 0, S 0,g = g D D if D 0. These limit cases can be viewed as motivation for the following constitutive law. III Perfectly plastic fluids. Let g = const 0 be fixed. We define S in terms of D by 1.6 S g if D = 0, S = g D D if D 0. With a slightly different notation, this constitutive law has been introduced by von Mises [24] for the first time cf. also [15], [18]. Incompressible continua which obey 1.6 are 3 also called von Mises solids. These continua cannot withstand deviatoric stresses S such that S g. We notice that 1.6 is equivalent to 1.7 { if S g, then D = 0, if S = g, then λ 0 : D = λs cf., e.g., [11], Chap VI, 1, Remark Minimization problems We start from the above power law model. Given any vector field u = u 1,..., u n in, we define S p = S p Du according to i resp. ii above. Inserting S p in place of S into 1.2 we obtain 1.2 ν Du p 2 Du + P = f. If 1 p 2, this system of PDEs has to be considered within the set {x : Dux 0}. From now on we continue our discussion for any space dimension n 2. We then consider the weak formulation of 1.1, 1.2, 1.3 and introduce the equivalent minimization problem. For notational simplicity, we restrict our discussion to the case 1 p n. The passage to the limit p 1 leads in a natural way to the minimization problem with 1.2 for the case p = 1 cf. also 1.6 above. Let W 1,r denote the usual Sobolev space. We set W 1,r 0 := { u W 1,r : u = 0 a.e. on and 2 { } W 1,r 0,div := u W 1,r 0 : div u = 0 a.e. in. We introduce the following Definition 2.1 Let t = np/n p and f L t 3 1 p n. Then u W 1,p 0,div is called a weak solution to 1.1, 1.2 with S = S p, 1.3 if ν Du p 2 Du : Dv dx = f v dx v W 1,p 0,div. 2 By bold capitals we denote spaces of functions with values in R n, i.e. L p = L p ; R n etc. 3 By t = t/t 1 we denote the conjugate of 1 t . 4 Cf. footnote 1 for the notation Du : Dv. } 4 The theory of monotone operators from a reflexive Banach space into its dual implies that for every f L np/n p there exists exactly one solution u W 1,p 0,div to 2.1. Remark 2.1 Let u W 1,p 0,div satisfy 2.1. Then there exists P Lp/p 1 such that P dx = 0, ν Du p 2 Du : Dw dx = f w dx + P div w dx w W 1,p 0. This follows from [13], Chap. III, or [20], Chap. II. As above, let t = np/n p and f L t 1 p n. Given v W 1,p 0,div, we define F p v := ν Dv p dx f v dx. p This functional is continuous and strictly convex. Korn s inequality implies that F p is coercive. Let us consider the problem P p minimize F p over W 1,p 0,div. By standard arguments, one readily proves the existence and uniqueness of a minimizer u p W 1,p 0,div for F p. Moreover, the following equivalence is valid: u p W 1,p 0,div fulfills 2.1 F pu p = min F p v. v W 1,p 0,div Now let us consider the minimum problem P p for the case p = 1. inequality fails in W 1,1 0 we introduce the space 5 Since Korn s LD := { u L 1 : D ij u L 1, i, j = 1,..., n }. It is easily seen that LD is a Banach space with respect to the norm n u LD := u L 1 + D ij u L 1. i,j=1 5 Cf. footnote 1. 5 Clearly, W 1,1 LD proper inclusion. We notice that the embedding theorems and the trace theorem for W 1,1 continue to hold for LD see [23], pp , , for details. Next we define LD 0,div := { u LD : div u = 0 a.e. in, u = 0 a.e. on }. This space is the natural energy space for the functional F 1. Let ν = const 0, and let f L n. For v LD 0,div we define 6 Fv F 1 v = ν We consider the minimization problem Dv dx P minimize F over LD 0,div. Since F0 = 0, the following two alternatives hold for P: 1 o inf Fv = 0 ; v LD 0,div 2 o inf Fv =. v LD 0,div Clearly, 1 o is equivalent to 2.2 f v dx ν f v dx. Dv dx v LD 0,div. This inequality represents a safe load condition on the pair ν, f cf. [23] for details within the field of perfectly plastic solids. If 2.2 is satisfied, then a concept of weak solution to 1.1, 1.2 with S as in 1.6, cf. Section 1., 1.3 can be introduced. A detailed discussion will appear in [7] and [17]. On the other hand, 2 o is equivalent to ṽ LD 0,div : Fṽ 0. Remark 2.2 Let ν = const 0 and let f L n satisfy 2.2. Let u p W 1,p 0,div 1 p n/n 1 verify 2.1. Choosing v = u p in 2.1 and applying Young s inequality we obtain for all 1 p n/n Fu p ν meas F p u p = p p 1 f u p dx. 6 Here, the constant ν represents the yield value of the perfectly plastic fluid under consideration cf. the constitutive laws 1.4 and 1.6 of Section 1., where this value was denoted by g. 6 Hence ν and in conclusion 2.4 Du p dx ν 1 1 meas p Moreover, from 2.3 and 2.4 it follows that 1 p Du p dx meas. f u p dx ν Du p dx p 3. The space BD 3.1 Basic notions Definition of BD. lim Fu p = lim F p u p = 0. p 1 p 1 Let R n be an open set. We introduce the following notations: B := σ-algebra of all sets A = B, B R n Borel, M := set of all signed measures m defined on B such that m +. Next, given u = u 1,..., u n L 1 loc, we identify u i with a distribution on and denote by j u i its partial derivative with respect to x j in the sense of distributions. Then we consider the distributions and introduce the following D ij u := 1 2 iu j + j u i, i, j = 1,..., n, Definition 3.1 The space BD is given by { BD := u L 1 : i, j {1,..., n} µ ij M s.t. D ij u, φ = φ dµ ij φ C c Clearly, the space LD can be identified with a subspace of BD, namely, for u LD there holds D ij u, φ = D ij uφ dx φ Cc, 7 }. where µ ij A = A D ij u dx A B. The elements of BD are called vector fields of bounded deformation. This space has been introduced in [21], [22] for the study of mathematical problems in the theory of plasticity of solids. It has been also introduced in [16]. We refer the reader to [2], [12] and [23] for the discussion of the basics of BD. Fine properties of the elements of this space have been investigated in [1], [14]. Our motivation for the use of the space BD is the study of the limit case p = 1 of a power law fluid cf. Sections 1. and 2. as well as the vanishing viscosity limit of a Bingham plastic see [7] and [17], respectively, where we recall that in our discussion the vector field u represents the velocity of an incompressible fluid. Therefore, in Section 3.2 we consider the subspace of those u BD such that div u = 0 and u n = 0 on in a sense to be specified. Let u BD and let µ ij M i, j = 1,..., n be as in Definition 3.1. We define µ µ 1n 3.1 µ :=... µ n1... µ nn Then µ is an R n2 -valued measure on B 7. For A B we set { µ A := µa k R n 2 : A k B pairwise disjoint, A = 3.2 k=1 = total variation of µ. It is well known that µ is a finite measure on B. } A k k=1 7 For sequences a ij,k k N R we define lim k a 11,k... a 1n,k.. a n1,k... a nn,k = a a 1n.. a n1... a nn in the sense of the Euclidian norm in R n2. 8 Equivalent characterization of BD. Let u L 1 loc. For open sets A we define { } 1 Du A := sup u i j φ ij + u j i φ ij dx : φ ij Cc 1 A, φ 1 in A. 2 It is easily seen that for every u LD 3.3 Du = By routine arguments, we obtain Proposition for every u BD there holds A Du dx. 1 BD = {u L 1 : Du + }; µ A = Du A µ and µ as in 3.1 and 3.2, respectively. Some basic results. From the definition of Du one easily deduces A open 3.4 if u k u in L 1 as k, then Du lim inf k Du k. Clearly, BD is a Banach space with respect to the norm u BD := u L 1 + Du. We present some results which will be used below. Proposition 3.2 Let R n be a bounded domain with Lipschitz boundary. Then 1 BD L p compactly if 1 p n/n 1, continuously if p = n/n 1. 2 Existence of a trace There exits a linear mapping : BD L 1 such that and u = u u C BD 3.5 u dh n 1 u BD. 8 8 Here H n 1 denotes the n 1-dimensional Hausdorff measure. A detailed discussion of measure and integration on k-dimensional Lipschitz-manifolds in R n 1 k n can be found in: Naumann, J.; Simader, C.G., Measure and integration on Lipschitz-manifolds. Preprint , Inst. f. Math. Humboldt-Univ. Berlin Available at: 9 3 Let B R n be an open ball such that B. For u BDB define u + := + u = trace of u on, u := u B\ = trace of u B\ on. Then 3.6 Du = where u + u dh n 1, ξ = {τ ij ξ} 1 i,j n, τ ij ξ = 1 2 ξ in j + ξ j n i, ξ = ξ 1,..., ξ n R n and where n = n 1,..., n n denotes the unit outward normal along. The proofs of these results can be found in [2], [12], [23]. We notice that besides 3.5, the trace mapping obeys the following continuity property: cf. [23], pp The space BD div for every sequence u k k N BD such that u k u in L 1, Du k Du as k there holds u k u in L 1 as k Let be of class C 1. Observing the embedding BD L n/n 1 cf. Proposition 3.2, Proposition 3.1, we define { } BD div := u BD : u φ dx = 0 φ W 1,n. Of course, u BD div implies 3.7 u ζ dx = 0 ζ Cc, i.e. div u = 0 in the sense of distributions in. We notice that 3.7 is, however, not sufficient to prove 4.8 below. To get an information on the normal component of u BD div along the boundary of we consider the space W q := {u L q : div u L q }, 1 q +. 10 Clearly, Wq is a Banach space with respect to the norm u Wq := u L q + div u L q. From [13], pp , or [19], Theorem 5.3, it follows that there exists a linear continuous and surjective mapping γ n : Wq W 1/q, q such that γ n u = u n u C 1, where n = n 1,..., n n again denotes the unit outward normal along for the case W 2 see also [20], p. 83 Lipschitz, and [23], pp of class C 2. Moreover, for all u W q and all φ W 1,q there holds the generalized Gauß-Green formula u φ dx + div uφ dx = γ n u, φ, where the bracket on the right-hand side denotes the dual pairing between γ n u W 1/q, q and with a slight abuse of notation the trace of φ cf. [19], Theorem 5.3. Now, the above mentioned embedding of BD gives BD div W n/n 1. From 3.7 and the generalized Gauß-Green formula it follows that u BD div verifies 3.8 γ n u, φ = 0 φ W 1,n. 9 Conversely, if u BD verifies 3.7 and 3.8 then u is of class BD div. We thus obtain the following equivalent definition BD div = {u BD : u satisfies 3.7 and 3.8}. 4. An approximation theorem for BD div The main result of our paper is the following 9 The surjectivity of the trace mapping of W 1,n implies that 3.8 is equivalent to γ n u, χ = 0 χ W n 1/n,n. 11 Theorem 4.1 Let R n be a bounded star-shaped domain with Lipschitz boundary. Then, for every u BD div there exists a sequence u k k N C c such that div u k = 0 in and k N, u k u in L n/n 1 as k, Du k dx Du + u dh n 1 as k. We notice that density theorems for BD are proved in [3], [12] and [23]. Before turning to the proof we make some Preliminaries. Let R n be a bounded domain which is star-shaped with respect to a point x 0 without loss of generality we may assume that x 0 = 0. It follows 1 if t t 1, then 1 t 1 t, 2 if is any sequence of reals such that +1 1 and lim k = 1, then 1 =. k=1 Next, for u L p 1 p +, define ũx := { ux for a.e. x, 0 for a.e. x R n \, ũ t x := ũtx for t 1, for a.e. x R n. Then ũ t x = 0 for a.e. x R n \ 1. If t is any sequence of reals such that +1 1 and lim k = 1, then by 2 ux p dx 0 as k \ and therefore 4.4 ũ tk u p dx = as k. 1 1 u x ux p dx + \ 1 ux p dx 0 12 Let ω C R n be a fixed function such that 0 ωx const, ωx = ω x x R n, suppω B 1 0, ω dx = 1. B 1 0 For ρ 0 and x R n define ω ρ x := 1 x ρ ω, n ρ ũ t ρ x := ω ρ x yũ t y dy = R n B ρ x ω ρ x yuty dy. 1 t Clearly, ũ t ρ C R n. Set d t := 1 2 dist 1 t, recall t 1. Then, for every 0 ρ d t and x with distx, d t we have 1 t B ρ x =. Thus 4.5 ũ t ρ x = 0 0 ρ d t, x with distx, d t. Now, let u L p 1 p + satisfy 4.6 u φ dx = 0 φ C R n. As above, let t 1. The substitution ξ = tx x R n implies ξ iff x 1. Next, for t any φ C R n, define ξ ψξ := φ, ξ R n. t Observing the definition of BD div, from 4.6 it follows 4.7 ũ t φ dx = utx φx dx = t 1 n uξ ψξ dξ = 0. 1 t Moreover, we consider the mollification ũ t ρ componentwise. It follows 4.8 divũ t ρ x = 0 0 ρ d t, x. 13 Indeed, the function φ: y φy = ω ρ x y is admissible in 4.7 notice that suppφ need not be included in. We find divũ t ρ x = div x ωρ x yũ t y dy R n = x ω ρ x y ũ t y dy R n = y ω ρ x y ũ t y dy R n = 0. Proof of the theorem. To begin with, we fix an open ball B R n such that B. Let u BD div. We define ũ and ũ t t 1 as above and consider the mollification ũ t ρ componentwise. Then we fix a sequence k N of reals such that +1 1 and lim k = 1. Since u L n/n 1 we obtain ũ tk ũ n/n 1 dx = ũ tk u n/n 1 dx 0 as k c.f 4.4. Set B d tk := dist,, k N. Clearly lim k d tk = 0. Then we take ρ k such that ρ k d tk, ũ tk ũ tk ρk L n/n 1 B 1 k, k N. Define We obtain ũ k x := ũ tk ρk x, x R n, u k := ũ k, k N. u k C c by 4.5, div u k = 0 in by 4.8, ũ k ũ in L n/n 1 B as k. Thus, as shown in , the sequence u k k N satisfies 4.1 and 4.2. It remains to prove 4.3. To this end, we show two inequalities. 14 Inequality 1. Since u k C c we have Du k dx = B Dũ k dx = Dũ k B cf Combining 4.12 and 3.4, 3.6 B in place of one finds lim inf Du k dx = lim inf Dũ k B k k 4.13 Inequality 2. Fix δ 0 such that Then Dũ B = Du + Du = Du + u dh n 1. δ := {x R n : distx, δ} B. B ρ x = for all 0 ρ δ/3 and all x B\ such that distx, 2δ/3. Since 0 and lim k = 1, there exists k 0 N such that for all k k d tk δ 3 and such that for all k k 0 x 4.15 B ρ = 0 ρ δ 3, x B\ such that distx, 2δ 3. Let φ ij Cc 1 i, j = 1,..., n satisfy φ 2 = n i,j=1 φ2 ij 1 in. For ρ 0 and x R n consider φ ij ρ x := ω ρ x yφ ij y dy = ω ρ x yφ ij y dy. R n B ρ x suppφ ij Then 4.16 n i,j= 1 φ ij ρ x 2 1 x R n, 15 and for i, j = 1,..., n 4.17 φ ij ρ x = 0 k k 0, 0 ρ δ 3, x B\ such that distx, 2δ 3. Now, for every k k 0 and i, j = 1,..., n we have [ uki j φ ij + u kj i φ ij dx = ω ρk x y j φ ij x dx ũ tk i y R n ] + ω ρk x y i φ ij x dx ũ tk j y dy [ j = φ ij ρk yũi y + i φ ij ρk yũj y] dy, R n where we used ω ρ x y = ω ρ y x and where an integration by parts as well as a change of integration and partial differentiation was made. Hence, recalling ũ y = 0 for all y R n \ 1, we obtain uki j φ ij + u kj i φ ij dx 4.18 = 1 = 1 t n k = 1 t n k To proceed, for z B we define B [u i y j φ ij ρk y + uj y i φ ij ρk y ] dy [ u i z j φ ij ρk z + u j z i φ ij ρk z ] dz [ũi z j φ ij ρk z + ũ j z i φ ij ρk z ] dz. ζ k ij z := φ ij ρk z = B ρk z suppφ ij k k 0, i, j = 1,..., n. It follows n ζ k rs z 2 1 z B, r,s=1 z ω ρk ξ φ ij ξ dξ and for i, j = 1,..., n recall that 0 ρ k d tk δ/3, cf. 4.9, 4.14; observe 4.16 and 4.15, 4.17 ζ k 2δ ij z = 0 z B\ such that distz, 3. 16 Thus, ζ k ij Cc 1 B. Finally, by the definition of ζ k ij, i φ ij ρk z = i ζ k ij z z B. Inserting this into the integral on the right-hand side of 4.18 we find for every i, j = 1,...,n and for all k k uki j φ ij + u kj i φ ij dx = t 1 n k 2 B t 1 n k sup Dũ B. ũi j ζ k ij + ũ j i ζ k ij dx { 1 ũi j η ij + ũ j i η ij dx : 2 B } η ij C 1 c B, η 1 in Thus 4.19 lim sup k Du k dx = lim sup Du k k Dũ B = Du + u dh n 1. Combining 4.13 and 4.19 we obtain The safe load condition in BD div Let ν = const 0 and let f L n. We define Gv := ν Dv + v dh n 1 f v dx, v BD div. This functional represents the relaxation of F with respect to BD div cf. [4], pp , ; [5], pp and [6]. Remark 5.1 To get more insight into the role of the boundary integral in the definition of Gv, let us define ξ t := ξ ξ nn, ξ R n 17 n = n 1,..., n n denoting the unit outward normal along. Then, for every v BD div C we have v = v t. Thus, by Proposition 3.2, 2, for every v BD d
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