Sandia National Labs SAND J. A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws

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1 Archive for Rational Mechanics and Analysis manuscript No. will be inserted by the editor) Q. Du M. D. Gunzburger R. B. Lehoucq K. Zhou A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws DRAFT date 15 May 2011 Received: date / Accepted: date Q. Du and K. Zhou were supported in part by the U.S. Department of Energy Office of Science under grant DE-SC M. Gunzburger was supported in part by the U.S. Department of Energy Office of Science under grant DE-SC Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy under contract DE-AC04-94AL Q. Du Department of Mathematics Penn State University Tel.: State College, PA 16802, USA Fax: qdu@math.psu.edu M. D. Gunzburger Department of Scientific Computing Florida State University Tallahassee FL , USA Tel.: Fax: gunzburg@fsu.edu R. B. Lehoucq Sandia National Laboratories P.O. Box 5800, MS 1320 Albuquerque NM , USA Tel.: Fax: rblehou@sandia.gov K. Zhou Department of Mathematics Penn State University State College, PA 16802, USA Tel.: Fax: zhou@math.psu.edu

2 2 Q. Du et al. Abstract A vector calculus for nonlocal operators is developed, including the definition of nonlocal generalized divergence, gradient, and curl operators and the derivation of their adjoint operators. Analogs of several theorems and identities of the vector calculus for differential operators are also presented. A subsequent incorporation of volume constraints gives rise to well-posed volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators. This is demonstrated via some examples. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. An application of the nonlocal calculus is to pose abstract balance laws with reduced regularity requirements. Keywords nonlocal vector calculus continuum mechanics peridynamics 1 Introduction We introduce a vector calculus for nonlocal operators that mimics the classical vector calculus for differential operators. We define, within a Hilbert space setting, generalized nonlocal representations of the divergence, gradient, and curl operators and deduce the corresponding nonlocal adjoint operators. Nonlocal analogs of the Gauss theorem and the Green s identities of the vector calculus for differential operators are also derived. The nonlocal vector calculus can be used to define nonlocal volume-constrained problems that are analogous to boundary-value problems for partial differential operators. In addition, we establish relationships between the nonlocal operators and their differential counterparts. A balance law postulates that the rate of change of an extensive quantity over any subregion of a body is given by the rate at which that quantity is produced in the subregion minus the flux out of the subregion through its boundary. Along with kinematics and constitutive relations, balance laws are a cornerstone of continuum mechanics. Difficulties arise, however, in, for example, the classical equations of continuum mechanics due to, e.g., shocks, corner singularities, and material failure, all of which are troublesome when defining an appropriate notion for the flux through the boundary of the subregion. The nonlocal vector calculus we develop has an important application to balance laws that are nonlocal in the sense that subregions not in direct contact may have a non-zero interaction. This is accomplished by defining the flux in terms of interactions between possibly disjoint regions of positive measure possibly sharing no common boundary. An important feature of the nonlocal balance laws is that the significant technical details associated with determining normal and tangential traces over the boundaries of suitable regions is obviated when volume interactions induce flux. Our nonlocal calculus, then, is an alternative to standard approaches for circumventing the technicalities associated with lack of sufficient regularity in local balance laws such as measure-theoretic generalizations of the Gauss- Green theorem see, e.g., [4, 18]) and the use of the fractional calculus see, e.g., [1,24]).

3 Nonlocal vector calculus, volume-constrained problems and balance laws 3 Preliminary attempts at a nonlocal calculus were the subject of [11, 12], which included applications to image processing 1 and steady-state diffusion, respectively. However the discussion was limited to scalar problems. In contrast, this paper extends the ideas in [11,12] to vector and tensor fields and beyond the consideration of image processing and steady-state diffusion. For example, the ideas presented here enable an abstract formulation of the balance laws of momentum and energy and for the peridynamic 2 theory for solid mechanics that parallels the vector calculus formulation of the balance laws of elasticity. The nonlocal vector calculus presented in this paper, however, is sufficiently general that we envisage application to balance laws beyond those of elasticity, e.g., to the laws of fluid mechanics and electromagnetics. The paper is organized as follows. The nonlocal vector calculus is developed in Sections 2 and 3. In Section 2, nonlocal generalizations of the differential divergence, gradient, and curl operators and of the corresponding adjoint operators are given as are nonlocal generalizations of several theorems and identities of the vector calculus for differential operators. In Section 3, we introduce the notion of nonlocal constraint operators that allow us to define nonlocal volume-constrained problems that generalize well-known boundaryvalue problems for partial differential operators. Amended versions of several of the theorems and identities of the nonlocal vector calculus are derived that account for the constraint operators. Connections between the nonlocal operators and their differential counterparts are made in Sections 4 and 5. In Section 4, we connect the two classes of operators in a distributional sense whereas, in Section 5, the connections are made between weighted integrals involving the nonlocal adjoint operators and weak representatives of their differential counterparts. The connections made in those two sections justify the use of the terminology nonlocal divergence, gradient, and curl operators to refer to the nonlocal operators we define. In Section 6, a brief review of the conventional notion of a balance law is provided after which an abstract nonlocal balance law is discussed; also, a brief discussion is given of the application of our nonlocal vector calculus to the peridynamic theory for continuum mechanics. 2 A nonlocal vector calculus We develop a nonlocal vector calculus that mimics the classical vector calculus for differential operators. The nonlocal vector calculus involves two types of functions and two types of nonlocal operators. Point functions refer to functions defined at points whereas two-point functions refer to functions defined for pairs of points. Point operators map two-point functions to point functions whereas two-point operators map point functions to two-point functions so that the nomenclature for operators refer to their ranges. Point and 1 The authors of [11] refer to [25] where a discrete nonlocal divergence and gradient are introduced within the context of machine learning. 2 Peridynamics was introduced in [20,22]; [23] reviews the peridynamic balance laws of momentum and energy and provides many citations for the peridynamic theory and its applications. See Section 6.1 for a brief discussion.

4 4 Q. Du et al. two-point operators are both nonlocal. Point operators involve integrals of two-point functions whereas two-point operators explicitly involve pairs of point functions. We now make more precise the definitions given above. To this end, for a positive integer d, let denote an open subset of R d. Points in R d are denoted by the vectors x, y, or z and the natural Cartesian basis is denoted by e 1,..., e d. Let n and k denote positive integers. Functions from or subsets of into R n k or R n or R are referred to as point functions or point mappings and are denoted by Roman letters, upper-case bold for tensors, lower-case bold for vectors, and plain-face for scalars, respectively, e.g., Ux), ux), and ux), respectively. Functions from or subsets of to R n k, or R n, or R are referred to as two-point functions or two-point mappings and are denoted by Greek letters, upper-case bold for tensors, lower-case bold for vectors, and plain-face for scalars, respectively, e.g., Ψx, y), ψx, y), and ψx, y), respectively. Symmetric and antisymmetric two-point functions satisfy, e.g., for the scalar two-point function ψx, y), we have ψx, y) = ψy, x) and ψx, y) = ψy, x), respectively. The dot or inner) product of two vectors u, v R n is denoted by u v R; the dyad or outer) product is denoted by u w R n k whenever w R k ; given a second-order tensor matrix) U R k n, the tensor-vector or matrix-vector) product is denoted by U v and is given by the vector whose components are the dot products of the corresponding rows of U with v. 3 For n = 3, the cross product of two vectors u and v is denoted by u v R 3. The Frobenius product of two second-order tensors A R n k and B R n k, denoted by A: B, is given by the sum of the component-wise product of the two tensors. The trace of B R n n, denoted by tr B ), is given by the sum of the diagonal elements of B. Inner products in L 2 ) and L 2 ) are defined in the usual manner. For example, for vector functions, we have u, v) = u v dx for ux), vx) R n µ, ν) = µ ν dydx for µx, y), νx, y) R n with analogous expressions involving the Frobenius product and the ordinary product for tensor and scalar functions, respectively. 2.1 Nonlocal point divergence, gradient, and curl operators The nonlocal point divergence, gradient, and curl operators map two-point functions to point functions and are defined as follows. These operators along with their adjoints are the building blocks of our nonlocal calculus. Definition 1 [Nonlocal point operators] Given the vector two-point function ν : R k and the antisymmetric vector two-point function 3 In matrix notation, the inner, outer, matrix-vector products are given by x y = x T y, x y = xy T, and U v = U T v.

5 Nonlocal vector calculus, volume-constrained problems and balance laws 5 α: R k, the nonlocal point divergence operator D ν ) : R is defined as D ν ) ) x) := νx, y) + νy, x) αy, x) dy for x. 1a) Given the scalar two-point function η : R and the antisymmetric vector two-point function β : R n, the nonlocal point gradient operator G η ) : R k is defined as G η ) x) := ηx, y) + ηy, x) ) βy, x) dy for x. 1b) Given the vector two-point function µ: R 3 and the antisymmetric vector two-point function γ : R 3, the nonlocal point curl operator C µ ) : R 3 is defined as C µ ) x) := γy, x) µx, y) + µy, x) ) dy for x. 1c) The nonlocal point operators D, G, and C map vectors to scalars, scalars to vectors, and vectors to vectors, respectively, as is the case for the divergence, gradient, and curl differential operators. Relationships between the nonlocal point operators and differential operators are made in Sections 4 and 5 where we demonstrate circumstances under which the nonlocal point operators are identified with the differential operators in the sense of distributions and as weak representations, respectively. For the sake of simplicity, in much of the rest of the paper, we introduce the following notation: α := αx, y) α := αy, x) ψ := ψx, y) ψ := ψy, x) u := ux) u := uy) u := ux) u := uy) and similarly for other functions Nonlocal integral theorems Definition 1 leads to integral theorems for the nonlocal vector calculus that mimic basic integral theorems of the vector calculus for differential operators;in particular, 2a) can be viewed as nonlocal Gauss theorem. The antisymmetry, with respect to x and y, of the integrands in the definitions of the nonlocal point operators 1) plays a crucial role. Theorem 1 [Nonlocal integral theorems] Let the functions ν, η, and µ and the operators D, G, and C be defined as in Definition 1. Then, we 4 For example, from 1a), we have D ν ) x) = ν + ν ) α dy.

6 6 Q. Du et al. have the following analogs of the integral theorems of the vector calculus for differential operators: D ν ) dx = 0 2a) G η ) dx = 0 2b) C µ ) dx = 0. 2c) Proof From 1a), we have that D ν ) dx = ν + ν ) α dy dx. Because α is antisymmetric, the integrand in the double integral is antisymmetric, i.e., we have that ν + ν ) α = ν + ν) α, from which it easily follows that that double integral vanishes. Thus, 2a) follows. The nonlocal integral theorems 2b) and 2c) follow in exactly the same way from 1b) and 1c), respectively. 5 The nonlocal integral theorems 2a) 2c) have the simple consequences listed in the following corollary that can be viewed as nonlocal integration by parts formulas. Corollary 1 [Nonlocal integration by parts formulas] Adopt the functions and operators appearing in Definition 1. Then, given the point functions ux): R, vx): R n, and wx): R 3, we have 6 ud ν ) dx + u u)α ) ν dydx = 0 3a) v G η ) dx + w C µ ) dx Proof Let ξ = uν; then, v v) β ) η dydx = 0 3b) γ w w) ) µ dydx = 0. 3c) ξ + ξ ) α = uν + u ν ) α = uν + ν ) α + u u)ν α so that, from 1a), we have ) Dξ) = uν + ν ) α + u u)ν α dy x. 5 Alternately, 1b) and 1c) can be derived from 1a); one simply chooses ν = ηb and ν = b µ, respectively, in 1a) where b is a constant vector; one also has to associate β and γ with α. 6 Whenever ν and α are scalar valued functions, the integration by parts formula 3a) appears in [13, Lemma 2.1].

7 Nonlocal vector calculus, volume-constrained problems and balance laws 7 Then, 2a) results in ) Dξ) dx = u ν + ν ) αdy dx + so that, by 1a) and the anti-symmetry of α, ud ν ) dx + u u )ν α dydx = 0. u u)ν α dydx = 0 A change of variables in the double integral then results in 3a). In a similar manner, 3b) and 3c) can be derived from 2a) along with 1b) and 1c), respectively. Alternately, 3b) and 3c) easily follow by setting, for an arbitrary constant vector b, ν = ηb and ν = b µ, respectively, in 3a) and also associating β and γ with α. 2.3 Nonlocal adjoint operators The adjoints of the nonlocal point operators are two-point operators and are defined as follows. Definition 2 Given a point operator L that maps two-point functions F to point functions defined over, the adjoint operator L is a two-point operator that maps point functions G to two-point functions defined over that satisfies ) G, LF ) L G), F ) = 0. 4) Here, the operators F and G may denote pairs of vector-scalar, scalar-vector, or vector-vector functions. Corollary 1 can be used to immediately determine the adjoint operators corresponding to the point operators defined in Definition 1. Corollary 2 [Nonlocal adjoint operators] Given the point function ux): R, the adjoint of D is the two-point operator given by D u ) x, y) = u u)α for x, y, 5a) where D u ) : R k. Given the point function vx): R n, the adjoint of G is the two-point operator given by G v ) x, y) = v v) β for x, y, 5b) where G v ) : R. Given the point function wx): R 3, the adjoint of C is the two-point operator given by where C w ) : R 3. C w ) x, y) = γ w w) for x, y, 5c)

8 8 Q. Du et al. Proof Let F = ν and G = u in 4). Then, 5a) immediately follows from 3a). Similarly, 5b) and 5c) are immediate consequences of 3b) and 3c), respectively, and 4). We can rewrite nonlocal integration by parts formulas in terms of the nonlocal adjoint operators given by Corollaries 1 and 2, respectively as follows: ud ν ) dx v G η ) dx w C µ ) dx D u ) ν dy dx = 0 G v ) η dy dx = 0 C w ) µ dy dx = 0. 6a) 6b) 6c) 2.4 Nonlocal Green s identities Nonlocal Green s identities can be derived from 4) by setting F = ΘL H), where L H) may be a scalar or vector or second-order tensor function in which cases Θ is a scalar or second-order tensor or fourth-order tensor function, respectively. This leads to the nonlocal Green s first identity G, LΘL H)) ) L G), ΘL H) ) = 0. 7) Suppose now that Θ is a symmetric tensor. Then, reversing the roles of G and H and subtracting the result from 7) yields the nonlocal Green s second identity G, L ΘL H) )) H, L ΘL G) )) = 0. 8) Using 7) and 8) immediately yields the following two sets of results. Corollary 3 [Nonlocal generalized) Green s first identities] Given the scalar point functions ux): R and vx): R and the two-point second-order tensor function Θx, y): R n n, then ud Θ D v) ) dx D u) Θ D v) dydx = 0. 9a) Given the vector point functions ux): R n and vx): R n and the two-scalar point function θx, y): R, then v G θg u) ) dx θg v)g u) dydx = 0. 9b) Given the vector point functions ux): R 3 and wx): R 3 and the two-point fourth-order tensor function Θx, y): R , then w C Θ : C u) ) dx C w) : Θ : C u) dydx = 0. 9c)

9 Nonlocal vector calculus, volume-constrained problems and balance laws 9 Corollary 4 [Nonlocal generalized) Green s second identities] We assume the same notation as in Corollary 3 and we also assume that the tensors Θ appearing in 9a) and 9c) are symmetric. Then, ud Θ D v) ) dx vd Θ D u) ) dx = 0 10a) v G θg u) ) dx u G θg v) ) dx = 0 10b) w C Θ : C u) ) dx u C Θ : C w) ) dx = 0. 10c) 2.5 Further observations and results about nonlocal operators In this subsection, we collect several observations and results that can be deduced from the definitions and results of Sections A nonlocal divergence operator for tensor functions A nonlocal divergence operator for tensor functions is defined by applying 1a) to each of the rows of the tensor. Definition 3 Given the tensor two-point function Ψ : R n k and the antisymmetric vector two-point function α: R k the nonlocal point divergence operator D t Ψ)x) : R n for tensors is defined as D t Ψ)x) := ) Ψ + Ψ α dy for x. 11a) A nonlocal Gauss theorem for tensor functions as well as the corresponding integration by parts formula and Green s identities can be derived and the adjoint operator Dt can defined in the same way as was done above for the nonlocal divergence operator D for vector functions, e.g., see Corollary 2. In particular, Definition 2 implies that, given the point function vx) : R n, D t v)x, y) = v v) α for x, y, 11b) where Dt v) : R n k, i.e., Dt maps a vector point function to a two-point second-order tensor function. 7 7 Despite our notational convention that upper case Greek and Roman fonts denotes point and two-point tensor points, respectively, we could, in a slight abuse of notation, abbreviate D tψ) by DΨ). In an analogous fashion, D t v) may be abbreviated by D v) when the argument is a vector point function. Note that this notational abuse is customary for the differential divergence and gradient operators for which is used to denote the divergence operator for both vectors and tensors and is used to denote the gradient operator on both scalars and vectors.

10 10 Q. Du et al Nonlocal vector identities Compositions of the point operators defined in Sections 2.1 and with the corresponding adjoint two-point operators derived in Section 2.3 lead to the following nonlocal vector identities. In this proposition, we set α = β = γ and, for the first, second, and fourth results, i.e., those involving nonlocal curl operators, we set n = k = 3. Proposition 1 The nonlocal divergence, gradient, and curl operators and the corresponding adjoint operators satisfy D C u) ) = 0 for u: R 3 12a) C D u) ) = 0 for u: R 12b) G u) = tr D t u) ) for u: R n 12c) D t D t u) ) G G u) ) = C C u) ) for u: R 3. 12d) Proof We prove 12d); the proofs of 12a) 12c) are immediate after direct substitution of the operators involved. Let u: R 3. Then, by 1c), 5c), 11a), and 11b) and recalling that α is an antisymmetric function, we have D t D t u) ) G G u) ) ) = u u) α + u u ) α α dy ) + u u) α + u u ) α α dy = 2 u u)α α) u u) α ) ) α dy = 2 α u u) α )) dy, where, for the last equality, we have used the vector identity a b c) = bc a) ca b). A simple computation show that the last expression is equal to C C u) ) so that 12d) is proved. Functions of the form C u) do not entirely comprise the null space of the operator D. In fact, for any antisymmetric two-point function νx, y), we have D ν ) = 0. However, functions of the form C u) are the only symmetric two-point functions belonging to the null space of D. Analogous statements can be made for the null space of the operator C and two-point functions of the form D u). The four identities in 12) are analogous to the vector identities associated with the differential divergence, gradient and curl operator: u) = 0, u) = 0, u = tr u), u) + u) = u),

11 Nonlocal vector calculus, volume-constrained problems and balance laws 11 respectively. Because ) =, =, and ) = ), these identities can be written in the form ) u) ) = 0, ) u) ) = 0, u = tr ) u ), ) u ) ) u ) = ) u ), which more directly correspond to 12a) 12d), respectively. In particular, the identities 12a), 12b), and 12d) suggest that D, G, and C can also be viewed as nonlocal analogs of the differential divergence, gradient and curl operators that, when operating on point functions, result in two-point functions. 8 Another set of results for the nonlocal operators that mimic obvious properties of the corresponding differential operators are given in the following proposition whose proof is a straightforward consequence of 6) and the definitions of the operators involved. Proposition 2 Let b and b denote a constant scalar and vector, respectively. Then, the nonlocal adjoint divergence, gradient, and curl operators satisfy D b) = 0, G b) = 0, and C b) = 0. Moreover, these three relationships are equivalent to the three nonlocal integral theorems 2a) 2c). These results do not hold for the point divergence, gradient, and curl operators, e.g., Db), Gb), and Cb) do not necessarily vanish for constants b and constant vectors b. However, there exist special situation for which Db) = 0 holds, and likewise for the other point operators. One such case occurs whenever is symmetric with respect to x = y Nonlocal curl operators in two and higher dimensions The nonlocal point and two-point curl operators defined in 1c) and 5c), respectively, were defined in three dimensions. These definitions can be generalized to arbitrary dimensions by replacing the vector cross product with the wedge product so that, e.g., instead of 5c) we would have C w): R r given by C w)x, y) := γ w w), where γx, y): R r with r a positive integer. This suggests a possible exterior calculus-based formalism. However, such a formalism is beyond the scope of this paper so that only the special cases of r = 3 and lower dimensions are considered and the vector cross product is retained. We can also define nonlocal point and two-point curl operators in two space dimensions, i.e., for R 2 ; in fact, we have two types of each kind of 8 Note that G C µ )) 0 and C G η )) 0 because of the nonlocality of the operators.

12 12 Q. Du et al. curl operator. 9 First, we assume, without loss of generality, that µ e 3 = 0, w e 3 = 0, and γ e 3 = 0 in 1c) and 5c). Then, for a vector two-point function µx, y): R 2, we can view C µ ) as the nonlocal scalar point function defined by C µ ) ) x) = µ2 + µ 2)γ 1 µ 1 + µ 1)γ 2 dy for x and, for a vector point function w : R 2, we can view C w) as the nonlocal scalar two-point function defined by C w)x, y) = w 2 w 2 )γ 1 w 1 w 1 )γ 2 for x, y. Next, assume, again without loss of generality, that µ = µe 3, w = we 3, and γ e 3 = 0 in 1c) and 5c). Then, for a scalar two-point function µx, y): R, we can view Cµ) as the nonlocal vector point function defined by Cµ)x) := µ + µ )γ 2 e 1 γ 1 e 2 ) dy for x and, for a scalar point function wx): R, we can view C w) as the nonlocal vector two-point function defined by C w)x, y) = w w)γ 2 e 1 γ 1 e 2 ) for x, y. 3 Constraint regions, constraint operators, and volume-constrained problems In addition to the divergence, gradient, and curl operators and integrals over a region in R n, the theorems and identities of the vector calculus for differential operators also involve operators acting on functions defined on the boundary of that region and integrals over that boundary surface. 10 The 9 This is analogous to the two types of curl operators in two dimensions, one operating on vectors the other on scalars, given by curl u = u1 x 2 u2 x 1 and curl u = u x 2 e 1 + u x 1 e For example, given a region R n having boundary, the divergence theorem for a vector-valued function u states that u dx = u n dx and the Green s generalized) first identity for scalar functions u and v states that, for tensor-valued constitutive functions Θ, u Θ v) dx + u Θ v) dx = uθ v) n dx. The nonlocal divergence theorem 2a) should correspond to the first of these relations and the nonlocal Green s first identity 9a) should correspond to the second. However, we see that neither 2a) or 9a) contain terms that correspond to the boundary integrals in the above two relations.

13 Nonlocal vector calculus, volume-constrained problems and balance laws 13 nonlocal theorems and identities developed in Section 2 do not contain terms analogous to the boundary integrals found in the corresponding differential operator versions nor do they involve auxiliary boundary operators. However, by viewing boundary operators in the vector calculus for differential operators as constraint operators, 11 it is a simple matter to rewrite the nonlocal vector theorems and identities so that they do include such terms. The reason it was not necessary to introduce constraint operators and boundary integrals in the theorems and identities of the nonlocal vector calculus is that, in the nonlocal case, boundary operators must operate on functions defined over measurable volumes, and not on lower-dimensional manifolds [12, 20]. As a result, the actions of these operators are, in a real sense, indistinguishable from those of the nonlocal operators we have already defined, except for the resulting domains. 3.1 Constraint regions We divide the region into disjoint open subsets s and c, i.e., we have = s c ) s c ), where ) denotes the closure. We refer to s as the solution domain 12 and c as the constraint domain. Note that no relation is assumed between s and c so that, e.g., the four configurations of Figure 1 are possible. The bottom-left sketch in that figure depicts a constraint region that is a strip following the boundary of s. Not surprisingly, this particular situation is most closely related to boundary surfaces, but, in the nonlocal case, constraint regions may take other guises, as depicted in Figure 1. This is another reason why we first introduced the nonlocal vector calculus without such regions or without constraint operators. 3.2 Nonlocal point constraint operators The constraint operators corresponding to each of the nonlocal point operators defined in Definition 1 are defined as follows, where the two-point functions α, β, Ψ, η, etc. are defined as in Definition Note that each definition also involves redefining, from to s, the domains resulting from the actions of the nonlocal point operators D, G, and C. Definition 4 [Nonlocal point constraint operators] Let = s c ) s c ). Corresponding to the point divergence operator D ν ) : s 11 In addition to trying to mimic more closely the theorems and identities of the vector calculus for differential operators, we introduce constraint regions and constraint operators because they are needed to describe nonlocal volume-constrained problems and showing their well posedness; see Section 3.5 for examples of nonlocal volume-constrained problems. 12 Why we use the terminology solution domain and constraint domain will become clear when, in Section 3.5, we introduce examples of nonlocal volumeconstrained problems that are analogous to differential boundary-value problems. 13 A constraint operator corresponding to the nonlocal divergence operator for tensors defined in 11a) can be defined in an entirely analogous manner.

14 14 Q. Du et al. Fig. 1 Four of the possible configurations for = s c) s c), where s and c denote the solution and constraint domains, respectively. R defined as in 1a) but for x restricted to s, we have the point constraint operator N ν): c R defined as N ν)x) := ν + ν ) α dy for x c. 13a) Corresponding to the point gradient operator G η ) : s R k defined as in 1b) but for x restricted to s, we have the point constraint operator Sη): c R k defined as Sη)x) := η + η )β dy for x c. 13b) Corresponding to the point curl operator C µ ) : s R 3 defined as in 1c) but for x restricted to s, we have the point constraint operator T µ): c R 3 defined as T µ)x) := γ µ + µ ) dy for x c. 13c) Note that the definitions of the point operators and the corresponding point constraint operators are defined using the same integral formulas but the former is defined for x s and the latter for x c. 3.3 Nonlocal integral theorems with constraint operators It is now a trivial matter to rewrite the nonlocal integral theorems given in Theorem 1 and the nonlocal integration by parts formulas given in Corollary 1 in terms of the nonlocal point operators and point constraint operators. We simply take advantage of the fact that the definitions of the two types of operators have similar definitions. Theorem 2 [Nonlocal integral theorems with constraint operators] Assuming the notations and definitions found in Definition 4, we have the

15 Nonlocal vector calculus, volume-constrained problems and balance laws 15 following analogs of the integral theorems of the vector calculus for differential operators: D ν ) dx = N ν) dx 14a) s c G η ) dx = Sη) dx 14b) s c C µ ) dx = T µ) dx. 14c) s c Corollary 5 [Nonlocal integration by parts formulas with constraint operators] Adopt the hypotheses of Theorem 2. Then, udν) dx + u u)ξ ) ν dydx = un ν) dx 15a) s c v Gη) dx + s w Cµ) dx s v v) β ) η dydx = v Sη) dx c γ w w) ) µ dydx = w T µ) dx. c 3.4 Nonlocal adjoint operators and Green s identities with constraint operators 15b) 15c) With the definition of the point constrained operators and the redefinition of the point operators, the definition of adjoint operators given in 4) can be rewritten as G, LF ) ) s L G), F ) = G, BF ) ) c, 16) where B is an operator that maps two-point functions F to point functions defined over c. As a result, the adjoint operators D, G, and C given in Corollary 2 remain unchanged. We can then rewrite nonlocal integration by parts formulas with constraint operators equations given by Corollary 5 as follows: ud ν ) dx D u ) ν dydx = un ν) dx 17a) s c v G η ) dx G v ) η dydx = v Sη) dx 17b) s c w C µ ) dx C w ) µ dydx = w T µ) dx. 17c) s c

16 16 Q. Du et al. The abstract Green s first and second identities 7) and 8) now take the form G, LΘL H)) ) s L G), ΘL H) ) = G, BΘL H)) ) c 18) and G, L ΘL H) )) H, L ΘL G) )) s s = G, B ΘL H) )) H, B ΘL G) )), c c 19) respectively. Likewise, the nonlocal Green s first identities 9a) 9c) now respectively take the form ud Θ D v) ) dx D u) Θ D v)ν dydx s = un Θ D v) ) 20a) dx c v G θg u) ) dx θg v)g u) dydx s = v S θg u) ) 20b) dx c w C Θ : C u) ) dx C w) : Θ : C u) dydx s = w T Θ : C u) ) 20c) dx c and the nonlocal Green s second identities 10a) 10c) now respectively take the form ud Θ D v) ) dx vd Θ D u) ) dx s s = un Θ D v) ) dx vn Θ D u) ) dx c c v G θg u) ) dx u G θg v) ) dx s s = v S θg u) ) dx u S θg v) ) dx c c w C Θ : C u) ) dx u C Θ : C w) ) dx s s = w T Θ : C u) ) dx u T Θ : C v) ) dx. c c 21a) 21b) 21c)

17 Nonlocal vector calculus, volume-constrained problems and balance laws Examples of nonlocal volume-constrained problems The nonlocal point operators, the corresponding nonlocal adjoint operators, and the corresponding nonlocal constraint operators given in 1), 5), and 13), respectively, can be used to define nonlocal volume-constrained problems. Here, we merely state problems involving scalar and vector secondorder operators. Let c = c1 c2, where 14 c1 c2 =, and let Θ 4, Θ 2, and θ denote fourth-order tensor, second-order tensor, and scalar two-point functions, respectively. The nonlocal volume-constrained problems D Θ 2 D u) ) = f in u = g in c1 N Θ 2 D u) ) 22a) = h in c2 D t Θ4 : Dt u) ) = f in u = g in c1 N t Θ4 : Dt u) ) 22b) = h in c2 C Θ 4 : C u) ) + G θg u) ) = f in u = g in c1 T Θ 4 : C u) ) + S θg u) ) 22c) = h in c2 are analogous to the second-order differential boundary-value problems K 2 u ) = f in u = g on 1 K2 u ) 23a) n = h on 2 K 4 : u ) = f in u = g on 1 K4 : u ) 23b) n = h on 2 K 4 : u ) k u ) = f in u = g on 1 n K 4 : u ) } = h 1 on k u = h 2, 2 23c) respectively, where = 1 2 denotes the boundary of with 1 2 = and where K 4, K 2, and k denote fourth-order tensor, second-order tensor, and scalar point functions, respectively. A special form of the nonlocal volume-constrained problem 22a) corresponding to a scalar-valued solution is studied in [12] by appealing to a variational formulation. Well-posedness results are provided there for the case in 14 Either c1 or c2 may be empty.

18 18 Q. Du et al. which the natural energy space used to define the variational problem) is equivalent to L 2 ), the space of square integrable functions. A rigorous connection to 23a) is demonstrated as well for such a case. Although the notion of a nonlocal vector calculus is not introduced, see the recent book [2] where the authors and their collaborators collect their substantial recent work on nonlocal diffusion and its relationship to 23a); in that work, the nonlocal operator in 22a) represents the infinitesimal generator. The two papers [3, 10] also consider various properties of the nonlocal operator in 22a). As shown in [7], more generally, the natural energy space associated with the nonlocal operators used in 22a) may be a strict subspace of L 2 ), for instance a fractional Sobolev space. For the latter cases, the nonlocal variational problems possess smoothing properties akin to that for elliptic partial differential equations but with possibly reduced order [7]. 4 Identification of nonlocal operators with differential operators in a distributional sense In this section, we identify, in a distributional sense, nonlocal operators with their differential counterparts. For the sake of brevity, we mostly consider the nonlocal point divergence operator D and its adjoint operator D. However, analogous results also hold for the nonlocal gradient and curl operators G and C, respectively, and their adjoints. Further connections between the nonlocal and differential operators are made in Section 5 where we demonstrate circumstances under which the nonlocal point operators are weak representations of the divergence, gradient, and curl differential operators. We begin with the identification of the nonlocal point divergence operator D with the divergence differential operator. This identification is subject to the understanding that the former operates on two-point functions while the latter operates on point functions. Proposition 3 Let ν C0 ), i.e., ν belongs to the space of compactly supported infinitely differentiable functions over the domain. Select the antisymmetric) distribution αy, x) = y δy x), 24a) where and δy x) denote the differential gradient and Dirac delta distribution, respectively. Then, D ν ) x) = x νx, x), 24b) where denotes the differential divergence operator. In particular, we have D ux) + uy) ) x) = x ux). 24c)

19 Nonlocal vector calculus, volume-constrained problems and balance laws 19 Proof We have x νx, x) = x νx, y) ) y=x + y νx, y) ) y=x = y νy, x) ) y=x + y νx, y) ) y=x = y νy, x) + νx, y) )) y=x = y νy, x) + νx, y) ) δy x) dy ) = νy, x) + νx, y) y δy x) dy = D ν ) x) so that the relations 24b) and 24c) now follow. Next, we identify the nonlocal Gauss theorem 2a) with the classical Gauss theorem for functions that are compactly supported within. Corollary 6 The nonlocal Gauss theorem 2a) reduces to νx, x) n x = 0. Proof The result is immediate from 24b) and 2a). The following proposition shows that the average of the action of D on a point function can be related, in a distributional sense, to the action of on that function. Proposition 4 Let u C0 ) and select α as in 24b). Then, D u) dy = x u. Proof Using 5a) we have that ) D u) dy = uy) ux) y δy x) dy ) = δy x) y uy) ux) dy = δy x) y uy) dy = x ux) = x ) ux). Now consider the composition of the nonlocal divergence operator and its adjoint, i.e., D D ), which, according to the next proposition, can be identified, in the sense of distributions, with the Laplace operator =. Proposition 5 Let u C 0 ) and select αx, y) 2 = 1 2 yδy x), where denotes the differential Laplace operator. Then, D D ux) ) = x ux).

20 20 Q. Du et al. Proof Using 1a) and 5a), we have D D u ) ) x) = 2 uy) ux) αx) 2 dy ) = uy) ux) y δy x) dy ) = δy x) y uy) ux) dy = δy x) y uy) dy = x ux) = x x ux) = x ) x ) ux), where the differential Green s second identity is used for the third equality. An immediate application is to relate the average of the product of operators D ) D ) to the weak Laplacian x x in a distributional sense. Corollary 7 Green s first identity 9a) with Θ = I reduces to x u x v dx = D u) D v) dydx. 5 Relations between weighted nonlocal operators and weak representations of differential operators The conventional differential calculus only involves operators mapping point functions to point functions; on the other hand, the nonlocal operators defined earlier map two-point functions to point functions or, in case of the nonlocal adjoint operators, point functions to two-point functions. To further demonstrate that the nonlocal operators correspond to nonlocal versions of the conventional divergence, gradient, and curl differential operators, we use the nonlocal operators D, G, and C to define, in Section 5.1, corresponding weighted operators that map point functions to point functions. We also show that the adjoint operators corresponding to the weighted operators are weighted integrals of the nonlocal adjoint operators D, G, and C. Then, in Section 5.2, the weighted operators are shown to be nonlocal versions of the corresponding differential operators. 5.1 Nonlocal weighted operators The nonlocal point operators defined in 1) induce new operators, referred to as weighted operators. Definition 5 Let ωx, y): R denote a non-negative scalar-valued two-point function. Let the operators D, G, and C be defined as in 1). Then,

21 Nonlocal vector calculus, volume-constrained problems and balance laws 21 given the point function ux): R n, the weighted nonlocal divergence operator D ω u): R is defined by D ω u)x) := D ωx, y)ux) ) x) for x. 25a) Given the point function ux): R, the weighted nonlocal gradient operator G ω u): R k is defined by G ω u)x) := G ωx, y)ux) ) x) for x. 25b) Given the point function wx): R 3, the weighted nonlocal curl operator C ω u): R 3 is defined by C ω w)x) := C ωx, y)wx) ) x) for x. 25c) The following result shows that the adjoint operators corresponding to the weighted nonlocal operators are determined as weighted integrals of the corresponding nonlocal two-point adjoint operators 5). Proposition 6 Let ωx, y): R denote a non-negative scalar twopoint function and let D, G, and C denote the nonlocal adjoint operators given in 5). The adjoint operator Dωu)x): R k corresponding to the weighted nonlocal divergence operator D ω is given by Dωu)x) = D u)x, y) ωx, y) dy for x 26a) for scalar point functions ux): R. The adjoint operator Gωv)x): R corresponding to the weighted nonlocal gradient operator G ω is given by Gωv)x) = G v)x, y) ωx, y) dy for x 26b) for vector point functions vx): R n. The adjoint operator Cωw)x): R 3 corresponding to the weighted nonlocal curl operator C ω is given by Cωw)x) = C w)x, y) ωx, y) dy for x 26c) for vector point functions wx): R 3. Proof We have that Dω u), u ) = = = ux)d ωx, y)ux) ) dx D u)x, y) ωx, y)ux) ) dydx ) ωx, y)d u)x, y) dy ux) dx,

22 22 Q. Du et al. where 25a) is used for the first equality and 4) and 5a) for the second. But, by definition, the adjoint operator D ω ) corresponding to the operator D ω ) satisfies u, Dω u) ) = D ωu), u ). Comparing the last two results yields 25a). The conclusions 25b) and 25c) are derived in a similar fashion. In direct analogy to the results of Section 2.5.1, we extend 25a and 26a to tensors and vectors, respectively. Definition 6 Let ωx, y): R denote a non-negative scalar twopoint function and let D t be given as in 11b). Given the tensor point function Ux): R n k, the weighted nonlocal divergence operator D t,ω U) : R n for tensors is defined by D t,ω U ) x) := Dt ωx, y)ux) ) x) for x. 27) Corollary 8 The adjoint operator Dt,ωu)x) : R n k corresponding to the operator D t,ω is given by ) Dt,ω u x) = Dt u)x, y) ωx, y) dy for x 28) for vector point functions u: R n. 5.2 Relationships between weighted operators and differential operators We have seen that the nonlocal operators satisfy many properties that mimic their differential counterparts. In particular, we established, in Section 4, that the nonlocal point operators can be identified with the corresponding differential operators in a distributional sense. In this section, we establish rigorous connections between the weighted operators and their differential counterparts by first introducing a cut-off or horizon parameter ε > 0 and then analyzing what occurs as ε 0, that is, in the local limit. To this end, we define B ε x) := {y R d : y x < ε}. Let = R d and let φ denote a positive radial function; select αx, y) = y x for x y y x { y x φ y x ) y B ε x) ω x y ) = 0 otherwise 29a) with the function φ satisfying a normalization condition y x 2 φ y x ) dy = d, 29b) B εx)

23 Nonlocal vector calculus, volume-constrained problems and balance laws 23 where d denotes the space dimension. Note that α is an antisymmetric function whereas ω is a symmetric ) function. We then have, for a scalar function u, that the components of G ) ω u and Dω u given by 25b) and 26a), respectively, are given by, for j = 1,..., d, ) d j ux) := ux + z) + ux) zj φ z ) dz 30a) B ε0) d j ux) := B ε0) where z j denotes the j-th component of z. The following result ensues. ux + z) ux) ) zj φ z ) dz, 30b) Lemma 1 Assume that ω and α are defined as in 29a). Let u: R d R and let d j ux) and d j ux) denote the j-th components of G ω u) and D ωu), respectively. Then, d j u = d j u. 31) Proof From 30) we have d j ux) + d j ux) = 2ux) from which the result directly follows. B ε0) y j φ y )dy = 0 Based on this lemma and the definition of the weighted operators, we have the following results. Corollary 9 Under the same conditions as in Lemma 1, we have D ω = G ω, G ω = D ω, and C ω = C ω. 32) Thus, under the hypotheses of this corollary, the identities in 32) mimic their counterparts in the vector calculus for differential operators. Identities such as those in Proposition 1 do not generally hold for the weighted operators due to their nonlocal properties. 15 Synergistic with the results of Section 4, we have the following. Corollary 10 Select the singular distribution z j φ z ) = j δz), where j u denotes the weak derivative of u with respect to x j. Then, d j = j and d j = j. The following lemma demonstrates that the components of the weighted gradient operator G ω or, by 32), D ω, converge, as ε 0, to the corresponding spatial derivatives. 15 The identities in Lemma 1 and Corollary 9 also hold for more general choices of α and ω, namely, those such that α is anti-symmetric and ω is a radial function with support in B εx).

24 24 Q. Du et al. Proposition 7 Let ω be defined as in 29a) and satisfy 29b). Then, for j = 1,..., d, the weighted operators d j and d j defined by 30) are bounded linear operators from H 1 R d ) to L 2 R d ). Moreover, if u H 1 R d ), then as ε 0 d j u j u L2 R d ) 0 d j u + j u L2 R d ) 0, 33a) 33b) where j u denotes the weak derivative of u with respect to x j. Moreover, if z 1+s φ z ) dz < for some 0 s 1, 33c) B ε0) then, for j = 1,..., d, the weighted operators d j and d j operators from H t R d ) to H t s R d ) for any t 0. are bounded linear Proof The conclusion 33a) is equivalent to d j u j u L 2 R d ) 0, where d j u and j u denote the Fourier transforms of d j u and j u, respectively. Under the condition 33c) for s [0, 1], if u H t R d ) for t 0, then d j u = = i = i B ε0) B ε0) B ε0) e iy ξ + 1) ûξ) y j φ y ) dy siny ξ) ûξ) y j φ y ) dy siny j ξ j ) cos k j y k ξ k ) + cosy j ξ j ) sin ) y k ξ k ) ûξ) y j φ y ) dy k j = iûξ) siny j ξ j ) cos y i ξ k ) y j φ y ) dy, B ε0) k j where the second and fourth equalities hold because of the symmetry of the domain of integration and the antisymmetry of the integrand with respect to the integration variable, respectively. Because for any 0 s 1, siny j ξ j ) cos k j y iξ k ) y j ξ j s, we have d j u ξ s ûξ) y j 1+s φ y ) dy B ε0) so that d j u H t s R d ). In particular, for t = s = 1, we obtain that, under the condition 29b), the weighted operators d j, j = 1,..., d, defined in 30a) are bounded linear operators from H 1 R d ) to L 2 R d ).

25 Nonlocal vector calculus, volume-constrained problems and balance laws 25 If u H 1 R d ), we have that d j u j u d j u + j u sinyj ξ j ) cos y k ξ k ) ûξ) y j φ y ) dy + ξj û k j B ε0) ξ j û B ε0) 2 ξ j û L 2 R d ), y 2 j φ y ) dy + ξ j û where the third inequality holds because sinx) x and cosx) 1. By Taylor s theorem, we have d j u = i j ξ j + yj B ε0)y 3 ξj 3 cosθ 1 )/6)1 + y k ξ k ) 2 cosθ 2 )/2) û y j φ y ) dy k j = iξ j û yj 2 φ y ) dy + I ε ξ) = iξ j û + I ε ξ), B ε0) where, for some θ 1 and θ 2, I ε ξ) := i yj 4 cosθ 1 )φ y ) dy ξj 3 û 6 B ε0) + i yj 2 y i ξ i ) 2 cosθ 2 )φ y ) dy ξ j û 2 k j B ε0) + i 12 B ε0) Because û is bounded a.e., we see that, for any ξ, I ε ξ) ε 2 ξ 3 j û + ε 2 k j Hence, we have y 4 j k j y k ξ k ) 2 cosθ 1 ) cosθ 2 )φ y ) dy ξ 3 j û. ξ k ) 2 ξ j û + ε 4 k j d j u j u a.e. for ξ R d as ε 0. Then, by the dominated convergence theorem, we obtain d j u j u L 2 R d ) 0 as ε 0. ξ k ) 2 ξ 3 j û 0 as ε 0. Lemma 1 implies the same results hold for the operator d j. The above lemma implies that if ω satisfies 33c) for s = 0, then the weighted operators are bounded operators from L 2 R d ) to L 2 R d ). More generally, for φ satisfying 33c) with positive s, the operators d j and d j actually map a subspace of L 2 R d ), for instance the fractional Sobolev space H s R d ), to L 2 R d ), or even map L 2 R d ) to H s R d ). We refer to [7] for related work. A direct consequence of Lemma 7 is the following result.

26 26 Q. Du et al. Proposition 8 Under the condition of Lemma 7, the weighted operators D ω, G ω, and C ω and their adjoint operators D ω, G ω, and C ω are bounded linear operators from H t R d ) to H t s R d ) for 0 s 1, where d = 3 for the weighted curl operators. Moreover, if u H 1 R d ) and u [H 1 R d )] d, D ω u) u Dωu) u G ω u) u Gωu) u C ω u) u Cωu) u, 34) where the convergence as ε 0 is with respect to L 2 R d ). Proof We prove the convergence of the operator D ω u) to the divergence differential operators. First, note that so that D ω u) = d d i u i and u = i=1 D ω u) u L 2 R d ) d i u i i=1 d d i u i i u i L 2 R d ) 0 as ε 0. i=1 The remaining results can be proved in a similar fashion. For the purpose of discussing nonlocal equations, we also need to consider combinations of the weighted operators such as D ω C 1 D ωu)), where C 1 x) is a tensor point function that, for example, describes the point property of a material. Note that the two-point property is involved in the definition of the weighted operators. In the next corollary, we illustrate that whenever the horizon ε goes to zero, the combinations of the weighted operators converge to their local counterparts. Corollary 11 Let u H 1 R d ), u [H 1 R d )] d, C 1 : R d R d R d in L R d R d ), and c 2 : R d R in L R d ). Then, D ω C1 D ωu) ) C 1 u) G ω c2 G ωu) ) c 2 u) C ω C1 C ωu) ) C 1 u) ), 35) where the convergence as ε 0 is with respect to H 1 R d ). 16 Proof Using Proposition 8 and a similar method of proof as for Lemma 7, the results are obtained. 16 Similar results can be obtained for the nonlocal divergence operator on tensors; in particular, we have that D t,ωc 3 : D t,ωu)) C 3 : u) with C 3 : R d R d R d R d R d in L R d R d R d R d ).

27 Nonlocal vector calculus, volume-constrained problems and balance laws 27 Let L denote a linear operator that commutes with the differential and nonlocal operators. We then have the following result. Lemma 2 Let L denote a linear operator that commutes with the differential and nonlocal operators. Then, if Lu H 1 R d ), D ω Lu) Lu G ω Lu) Lu C ω Lu) Lu D ωlu) Lu G ωlu) Lu C ωlu) Lu, where the convergence as ε 0 is with respect to L 2 R d ). 36) If L is selected as a differential operator with constant coefficients, or its formal inverse, we can observe convergence in either stronger or weaker norms. 6 Nonlocal balance laws Sections 2 and 3 introduced a nonlocal vector calculus. Combined with the analytical theory in sections 4 and 5, a potentially powerful formalism for instantiating the nonlocal balance laws of continuum mechanics is now at hand. We start by discussing the notion of an abstract balance law as given by P) = Q ), R d 37) postulating that the production of an extensive quantity over the open subregion is equal to the flux of the quantity through the boundary of that subregion. The additive quantities P and Q are expressed in terms of linear functionals over and, respectively. The purpose of this section is to demonstrate that the flux is given by the volume interaction between and R n \. Subsection 6.1 demonstrates that the flux induced by the peridynamic nonlocal continuum theory is given by a volume interaction and is elegantly phrased using the nonlocal calculus. However, assuming sufficient regularity, the volume interaction may be relegated to that on the surface. In particular, suppose that Q ) = q dh d 1, 38) where q and H d 1 are the flux density and a Hausdorff measure over the surface, respectively; see [5, Chap. II] for a discussion. Then, under general conditions, q dh d 1 = q n dh d 1 = divq dx 39) for some vector function q, where n denotes the outward unit normal vector for. The relations 38) and 39) are abstract versions of Cauchy s postulate and theorem, respectively. The relation 39) establishes the fundamental