A NONLOCAL VECTOR CALCULUS, NONLOCAL VOLUME-CONSTRAINED PROBLEMS, AND NONLOCAL BALANCE LAWS

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1 Mathematical Models and Methods in Applied Sciences c World Scientific Publishing Company A NONLOCAL VECTOR CALCULUS, NONLOCAL VOLUME-CONSTRAINED PROBLEMS, AND NONLOCAL BALANCE LAWS QIANG DU Department of Mathematics, Pennsylvania State University University Park, PA qdu@math.psu.edu MAX GUNZBURGER Department of Scientific Computing, Florida State University Tallahassee FL gunzburg@fsu.edu R. B. LEHOUCQ Sandia National Laboratories P.O. Box 5800, MS 1320, Albuquerque, NM rblehou@sandia.gov KUN ZHOU Department of Mathematics, Pennsylvania State University University Park, PA zhou@math.psu.edu Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx) A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoint operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are also presented. 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. The operators of the nonlocal calculus are used to define volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application discussed is posing abstract nonlocal balance laws and deriving the corresponding nonlocal field equations; this is demonstrated for heat conduction and the peridynamics model for continuum mechanics. Keywords: nonlocal operators, vector calculus, volume-constrained problems, balance laws, peridynamics, nonlocal diffusion AMS Subject Classification: 26B12, 26B15, 26B20, 45A05, 45P05, 45E99, 46F12 1

2 2 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou 1. Introduction Our principal goal is to develop a vector calculus for nonlocal operators that mimics the classical vector calculus for differential operators. We also show how the nonlocal vector calculus can be used to define nonlocal balance laws and volume-constrained problems that mimic boundary-value problems for differential operators. Nonlocal analogs of the divergence, gradient, and curl operators are defined and the corresponding nonlocal adjoint operators are deduced. Nonlocal analogs of the Gauss theorem and Green s identities of the vector calculus for differential operators are also derived. Relationships between the nonlocal operators and their differential counterparts are established. The nonlocal vector calculus can be used to define nonlocal volume-constrained problems that are analogous to boundaryvalue problems for partial differential operators. In addition, the nonlocal vector calculus has an important application to balance laws a that are nonlocal in the sense that subregions not in direct contact may have a non-vanishing interaction. This is accomplished by defining a nonlocal flux in terms of interactions between regions having positive measure, possibly not sharing a common boundary. As a result, the nonlocal vector calculus provides an alternative to standard approaches for circumventing the technicalities associated with the lack of sufficient regularity in local balance laws. Preliminary attempts at a nonlocal calculus are found in Refs. 11 and 12 which include applications to image processing and steady-state diffusion, respectively. In particular, in Ref. 26, which is cited in Ref. 11, a discrete nonlocal divergence and gradient are introduced within the context of machine learning; see also Refs. 4, 14, and 16 where a discrete calculus is also discussed. However, the discussion in those papers is limited to scalar problems. In contrast, this paper extends the ideas in Refs. 11 and 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 in the peridynamic continuum model b for solid mechanics that parallels the classical vector calculus formulation of the balance laws of elasticity. In fact, the nonlocal vector calculus presented in this paper 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 remainder of this section is devoted to establishing notation. In Section 2, the notions of local and nonlocal fluxes into or out of a region are compared and contrasted. The next two sections contain our prina A balance law postulates that the rate of change of an extensive quantity over any domain is given by the rate at which that quantity is produced in the domain minus the flux out of the domain. b The peridynamics continuum model was introduced in Refs. 21 and 23; Ref. 24 reviews the peridynamic balance laws of momentum and energy and provides many citations for the peridynamic model and its applications. See Section 7.3 for a brief discussion.

3 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws 3 cipal contributions. In Section 3, nonlocal divergence, gradient, and curl operators are introduced as are the corresponding adjoint operators, several vector identities, and other results about the operators. The nonlocal vector calculus is developed in Section 4; in particular, nonlocal integral theorems and nonlocal Green s identities are derived. In Section 5, connections between the nonlocal operators and distributional and weak representations of the associated classical differential operators are made. Sections 6 and 7 deal with applications of the nonlocal vector calculus. In Section 6, examples are given of nonlocal volume-constrained problems formulated in terms of the nonlocal operators. Then, in Section 7, a brief review of the conventional notion of a balance law is provided after which abstract nonlocal balance laws are discussed. The notion of nonlocal fluxes discussed in Section 2 is used in developing nonlocal balance laws and the vector calculus developed in Section 4 plays a crucial role in transforming balance laws into field equations. Also, in Section 7, a brief discussion is given of the application of our nonlocal vector calculus to the peridynamic theory for continuum mechanics. Throughout, wherever it is illuminating, we associate the definitions and results of the nonlocal vector calculus with the analogous definitions and results of the classical differential vector calculus Notation We have need of 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 twopoint operators are both nonlocal. Point operators involve integrals of two-point functions whereas two-point operators explicitly involve point functions evaluated at two different points. We now make more precise the definitions given above. Let m, k, and n denote positive integers. Points in are denoted by the vectors x, y, or z and the natural Cartesian basis is denoted by e 1,..., e n. For any domain, functions from into R m k or R m 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., U(x), u(x), and u(x), respectively. Functions from into R m k or R m 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., Ψ(x, y), ψ(x, y), and ψ(x, y), respectively. Symmetric and antisymmetric scalar two-point functions ψ(x, y) satisfy ψ(x, y) = ψ(y, x) and ψ(x, y) = ψ(y, x), respectively, and similarly for vector and tensor two-point functions. For the sake of notational simplicity, in much of the rest of the paper we adopt

4 4 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou the following notation: α := α(x, y) α := α(y, x) ψ := ψ(x, y) ψ := ψ(y, x) u := u(x) u := u(y) u := u(x) u := u(y) and similarly for other functions. The dot (or inner) product of two vectors u, v R m is denoted by u v R; the dyad (or outer) product is denoted by u w R m k whenever w R k ; given a second-order tensor (matrix) U R k m, 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. c In R 3, the cross product of two vectors u and v is denoted by u v R 3. The Frobenius product of two secondorder tensors A R m k and B R m k, denoted by A: B, is given by the sum of the element-wise product of the two tensors, i.e., A: B = i=1,...,m, j=1,...,k A ijb ij. The trace of B R m m, 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 u(x), v(x) (µ, ν) = µ ν dydx for µ(x, y), ν(x, y) with analogous expressions involving the Frobenius product and the ordinary product for tensor and scalar functions, respectively. Many of the results in the paper require that domains under consideration are Lebesgue measurable and functions are Lebesgue integrable with their pointwise properties interpreted in the Lebesgue measure sense, i.e., as holding almost everywhere. 2. Local and nonlocal fluxes and action-reaction principles A key concept in the development of a vector calculus is the notion of a flux which accounts for the interaction of points in a domain with points outside the domain. As a result, the notion of a flux is also fundamental to the understanding of balance laws in mechanics, heat transfer, and many other settings; see Section 7. In the classical setting of local interactions, that interaction occurs at the boundary of the domain whereas, in the nonlocal case, the interaction occurs over volumes external to the domain. In order to compare and contrast the notion of a nonlocal flux with the classical local flux, we begin by briefly reviewing the latter notion. c In matrix notation, the inner, outer, matrix-vector products are given by x y = x T y, x y = xy T, and U v = Uv.

5 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws Local fluxes Let 1 and 2 denote two disjoint open regions having Lipschitz boundaries. If 1 and 2 have a nonempty common boundary 12 = 1 2, then, for a vector-valued function q(x), the expression 12 q n 1 da (2.1) represents the classical local flux out of 1 into 2, where n 1 denotes the unit normal on 12 pointing outward from 1 and da denotes a surface measure in ; q n 1 is referred to as the flux density along 12 in the direction of the normal vector n 1 to the surface at that point. Because the local flux density q n 1 at a point is defined in terms of any orientable surface passing through that point, the flux conveys a notion of direction out of and into regions and is a proxy for the interaction between 1 and 2, i.e., the flux is an oriented interaction between two domains. An important observation is that the local flux from 1 into 2 occurs across their common boundary and that if the two disjoint regions have no common boundary, then the flux from one to the other is zero. The classical flux (2.1) is then deemed to be local because there is no interaction between 1 and 2 when separated by a finite distance. The classical flux satisfies the action-reaction principle d q n 1 da + q n 2 da = 0, (2.2) where, of course, 12 = 21 and n 2 = n 1 denotes the unit normal on 12 pointing outward from 2. In words, the flux 12 q n 1 da from 1 into 2 across their common boundary 12 is equal and opposite to the flux 21 q n 2 da from 2 into 1 across that same surface. The vector q is often expressed in terms of an intensive variable through a constitutive relation. e 2.2. Nonlocal fluxes For any point x, for integrable ψ(x, y), we identify ψ(x, y) dy (2.3) as the nonlocal flux density f at x into a measurable volume. Because of nonlocality, x does not necessarily have to belong to the closure of. The nonlocal flux density (2.3) has units of an intensive quantity per unit volume, in contrast to the local flux density q n that has units of an intensive quantity per unit area. In (2.3), d An example is in mechanics for which Newton s third law, i.e., the force exerted upon an object is equal and opposite to the force exerted by the object, is an action-reaction archetype. e For example, if q n 1 denotes the heat flux density, then q is related to the temperature via the Fourier heat law; see Section f In Section 3.2, we show how ψ can be expressed as an inner product of two vectors so that (2.3) mimics the local flux density q n.

6 6 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou ψ(x, y) may be viewed as a flux density per unit volume; no such notion exists in the local case because the flux density is not defined in terms of an integral as is the nonlocal flux density (2.3). The following result is fundamental to the development of the nonlocal calculus. Here and throughout, we assume that domains such as 1, 2, and are measurable. Theorem 2.1. The following are equivalent: (i) the antisymmetry of a two-point function ψ(x, y) = ψ(y, x) for almost all x, y (2.4a) (ii) the alternating or no self-interaction principle ψ(x, y) dydx = 0 (2.4b) (iii) the nonlocal action-reaction principle ψ(x, y) dy dx + ψ(x, y) dy dx = 0 1, 2 (2.4c) (iv) additivity for disjoint regions ψ(x, y) dydx 1 2 \( 1 2) = 1 ψ(x, y) dydx + \ 1 2 \ 2 ψ(x, y) dydx 1, 2 such that 1 2 =. Similar results hold for antisymmetric vector and tensor two-point functions. (2.4d) Proof. A change of variables under integration followed by change in the order of integration shows that ψ(x, y) dy dx = ψ(y, x) dy dx so that 2 ( ) ψ(x, y) dy dx = ψ(x, y) + ψ(y, x) dy dx. Thus, obviously, (i) implies (ii). This relation also shows that (ii) implies (i) because if (ii) holds, we have ( ) ψ(x, y) + ψ(y, x) dy dx = 0 R n which implies that ψ(x, y) + ψ(y, x) = 0 almost everywhere. Setting F( 1, 2 ) = 1 2 ψ(x, y) dydx, we easily see that (2.4a) and (2.4b) are satisfied. Then, the equivalence of (ii) and (iii) follows from Proposition (7.1) and the equivalence of (ii) and (iv) follows from Proposition (7.3).

7 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws 7 With (2.3) denoting a nonlocal flux density, for any two open regions 1 and 2, we identify ψ(x, y) dydx (2.5) 2 1 as a scalar interaction or nonlocal flux from 1 into 2. As in the local case, the nonlocal flux is associated with the notion of an orientation so that nonlocal fluxes are again oriented interactions between two domains. Because we require no self interactions, i.e., the flux from a region into itself vanishes so that (2.4b) holds whenever 1 = 2, by Theorem 2.1 we equivalently require that ψ : ( 1 2 ) ( 1 2 ) R be an antisymmetric function. By (2.3), 2 ψ(x, y) dy is the nonlocal flux density at a point x 1 into the region 2. As is the case for the local flux density q n 1, the nonlocal flux density 2 ψ(x, y) dy is related to an intensive variable through a constitutive relation; see Section Based on the above discussion, we see that (2.4c) is the nonlocal analogue of (2.2). In words, (2.4c) states that the flux (or interaction) from 1 into 2 is equal and opposite to the flux (or interaction) from 2 into 1. The flux is nonlocal because, by (2.5), the interaction may be nonzero even when the closures of 1 and 2 have an empty intersection. This is in stark contrast to classical local interactions for which we have seen that the interaction between 1 and 2 vanishes if their closures have empty intersection, i.e., if they have no common boundary. 3. Nonlocal operators The nonlocal vector calculus developed in Section 4 involves nonlocal operators that mimic the classical local differential divergence, gradient, and curl operators. An important distinction between local and nonlocal operators is that the adjoint operators for the former involve the same operators, i.e., the adjoint of is, of is, and of is, whereas the adjoint of nonlocal operators involve differently defined nonlocal gradient, divergence, and curl operators. See Section 3.3 for a further discussion Motivating the definitions of nonlocal operators To provide motivation for the definitions given in Definition 3.1 for the basic operators of the nonlocal vector calculus, we recall that the Gauss theorem q dx = q n da (3.1) shows that the integral of the divergence of a vector q over any region is equal to the flux out of that region. Of course, the latter is the integral over of the flux density q n. We adopt the same word definition for the nonlocal divergence, i.e., with D denoting the nonlocal divergence operator, we have that the integral of the nonlocal divergence of a vector ν over any region is equal to the flux out

8 8 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou of that region. In general, a point x can interact with all other points y so that, by (2.3), the nonlocal flux density out of a point x is given by ψ(x, y) dy. Then, according to our definition of a the nonlocal divergence operator, we have from (2.5) that D(ν)(x) dx = ψ(x, y) dydx. (3.2) In (3.1), the flux density q n is obviously related to the vector q, i.e., to the vector operand of the local differential divergence operator. In the nonlocal case, to define the action of the nonlocal divergence operator on the vector ν, we have to analogously determine how the operand ν of the nonlocal divergence operator D is related to ψ(x, y), where, because of non-locality, ν(x, y) is a vector two-point function. The information we have in hand to determine this relation is: because we assume no self interactions, (2.4a) requires that the scalar function ψ(x, y) be antisymmetric because of our definition of the nonlocal divergence operator, (3.2) relates D to ψ because we emulate the fact that is a linear operator, ψ(x, y) and therefore also D are linear in ν(x, y). (3.3a) (3.3b) (3.3c) This information is sufficient to apply the Schwartz Kernel Theorem (see, e.g., Refs. 8, 10, 20) to determine the relation between ψ and ν that leads to a definition of the nonlocal divergence operator. Before justifying this claim, we need to introduce some notation. Let U = [C ( )] k and V = C ( ) so that U consists of vector two-point functions whereas V consists of scalar point functions. The dual spaces are given by U = [D ( )] k and V = D ( ), respectively, where D denotes the space of distributions. The corresponding duality pairings are given as follows: ν, µ U,U = ν(z, y) µ(z, y) dzdy and v, u V,V = v(x)u(x) dx for ν U, µ U and v V, u V, respectively. We also use the product space U V, its dual (U V ), and, for κ U V and ρ (U V ), the corresponding duality pairing κ, ρ U V,(U V ) = κ(x, y, z) ρ(x, y, z) dzdydx.

9 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws 9 Specialized to our needs, the Schwartz Kernel Theorem is given as follows. Theorem 3.1. [Schwartz Kernel Theorem] Let D : U V denote a continuous linear map. Then, there exists a unique kernel ρ(x, y, z) (U V ) such that v, Dν V,V = vν, ρ (U V ),(U V ) v V, ν U or, using the definitions of the duality pairings, v(x)(dν)(x) dx = ρ(x, y, z) ν(z, y)v(x) dzdydx v V, ν U. (3.4) Corollary 3.1. Assume that (3.3) holds. The nonlocal flux density per unit volume ψ(x, y) is uniquely expressed in terms of the vector ν(x, y) by ψ(x, y) = 1 ( ) ρ(x, y, z) ν(z, y) ρ(y, x, z) ν(z, x) dz x, y R n (3.5) 2 and the nonlocal divergence operator D is uniquely expressed in terms of the vector ν(x, y) by (Dν)(x) = 1 ( ) ρ(x, y, z) ν(z, y) ρ(y, x, z) ν(z, x) dzdy 2 (3.6) x. Proof. Because is arbitrary, we conclude from (3.4) that (Dν)(x) = ρ(x, y, z) ν(z, y) dzdy x. (3.7) Comparing (3.2) and (3.7), we see that ψ(x, y) = ρ(x, y, z) ν(z, y) dz x, y. (3.8) The antisymmetry of ψ(x, y) with respect to x and y requires that ρ(y, x, z) ν(z, x) = ρ(x, y, z) ν(z, y) x, y, z (3.9) so that (3.5) and (3.6) follow from (3.8) and (3.7), respectively. The general definition for the nonlocal divergence is thus given by (3.6). We have found that we can simplify that definition and still develop a nonlocal vector calculus that meets our needs; however, it is possible that in some applications, retaining the more general definition (3.6) could prove useful. Our simplifying assumption is to let ρ(x, y, z) = 2α(x, y)δ(x z).

10 10 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou Substituting into (3.6), we obtain (Dν)(x) = 1 ( ) ρ(x, y, z) ν(z, y) ρ(y, x, z) ν(z, x) dzdy 2 R n ( ) = α(x, y) ν(z, y)δ(x z) α(y, x) ν(z, x)δ(y z) dzdy R n ( ) = α(x, y) ν(x, y) α(y, x) ν(y, x) dy At this point there are no restrictions on the two-point function α(x, y) as far as its symmetry or antisymmetry with respect to its two arguments. However, in Section 5, we provide reasons for why α(x, y) should be antisymmetric so that we assume this is true from the outset, leading to the definition ( ) (Dν)(x) = ν(x, y) + ν(y, x) α(x, y) dy (3.10) with α(x, y) antisymmetric. We now have that ψ = (ν + ν ) α; note that the integrability of ψ does not imply that α is necessarily integrable. For example, the integral appearing in (3.10) could be interpreted as the L 2 ( ) duality pairing of α with ν which would allow for α to be nonintegrable or to be a distribution Nonlocal point divergence, gradient, and curl operators Based on the discussion of Section 3.1, the nonlocal point divergence, gradient, and curl operators mapping two-point functions to point functions are defined in terms of their action on two-point functions as follows. These operators along with their adjoints are the building blocks of our nonlocal calculus. Definition 3.1. [Nonlocal operators] Given the vector two-point function ν(x, y): R k and the antisymmetric vector two-point function α(x, y): R k, the action of the nonlocal point divergence operator D on ν is defined as D ( ν ) ( (x) := ν + ν ) α dy for x, (3.11a) where D ( ν ) : R. Given the scalar two-point function η(x, y): R and the antisymmetric vector two-point function β(x, y): R k, the action of the nonlocal point gradient operator G on η is defined as G ( η ) ( (x) := η + η ) β dy for x, (3.11b) where G ( η ) : R k. Given the vector two-point function µ(x, y): R 3 and the antisymmetric vector two-point function γ(x, y): R 3, the action of the nonlocal point curl operator C on µ is defined as C ( µ ) (x) := γ ( µ + µ ) dy for x, (3.11c)

11 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws 11 where C ( µ ) : R 3. 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 Section 5 where we demonstrate circumstances under which the nonlocal point operators are identified with the corresponding differential operators in the sense of distributions and also as weak representations. Because the integrands in (3.11a) (3.11c) are antisymmetric, (2.4b) immediately implies that D(ν) dx = 0, G(η) dx = 0, and C(µ) dx = 0. (3.12) These relations may be viewed as free-space nonlocal integral theorems. g 3.3. Nonlocal adjoint operators The adjoint operators corresponding to nonlocal point operators are two-point operators that are defined as follows. Definition 3.2. Given a point operator Q that maps two-point functions F to point functions defined over, the adjoint operator Q is a two-point operator that maps point functions G to two-point functions defined over that satisfies ( G, Q(F ) ) ( Q (G), F ) = 0, (3.13) where, for Q = D, G, or C, we have that F and G denote pairs of vector-scalar, scalar-vector, or vector-vector functions, respectively. Definition 3.13 can be used to determine the nonlocal adjoint two-point operators corresponding to the nonlocal point operators introduced in Definition 3.1. Proposition 3.1. [Nonlocal adjoint operators] Given the scalar point function u(x): R, the adjoint of D is the two-point operator whose action on u is given by D ( u ) (x, y) = (u u)α for x, y, (3.14a) where D ( u ) : R k. Given the vector point function v(x): R k, the adjoint of G is the two-point operator whose action on v is given by G ( v ) (x, y) = (v v) β for x, y, (3.14b) g For example, D(ν) dx = 0 can be viewed as a nonlocal analog of the free space classical local Gauss theorem u dx = 0.

12 12 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou where G ( v ) : R. Given the vector point function w(x): R 3, the adjoint of C is the two-point operator whose action on w is given by where C ( w ) : R 3. Proof. Let ξ = uν. Then, C ( w ) (x, y) = γ (w w) for x, y, (3.14c) (ξ + ξ ) α = (uν + u ν ) α = u(ν + ν ) α + (u u)ν α (3.15) so that, from (3.11a) and the first equation in (3.12), we have ( ) 0 = D(ξ) dx = u(ν + ν ) α + (u u)ν α dydx R n = u (ν + ν ) α dydx + (u u)ν α dydx R n = ud(ν) dx + (u u)ν α dydx, (3.16) where the last equality follows because, due to the antisymmetry of α, we have that (u u)(ν ν ) α is an antisymmetric two-point function so that, by (2.4b), (u u)(ν ν ) α dydx = 0. (3.17) Associating F with ν, G with u, and Q with D, we see that (3.16) is exactly of the form (3.13) with Q = D, where D is given by (3.14a). In a similar manner, (3.14b) can be derived from (3.11b) and the second equation in (3.12) and (3.14c) can be derived from (3.11c) and the third equation in (3.12). The operator G, the negative of the adjoint of the nonlocal gradient operator G, can be used to define a second form for a nonlocal divergence operator. Similarly, the pairs G, D and C, C serve to define two forms for the nonlocal gradient and curl operators, respectively. It is not surprising that there are two forms for each operator because, in the nonlocal case, we have two different types of functions, i.e., one-point and two-point functions. Naturally, one needs two sets of operators, one operating on two-point functions, i.e., D, G, and C, and the other operating on one-point functions, i.e., D, G, and C. h Furthermore, the composition of operators requires the two sets of operators. For example, if one wants to compose a divergence and a gradient operator, the composition D(Gη) (where η denotes a two-point function) involves the application of D to the one-point function Gη, whereas in general, one expects D to operate on two-point functions. On the other hand, the composition D( D u) (where u denotes a one-point function) involves the application of D to the two-point function D u. h Of course, in the local case, one only deals with one-point functions so that there is no need to define two different sets of operators.

13 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws 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 3.2 and A nonlocal divergence operator for tensor functions and a nonlocal gradient operator for vector functions A nonlocal divergence operator for tensor functions is defined by applying (3.11a) to each row of the tensor. Definition 3.3. Given the tensor two-point function Ψ: R m k and the antisymmetric vector two-point function α: R k, the action of the nonlocal point divergence operator D for tensors on Ψ is defined as ( D(Ψ)(x) := Ψ + Ψ ) α dy for x, (3.18a) where D(Ψ)(x): R m. Definition 3.2 implies that, given the vector point function v(x) : R m, the action of the adjoint of D on v is given by D (v)(x, y) = (v v) α for x, y, (3.18b) where D (v) : R m k, i.e., D maps a vector point function to a secondorder tensor two-point function. One can also extend Definition (3.11b) of the nonlocal gradient operator of a two-point scalar function to define the nonlocal gradient operator G v acting on a two-point vector function ψ(x, y) as the point tensor function G(ψ)(x) := (ψ + ψ) β dy. The corresponding nonlocal adjoint operator G acting on a point tensor function U(x) is then given by the two-point vector function Nonlocal vector identities G (U)(x, y) = (U U) β. Compositions of the point operators defined in Sections 3.2 and with the corresponding adjoint two-point operators derived in Sections 3.3 and 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 m = k = 3; we also set m = k for the third result. i i The four identities in (3.19) 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),

14 14 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou Proposition 3.2. The nonlocal divergence, gradient, and curl operators and the corresponding adjoint operators satisfy D ( C (u) ) = 0 for u: R 3 (3.19a) C ( D (u) ) = 0 for u: R (3.19b) G (u) = tr ( D (u) ) for u: R k (3.19c) D ( D (u) ) G ( G (u) ) = C ( C (u) ) for u: R 3. (3.19d) Proof. We prove (3.19d); the proofs of (3.19a) (3.19c) are immediate after direct substitution of the operators involved. Let u: R 3. Then, by (3.11c), (3.14c), (3.18a), and (3.18b) and recalling that α is an antisymmetric function, we have D ( D (u) ) G ( G (u) ) ( = (u u) α + (u u ) α ) α dy + = 2 = 2 ( (u u) α + (u u ) α ) α dy ( (u u)(α α) ( (u u) α ) ) α dy ( α ( (u u) α )) dy, where, for the last equality, we have used the vector identity a (b c) = b(c a) c(a b). A simple computation shows that the last expression is equal to C ( C (u) ) so that (3.19d) is proved. Functions of the form C (u) do not entirely comprise the null space of the operator D. In fact, it is obvious that for any antisymmetric two-point function ν(x, 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). Note that, because of the nonlocality of the operators, G ( C ( µ )) 0 and C ( G ( η )) 0. Another set of results for the nonlocal operators that mimic obvious properties of the corresponding differential operators are given in the following proposition respectively. Because ( ) =, =, and ( ) = ( ), these identities can be written in the form (( ) u) ) = 0, ( ( ) u) ) = 0, u = tr ( ( ) u ), (( ) u ) ( ( ) u ) = ( ( ) u ) that more directly correspond to (3.19a) (3.19d), respectively. In particular, the identities (3.19a), (3.19b), and (3.19d) suggest that D, G, and C can also be viewed as nonlocal analogs of the differential gradient, divergence, and curl operators, respectively, that, when operating on point functions, result in two-point functions.

15 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws 15 whose proof is a straightforward consequence of the results given in Proposition 3.1. Proposition 3.3. Let b and b denote a constant scalar and vector, respectively. Then, the adjoints of the nonlocal divergence, gradient, and curl operators satisfy D (b) = 0, G (b) = 0, and C (b) = 0. These results do not hold for the point divergence, gradient, and curl operators, i.e., D(b), G(b), and C(b) do not necessarily vanish for constants b and constant vectors b Nonlocal curl operators in two and higher dimensions The nonlocal point and two-point curl operators defined in (3.11c) and (3.14c), respectively, are 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 (3.14c) we would have C (w): R r given by C (w)(x, y) = γ (w w), where γ(x, 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 r = 2 are considered and the vector cross product is retained. We next 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 curl operator. j First, we assume, without loss of generality, that µ e 3 = 0, w e 3 = 0, and γ e 3 = 0 in (3.11c) and (3.14c). Then, for a vector two-point function µ(x, 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 R n 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 (3.11c) and (3.14c). Then, for a scalar two-point function µ(x, y): R, j This is analogous to the two types of differential curl operators in two dimensions, one operating on vectors, the other on scalars, respectively given by curl u = u 1 x 2 u 2 x 1 and curl u = u x 2 e 1 + u x 1 e 2.

16 16 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou we can view C(µ) as the nonlocal vector point function defined by C(µ)(x) = (µ + µ )(γ 2 e 1 γ 1 e 2 ) dy for x and, for a scalar point function w(x): 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. 4. A nonlocal vector calculus We develop a nonlocal vector calculus that mimics the classical vector calculus for differential operators. In the classical calculus, interactions are local which is why, e.g., in the Gauss theorem, the contribution coming from interactions with points outside a domain appears in the form of a flux through the boundary. If interactions are nonlocal, then points outside of interact with points in so that that interaction cannot be accounted for merely by an integral along the boundary. In fact, one must account for interactions with points in the complement domain \. To treat the most general case, we introduce the notion of an interaction domain. Definition 4.1. Given an open subset, the interaction domain corresponding to is given by I := {y \ such that α(x, y) 0 for some x } (4.1) so that I consists of those points outside of that interact with points in. For simplicity, we assume that the corresponding interactions domains for β and γ are also given by (4.1). No assumption is made about the geometric relation between and I so that, e.g., the four configurations of Figure 1 are possible. We have that the situation I = \ is allowable as is = in which case, of course, I =. I I I I Fig. 1. Four of the possible configurations for (the shaded regions) and I (the white regions).

17 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws 17 From (4.1), we see that points in \ ( I ) do not interact with points in, i.e., and, by antisymmetry, α(x, y) = 0 for x and y \ ( I ) (4.2) α(x, y) = 0 for y and x \ ( I ). We do not assume that points in I interact only with points in I, i.e., (4.2) does not hold for x I, so that, in general, points in I may interact with points in \ ( I ) as well, Nonlocal interaction operators We want to define operators that can be used to account for the interaction of points in I with points in I. To this end, we obtain, successively using the antisymmetry of α and (2.4b), the linearity of the integral operator, (4.2), and (3.11a), 0 = (ν + ν ) α dydx I I = (ν + ν ) α dydx + (ν + ν ) α dydx I I I (4.3) = (ν + ν ) α dydx + (ν + ν ) α dydx I I = D(ν)dx + (ν + ν ) α dydx. I I We could not express the last term in terms of D because points in I can interact with points in \ ( I ). By (2.5), we identify the last term in (4.3) as the flux from I into I and I (ν + ν ) α dy as the flux density at x I into I. Thus, we could use I (ν + ν ) α dy to define the operator we are seeking. In (4.5a), we instead choose I (ν + ν ) α dy for this purpose because, using the antisymmetry of α, (2.1), and (2.4c), (ν + ν ) α dydx = (ν + ν ) α dydx I I I (4.4) = (ν + ν ) α dydx I i.e., it is convenient to have the integral over I of the nonlocal operator we seek to be the flux out of into I. Similar discussions can be made for the nonlocal gradient and curl operators. Thus, we define the nonlocal interaction operators as follows. Definition 4.2. [Nonlocal interaction operators] Given a domain, let the interaction domain I be defined by (4.1). Then, corresponding to the

18 18 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou point divergence operator D ( ν ) : R defined in (3.11a), we define the point interaction operator N (ν): I R through its action on ν by N ( ν ) (x) := (ν + ν ) α dy for x I. (4.5a) I Corresponding to the point gradient operator G ( η ) : R k defined in (3.11b), we define the point interaction operator S(η): I R k through its action on η by S ( η ) (x) := (η + η )β dy for x I. (4.5b) I Corresponding to the point curl operator C ( µ ) : R 3 defined in (3.11c), we define the point interaction operator T (µ): I R 3 through its action on µ by T ( µ ) (x) := γ (µ + µ ) dy for x I. (4.5c) I 4.2. Nonlocal integral theorems Definitions 3.1 and 4.2 lead to integral theorems of the nonlocal vector calculus that mimic the basic integral theorems of the vector calculus for differential operators. Theorem 4.1. [Nonlocal integral theorems] Assuming the notations and definitions found in Definitions 3.1 and 4.2, we have k D(ν) dx = N (ν) dx (4.6a) I G(η) dx = S(η) dx (4.6b) I C(µ) dx = T (µ) dx. (4.6c) I Proof. The proof is contained in (4.3) because, by (4.5a), the last term is equal to I N (ν) dx. In a similar manner, (4.6b) and (4.6c) can be derived. Alternately, they can be derived from (4.6a); one simply chooses ν = ηb and ν = b µ, respectively, in that equation, where b is a constant vector; one also has to associate β and γ with α. From (4.4), we have that I N ( ν ) dx is the nonlocal flux from into I. Thus, in words, (4.6a) states that the integral of the nonlocal divergence of ν over is k The nonlocal integral theorems (4.6) are analogous to the classical differential integral theorems given by v dx = v n dx, v dx = vn dx, and v dx = n v dx for functions v and v defined on, with n = 3 for the third one, for which the integrals are well defined.

19 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws 19 equal to the total flux out of into I. l Similar interpretations of (4.6b) and (4.6c) hold. The nonlocal integral theorems (4.6a) (4.6c) are action-reaction principles. For example, from (4.6a) and the assumptions made, we have that 0 = = = D ( ν ) dx N ( ν ) dx I (ν + ν ) α dydx + R n (ν + ν ) α dydx + I I I (ν + ν ) α dydx I (ν + ν ) α dydx (4.7) which is exactly of the form (2.4c) with 1 =, 2 = I, and ψ = (ν + ν ) α; (4.7) simply states that the flux out of into I is equal and opposite to the flux from I into. The nonlocal integral theorems (4.6a) (4.6c) have the simple consequences listed in the following corollary that can be viewed as providing nonlocal integration by parts formulas. Corollary 4.1. [Nonlocal integration by parts formulas] Adopt the hypotheses of Theorem 4.1 and let the nonlocal adjoint operators be given as in Proposition 3.1. Then, given the point functions u(x): R, v(x): R k, and w(x): R 3, we have m ud ( ν ) dx D ( u ) ν dydx = I I v G ( η ) dx G ( v ) η dydx = I I w C ( µ ) dx C ( w ) µ dydx = I I un (ν) dx I v S(η) dx I I w T (µ) dx. (4.8a) (4.8b) (4.8c) l This observation is analogous to the observation for the classical Gauss theorem v = v n da that, by (2.1), the integral of the local divergence of v over is equal to the total flux out of. m If ν and α are scalar-valued functions and for free space, a version of the integration by parts formula (4.8a) appears in Lemma 2.1 of Ref. 13. The classical analog of (4.8a) is given by, for a scalar function u(x) and vector function v(x), and similarly for (4.8b) and (4.8c). u v dx + v u dx = uv n da

20 20 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou Proof. Let ξ = uν; then, from (3.11a) and (3.15), we have ( ) D(ξ) = u(ν + ν ) α + (u u)ν α dy R n = u (ν + ν ) α dy + (u u)ν α dy = ud(ν) + (u u)ν α dy R n = ud(ν) + (u u)ν α dy I x, (4.9) where the last equality follows from (4.1). Also, from (3.15) and (4.5a), we have ( ) N (ξ) = u(ν + ν ) α + (u u)ν α dy I (4.10) = un (ν) (u u)ν α dy x I. I Then, 0 = D(ξ) dx N ( ξ ) dx I = ud(ν) dx un ( ν ) dx I + (u u)ν α dydx + I = ud(ν) dx un ( ν ) dx + I I I (u u)ν α dydx I (u u)ν α dydx, I where the first equality follows from (4.6a), the second from (4.9) and (4.10), and the third from the linearity of the integration operator. Then, (4.8a) follows from the antisymmetry of α, the fact that, similar to (3.17), we have that I I (u u)(ν ν ) α dydx = 0, and (3.14a). In a similar manner, (4.8b) and (4.8c) can be derived from (4.6a) along with (3.11b) and (3.11c), respectively. Alternately, (4.8b) and (4.8c) easily follow by setting, for an arbitrary constant vector b, ν = ηb and ν = b µ, respectively, in (4.8a) and also associating β and γ with α Nonlocal Green s identities Nonlocal (generalized) Green s identities are simple consequences of Corollary 4.1. For the following two corollaries, we carry over the notations, definitions, and results obtained above. Corollary 4.2. [Nonlocal (generalized) Green s first identities] Given the scalar point functions u(x): R and v(x): R and the two-point second-

21 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws 21 order tensor function Θ(x, y): R k k, then n ud ( Θ D (v) ) dx D (u) (Θ D (v) ) dydx I I = un ( Θ D (v) ) dx. I (4.11a) Given the vector point functions u(x): R k and v(x): R k and the two-point scalar function θ(x, y): R, then v G ( θg (u) ) dx θg (v)g (u) dydx I I = v S ( θg (u) ) (4.11b) dx. I Given the vector point functions u(x): R 3 and w(x): R 3 and the two-point second-order tensor function Θ(x, y): R 3 3, then w C ( Θ C (u) ) dx C (w) C ( Θ C (u) ) dydx I I = w T ( Θ C (u) ) (4.11c) dx. I Proof. The results (4.11a) (4.11c) follow by setting ν = Θ D (u) in (4.8a), η = θg (u) in (4.8b), and µ = Θ C (u) in (4.8c), respectively. Corollary 4.3. [Nonlocal (generalized) Green s second identities] We assume the same notation as in Corollary 4.2 and we also assume that the tensor Θ appearing in (4.11a) and (4.11c) is symmetric. Then, ud ( Θ D (v) ) dx vd ( Θ D (u) ) dx = un ( Θ D (v) ) dx vn ( Θ D (u) ) (4.12a) dx I I n We have that (4.11a) and (4.12a) are the nonlocal analogs of the local classical (generalized) first Green s identity u (C v) dx + u (C v) dx = un (C v) da and of the local classical (generalized) second Green s identity u (C v) dx v (C u) dx = un (C u) da vn (C u) da, respectively, and similarly for (4.11b), (4.12b), (4.11c) and (4.12c). Here, C(x) denotes a secondorder tensor function with C(x) symmetric for the second identity. These are generalizations of the classical local Green s identities for which C is the identity tensor.

22 22 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou v G ( θg (u) ) dx u G ( θg (v) ) dx = v S ( θg (u) ) dx u S ( θg (v) ) dx I I w C ( Θ : C (u) ) dx u C ( Θ : C (w) ) dx = w T ( Θ : C (u) ) dx u T ( Θ : C (v) ) dx. I I (4.12b) (4.12c) Proof. The result (4.12a) is obtained by reversing the roles of u and v in (4.11a) and then subtracting the result from (4.11a). The results (4.12b) and (4.12c) are obtained from (4.11b) and (4.11c), respectively, in a similar manner Special cases of the vector calculus We consider some special cases of the vector calculus developed in Sections The free space vector calculus As was mentioned previously, the vector calculus allows for the choice = in which case we also simply set I = in all definitions and results. The resulting nonlocal integral theorems have already been stated; see (3.12). Corresponding to the nonlocal divergence operator D, we have the nonlocal generalized Green s first identity ud ( Θ D (v) ) dx D (u) Θ D (v)ν dydx = 0 and the generalized nonlocal Green s second identity ud ( Θ D (v) ) dx vd ( Θ D (u) ) dx = 0 corresponding to (4.11a) and (4.12a), respectively, and similarly for the other operators The vector calculus for interactions of infinite extent In case points in interact with all points in, we have that I = \ and all definitions and results remain unchanged. In this case, N (ν) = D(ν) on I = \ The vector calculus for localized kernels An important special case is that of localized kernels for which we have that α(x, y) = 0 if y x ε (4.13)

23 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws 23 and similarly for β and γ; here, ε > 0 denotes a cut-off or interaction or horizon o parameter which is not necessarily small and which defines the extent of interactions. We then have that We also define the domain I = {y \ : y x < ε for x }. := {x : y x < ε for y \ } = {x : y x < ε for y I }, where the last equality follows from (4.13). See Figure 2 for an illustration. ε ε Fig. 2. For a localized kernel, the domain and the interaction regions I and whose thicknesses are given by the horizon ε. From (4.1), (4.5a), and (4.13), we have that N (ν) dx = (ν + ν ) α dydx = (ν + ν ) α dydx I I I I = (ν + ν ) α dydx = (ν + ν ) α dydx. I I The first term is the nonlocal flux from into I ; according to the last term, only points in contribute to that flux. Similar observations can be made for the operators S and T. Of course, all the definitions and results of Sections hold for the case of localized kernels. I - 5. Relations between nonlocal and differential operators In this section, we choose particular kernel functions α, β, and γ that lead to identifications of the nonlocal operators with their differential counterparts. In Section 5.1, the identification is made in a distributional sense whereas, in Section 5.2, we demonstrate circumstances under which the nonlocal point operators are weak o The terminology horizon is introduced in the context of cutt-off parameters in peridynamic models for continuum mechanics; see, e.g., Refs. 21 and 24.

24 24 Q. Du, M. Gunzburger, R. Lehoucq, K. Zhou representations of the divergence, gradient, and curl differential operators. 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. We then consider, in Section 5.3, a connection between the nonlocal and local Gauss theorems made with the help of two results given in Ref. 17. Throughout this section, we set k = n in Proposition 3.1, Theorem 3.1, and all subsequent results Identification of nonlocal operators with differential operators in a distributional sense 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 whereas the latter operates on point functions. Where there might be ambiguity in applying differential operators to two-point function, we identify explicitly the variable of differentiation, e.g., y ν(x, y) denotes the divergence of ν(x, y) with respect to y. We consider special cases of localized kernels (see Section (4.4.3)) for which ε 0. As such, the results and proofs provided in this subsection are purely formal excepting Proposition 5.1; a limiting process involving sequences of kernels similar to that presented in the proof of Proposition 5.1 would make all other proofs rigorous. We first relate, the nonlocal divergence with the local differential divergence. Let ρ ε = ρ ε (x) be a standard Gaussian with variance ε. Proposition 5.1. Let ν [C 0 ( )] n, i.e., the components of ν belongs to the space of compactly supported infinitely differentiable functions. Select the (antisymmetric) distribution α(x, y) = y ρ ε (y x), (5.1) where y denotes the differential gradient with respect to y. Then, lim D( ν ) (x) = ν(x, x) x, (5.2) ε 0 where denotes the differential divergence operator. Proof. First, by the chain rule, we have ν(x, x) = ( x ν(x, y) ) y=x + ( y ν(x, y) ) y=x = ( y ν(y, x) ) y=x + ( y ν(x, y) ) y=x = ( y (ν(y, x) + ν(x, y) )) y=x.

25 A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws 25 Then, given the convergence of ρ ε to the Dirac delta measure δ in the sense of distributions, we obtain ν(x, x) = ( y (ν(y, x) + ν(x, y) )) y=x = lim y (ν(y, x) + ν(x, y) ) ρ ε (y x) dy ε 0 R n ( ) = lim ν(y, x) + ν(x, y) y ρ ε (y x) dy ε 0 which gives (5.4) by the definition given in (3.11a). For any x, given that in the sense of distributions y ρ ε (y x) converges to y δ(y x) with δ the Dirac delta measure, we may let and rewrite (5.2) formally as α(x, y) = y δ(y x), (5.3) D ( ν ) (x) = ν(x, x) x. (5.4) We may further extend (5.4) in the sense of distributions to a more general function space for ν, again via a passage to limit or density argument. Along the same spirit, the following proposition relates D formally to = ( ). Proposition 5.2. Let u C 0 ( ) and select α as in (5.3). Then, for any ν [C ( )] n, D (u) ν dy = ν(x, x) u(x) x. (5.5) Proof. Starting with (3.14a) and (5.3), we have that D ( ) (u) ν dy = u(y) u(x) ν(x, y) y δ(y x) dy R n ( = δ(y x) y ν(x, y) ( u(y) u(x) )) dy R n = δ(y x)ν(x, y) y u(y) dy = ν(x, x) u(x) = ν(x, x) ( ) u(x) Choosing ν to be the natural Cartesian unit vectors in, we have, from (5.5), that D (u) dy = u(x) = ( ) u(x) x, (5.6)