new versions of the colombeau algebras

23
Math. Nachr. 278, No. 11, 1318 – 1340 (2005) / DOI 10.1002/mana.200310309 New versions of the Colombeau algebras V. M. Shelkovich 1 1 St.-Petersburg State Architecture and Civil Engineering University, 2 Krasnoarmeiskaya 4, 190005, St. Petersburg, Russia Received 16 March 2003, revised 7 October 2004, accepted 4 April 2005 Published online 5 August 2005 Key words Colombeau theory, product of distributions, polyharmonic regularization, analytic regularization, associated homogeneous distributions, asymptotic distributions MSC (2000) 46F05, 46F10 We construct some versions of the Colombeau theory. In particular, we construct the Colombeau algebra generated by harmonic (or polyharmonic) regularizations of distributions connected with a half-space and by analytic regularizations of distributions connected with an octant. Unlike the standard Colombeau’s scheme, our theory has new generalized functions that can be easily represented as weak asymptotics whose coefficients are distributions, i.e., in form of asymptotic distributions. The algebra of asymptotic distributions generated by the linear span of associated homogeneous distributions (in the one-dimensional case) which we constructed earlier [9] can be embedded as a subalgebra into our version of Colombeau algebra. The representation of distributional products in the form of weak asymptotic series proved very useful in solving problems which arise in the theory of discontinuous solutions of hyperbolic systems of conservation laws [10]–[16], [49] and [50]. c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction 1. There is a class of problems in mathematical physics where products of distributions (generalized functions) should be defined (see [6], [8], [14], [19], [40] and the references therein). Such problems arise, for instance, in the theory of singular solutions of nonlinear partial differential equations [9]–[16], [17], [19], [40], [41], [49] and [50]. So the construction of nonlinear theories of distributions is of utmost importance. The solution of this problem requires a development of special analytical methods, construction of differential algebras contain- ing the space of distributions, and a development of singular asymptotic construction technique. Besides, the development of nonlinear theories of distributions is of great interest in itself. The first nonlinear “sequential” theories of distributions were constructed in papers by Ya. B. Livchak [34]– [36] (see also Section 2). These excellent papers are full of deep ideas. Some ideas using the Livchak theory can be found in the papers by C. Schmieden, D. Laugwitz [44], and D. Laugwitz [32] and [33]. A particular case of the Livchak’s algebra L was introduced later by Yu. V. Egorov [17]. In order to construct an algebra of generalized functions Ya. B. Livchak used sequences of real-valued measurable functions and Yu. V. Egorov— sequences of smooth functions. The point value characterization of Livchak’s generalized function f (x) ∈L is given by (2.2) (see [36, §8]) and is an analog of the point value characterization introduced by M. Oberguggenberger and M. Kunzinger [42, Remark 2.11.] in Egorov’s theory. Some of the nonlinear “continuous” theories of distributions are associated with the name of J. Colombeau [8] (see [17], [19], [40] and Section 3). J. Colombeau [8] constructed an associative and commutative algebra G of new generalized functions where the Schwartz space of distributions D (R n ) and the space C (R n ) were embedded, and the last embedding was an algebra monomorphism. In Colombeau algebra G a product of dis- tributions can be defined, which, in the general case, presents a new generalized function from G. Colombeau generalized functions in this theory are connected with Schwartz distributions through the concept of associated distribution. It is well-known that Colombeau’s theory has applications in mathematical physics [19], [40] and e-mail: [email protected], Phone: +7 (812) 2517549, Fax: +7 (812) 3165872 c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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Math. Nachr. 278, No. 11, 1318 – 1340 (2005) / DOI 10.1002/mana.200310309

New versions of the Colombeau algebras

V. M. Shelkovich∗1

1 St.-Petersburg State Architecture and Civil Engineering University, 2 Krasnoarmeiskaya 4, 190005, St. Petersburg, Russia

Received 16 March 2003, revised 7 October 2004, accepted 4 April 2005Published online 5 August 2005

Key words Colombeau theory, product of distributions, polyharmonic regularization, analytic regularization,associated homogeneous distributions, asymptotic distributions

MSC (2000) 46F05, 46F10

We construct some versions of the Colombeau theory. In particular, we construct the Colombeau algebragenerated by harmonic (or polyharmonic) regularizations of distributions connected with a half-space and byanalytic regularizations of distributions connected with an octant. Unlike the standard Colombeau’s scheme,our theory has new generalized functions that can be easily represented as weak asymptotics whose coefficientsare distributions, i.e., in form of asymptotic distributions. The algebra of asymptotic distributions generated bythe linear span of associated homogeneous distributions (in the one-dimensional case) which we constructedearlier [9] can be embedded as a subalgebra into our version of Colombeau algebra. The representation ofdistributional products in the form of weak asymptotic series proved very useful in solving problems whicharise in the theory of discontinuous solutions of hyperbolic systems of conservation laws [10]–[16], [49] and[50].

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

1. There is a class of problems in mathematical physics where products of distributions (generalized functions)should be defined (see [6], [8], [14], [19], [40] and the references therein). Such problems arise, for instance,in the theory of singular solutions of nonlinear partial differential equations [9]–[16], [17], [19], [40], [41], [49]and [50]. So the construction of nonlinear theories of distributions is of utmost importance. The solution ofthis problem requires a development of special analytical methods, construction of differential algebras contain-ing the space of distributions, and a development of singular asymptotic construction technique. Besides, thedevelopment of nonlinear theories of distributions is of great interest in itself.

The first nonlinear “sequential” theories of distributions were constructed in papers by Ya. B. Livchak [34]–[36] (see also Section 2). These excellent papers are full of deep ideas. Some ideas using the Livchak theory canbe found in the papers by C. Schmieden, D. Laugwitz [44], and D. Laugwitz [32] and [33]. A particular case of theLivchak’s algebra L was introduced later by Yu. V. Egorov [17]. In order to construct an algebra of generalizedfunctions Ya. B. Livchak used sequences of real-valued measurable functions and Yu. V. Egorov — sequences ofsmooth functions. The point value characterization of Livchak’s generalized function f(x) ∈ L is given by (2.2)(see [36, §8]) and is an analog of the point value characterization introduced by M. Oberguggenberger and M.Kunzinger [42, Remark 2.11.] in Egorov’s theory.

Some of the nonlinear “continuous” theories of distributions are associated with the name of J. Colombeau[8] (see [17], [19], [40] and Section 3). J. Colombeau [8] constructed an associative and commutative algebraG of new generalized functions where the Schwartz space of distributions D′(Rn) and the space C∞(Rn) wereembedded, and the last embedding was an algebra monomorphism. In Colombeau algebra G a product of dis-tributions can be defined, which, in the general case, presents a new generalized function from G. Colombeaugeneralized functions in this theory are connected with Schwartz distributions through the concept of associateddistribution. It is well-known that Colombeau’s theory has applications in mathematical physics [19], [40] and

∗ e-mail: [email protected], Phone: +7 (812) 2517549, Fax: +7 (812) 3165872

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Math. Nachr. 278, No. 11 (2005) / www.mn-journal.com 1319

[41]. However, due to the fact that elements of the Colombeau algebra have no explicit functional interpretation,these applications are rather limited.

There are papers where theories of binary multiplication of distributions are considered. In the papers ofV. K. Ivanov (see, for example, [21]–[23]) such problems are systematically studied. In the papers by V. K. Ivanovand T. A. Verjbalovich [23], T. A. Verjbalovich [53], V. A. Kakichev and V. M. Shelkovich [24] (see Remark 6.3)and A. M. Kytmanov [30] multidimensional theories were constructed where the Itano’s counterexamples [20]do not arise. In [23] and [53] analytic regularizations of distributions are constructed by the Cauchy integral rep-resentation, in [24] analytic regularizations of distributions are constructed by the Bochner–Vladimirov integralrepresentation (see also [25]), in [30] harmonic regularizations of distributions are constructed by the Poissonintegral representation. Later, a theory of binary multiplication of distributions using harmonic representation ofdistribution was constructed by V. Boie [5] and by Li Bang-He and Li Ya-Qing [3]. These binary theories give ananswer to a problem posed by M. Oberguggenberger [40, Problem 27.1], namely, to use harmonic regularizationsto define a product of two distributions.

There is another approach associated with the names of Ya. B. Livchak [36], Li Bang-He [1], V. K. Ivanov[22]. The main idea of this approach is that a product f(x)g(x) of distributions f(x) and g(x) from a certainclass is defined as a weak asymptotics of the product f(x, ε)g(x, ε) as ε → +0, where f(x, ε) and g(x, ε)are corresponding regularizations of distributions, ε > 0 is the approximation parameter. The developmentof this approach in [46]–[48] led to the construction of associative and commutative differential algebra E∗

1 of

asymptotic distributions, generated by the linear span E1 of distributions δ(m−1)(x − ck), P(

1(x−ck)m

), m =

1, 2, . . . , ck ∈ R, k = 1, . . . , s. Later, in [9] associative and commutative differential algebra E∗ of asymptoticdistributions, generated by the linear span of associated homogeneous distributions was constructed (for details,see Section 7). Elements of algebraE∗

1 (products of distributions) are asymptotic series (1.1) of Laurent–Hartogstype in the powers in ε, where ε is a small parameter, the coefficients of these series are distributions from E1.Elements of the algebra E∗ (for details, see Section 7) can be represented in the form of weak asymptotic seriesof powers in ε and log ε

f∗ε (x) =

∞∑m=0

n0(m)∑n=0

fαm,n(x)εαm logn0−nε , ε → +0 , (1.1)

where the coefficients fαm,n(x) are associated homogeneous distributions, αm ∈ R is an increasing sequence,n0(m) ∈ N0. In [27]–[29] we showed that algebras of asymptotic distributions can be realized as subspaces inthe locally convex spaces of infinite-dimensional Schwartz vector-valued distributions

f∗(x) =(f(αm,n)(x)

), αm ∈ R , n ∈ N0 , (1.2)

where distributions f(αm,n)(x) ∈ A are coefficients in the weak asymptotic series (1.1).The representation of distributional products in the form of weak asymptotic series (1.1) proved very useful

in the theory of discontinuous solutions of nonlinear equations. In [9], using the algebra of asymptotic distri-butions E∗, Maslov’s hypothesis [39] that smooth self-similar solutions of nonlinear equations correspond tothree algebras of singularities with finitely many generators was studied. For instance, this construction wasused to develop the weak asymptotics method (the summary of this method see in [14]). The basis of the weakasymptotics method is constructing singular superpositions of distributions (multiplication of distributions) (seeSection 8). By this method the dynamics of propagation and interaction of nonlinear waves (infinitely narrowδ-solitons, shocks, δ-shocks) was studied [11]–[16], [49] and [50].

The goal of this paper is to make a step on the way to construct a nonlinear theory of distributions similar toL. Schwartz’s theory.

In Sections 2 and 3 the nonlinear theories of distributions of Livchak and Colombeau are described, respec-tively. In Sections 4–6 we construct some new versions of the Colombeau algebras. In Section 4 a new versionof the Colombeau algebra generated by m-regularizations (4.5) of distributions is constructed. In Sections 5and 6 we solve two problems which are related to the problem of M. Oberguggenberger [40, Problem 27.1]mentioned above. Namely, we construct Colombeau type algebras generated by both harmonic (polyharmonic)regularizations of distributions connected with a half-space (Section 5) and analytic regularizations of distribu-tions connected with an octant (Section 6), respectively. The result of Section 5 was first obtained in [45] in adifferent form.

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1320 Shelkovich : New versions of the Colombeau algebras

Since different classes of functions are used to construct the aforementioned versions of Colombeau’s algebras,these versions do not coincide. However, their intersections are not empty. The problem of relation between theseversions needs a special study.

Unlike the standard Colombeau scheme, our new generalized functions can be easily represented as weakasymptotic series whose coefficients are distributions. In Section 7 we show that the algebra of asymptoticdistributions E∗ from [9] can be embedded as a subalgebra into our version of the Colombeau algebra (in theone-dimensional case). In spite of the fact that algebra of asymptotic distributions E∗ is one-dimensional, asmentioned above, it is an efficient tool for constructing singular solutions of nonlinear (multidimensional) PDEs[11]–[16], [49] and [50]. The point is that physically interesting processes usually occur on the wave frontcarrying singularities. As a rule, such wave front Γt =

x : S(x, t) = 0

is a smooth surface of codimension 1

in the space Rnx × Rt. Therefore, in order to construct singular solutions of nonlinear systems we need an

algebra of generalized functions generated by a class of one-dimensional distributions f(ξ), where ξ = S(x, t).In Section 8 we illustrate how our approach can be used for describing the dynamics of nonlinear waves.

In view of what was mentioned above the attempt to find a relation between Colombeau’s algebras and thealgebras of asymptotic distributions made in this paper seems fruitful. Our construction enables one to turnour versions of algebras into the infinite-dimensional space of vector-valued distributions with a locally convextopology similar to [27]–[29]. If the given construction could be applied to the whole Colombeau algebra G, thisalgebra, like E∗, would turn into a space of infinite-dimensional vector-valued distributions (1.2) which wouldbridge the gap between J. Colombeau’s theory and that of L. Schwartz’s distributions.

2. Denote by N, R, R+, C the sets of positive integers, real numbers, positive real numbers and complexnumbers, respectively, and set N0 = 0 ∪ N. For α = (α1, . . . , αn) ∈ Nn

0 and x = (x1, . . . , xn) ∈ Rn weassume α! = α1! . . . αn!, |α| =

∑nk=1 αk and xα = xα1

1 . . . xαnn . We shall denote partial derivatives of the order

|α| by ∂αx = ∂|α|

∂x1α1 ...∂xn

αn . Here and in what follows∫ · dx denotes an improper integral

∫Rn · dx.

2 Livchak’s theory

1. Here we describe one version of Livchak’s theory [34]–[36]. Let M be the set of all sequences fn(x) ofreal-valued measurable functions on Rk. A sequence fn(x) ∈ M is called an almost zero if there exists n0

such that fn(x) = 0 for all n ≥ n0. The set of all almost zero sequences we denote by T (M). The quotientalgebra

L = M/T (M)

is called the Livchak algebra of generalized functions on Rk. The equivalence class containing a sequence fnis called its thin limit and is denoted by T limn fn(x). Thus, we introduce a generalized function as a thin limitf(x) = T limn fn(x) of a sequence of real-valued measurable functions fn(x) on Rk [36, §8].

Let φ(y1, . . . , ym) be a Borel function on Rm and let fr(x) = T limn frn(x) ∈ L, r = 1, . . . ,m, be general-

ized functions. Then we introduce a generalized function

φ(f1(x), . . . , fm(x)) def= T limnφ(f1

n(x), . . . , fmn (x)) .

One can prove that the generalized function φ(f1(x), . . . , fm(x)) is independent of the choice of sequencesf r

n(x), r = 1, . . . ,m. Thus, in particular, we define the operation of multiplication of distributions, sinceφ(y1, y2) = y1y2 is a Borel function on R2.

2. The change of variables. If ψ(x) is a measurable mapping Rk → Rk and f(x) = T limn fn(x) ∈ L is ageneralized function then one can define a generalized function

g(x) = f(ψ(x)

) def= T limnfn

(ψ(x)

).

Partial derivatives of generalized functions. Let f(x) = T limn fn(x) ∈ L. Then we define partial derivatives

∂f(x)∂xj

def= T limn

∂nfn(x)∂xj

, (2.1)

where

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Math. Nachr. 278, No. 11 (2005) / www.mn-journal.com 1321

∂nfn(x)∂xj

= n(fn

(x1, . . . , xj−1, xj + n−1, xj+1, . . . , xk

)−fn

(x1, . . . , xj−1, xj , xj+1, . . . , xk

)),

j = 1, . . . , k. Since

∂nfn(x)∂xj

is a measurable sequence Definition (2.1) describes the generalized function

∂f(x)∂xj

uniquely.

The point value of generalized functions. Let Sk be the set of all sequences xn = (x1n, . . . , x

kn),

(x1n, . . . , x

kn) ∈ Rk. Each vector a ∈ Rk can be identified with the stationary sequence from Sk, i.e., Rk ⊂ Sk.

The sequence xn ∈ Sk is called an almost zero if there exists n0 such that xn = 0 for all n ≥ n0. The set ofall almost zero sequences we denote by T (Sk). Let

Sk

= Sk/T (Sk) .Let

V kn =

x = (x1, . . . , xk) ∈ Rk : 0 ≤ xj ≤ n−1, j = 1, . . . , k

, n ∈ N .

If f(x) = T limn fn(x) ∈ L, xn ∈ Sk, x = T limn xn ∈ Sk, we associate the generalized function f(x) with

the function f∗(x) on Sk

by the formula

f∗(x) def= T limnfn(tn, xn) , x ∈ S

k, (2.2)

where

fn(tn, xn) = fn(xn + tn) , tn ∈ V kn .

The generalized number f∗(x) will be called the value at the point x of the function f∗(x).One can prove that if f(x) = T limn fn(x), g(x) = T limn gn(x) are generalized functions on Rk and

f∗(x) = g∗(x) for any x ∈ Sk

then f(x) = g(x) [36, §8]. It is clear that if f(x) = g(x) then f∗(x) = g∗(x) for

any x ∈ Sk.

Note that there exist generalized functions f(x) = T limn fn(x), g(x) = T limn gn(x) such that f(x) = g(x)and f∗(x) = g∗(x) for all x ∈ Rk.

Moreover, for the Livchak generalized functions operations of the convolution, integration and Fourier-trans-form are defined.

3 Standard Colombeau’s theory

1. Here we give one of the standard constructions of the Colombeau theory [8, Ch. 1], [17], [19, Ch. 1] and[40, III]

Let D(Rn) be the space of test functions with compact supports. We define a subset of test-functions

A0(Rn) =ψ(x) ∈ D(Rn) :

∫ψ(x) dx = 1

,

Aq(Rn) =ψ(x) ∈ A0(Rn) :

∫xαψ(x) dx = 0, 1 ≤ |α| ≤ q

, q ∈ N .

Obviously, D(Rn) ⊃ A0(Rn) ⊃ A1(Rn) ⊃ . . .Aq(Rn) ⊃ . . . and Aq(Rn) = ∅ for all q ∈ N0. Let us

consider a differential algebra E(Rn) = C∞(Rn)A0(Rn) consisting of all mappings f : A0(Rn) → C∞(Rn).Alternatively, one may view E(Rn) as the set of all maps f : A0(Rn) × Rn → C, where f(ψ, x) is smoothin the second variable x ∈ Rn, and the first member ψ ∈ Aq(Rn) is considered as a parameter (not as a testfunction).

Let us define a function ψε(x) = 1εn ψ

(xε

)for ψ(x) ∈ A0(Rn). Let EM (Rn) be a differential subalgebra of

moderate elements f(ψ, x) ∈ E(Rn), such that for any compact K ⊂ Rn and any α ∈ Nn0 there exists N ∈ N,

such that for all ψ ∈ AN there holds the inequality

supx∈K

|∂αx f(ψε, x)| ≤ Cε−N , for all ε ∈ (0, ε0) , (3.1)

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1322 Shelkovich : New versions of the Colombeau algebras

where C > 0 is a constant and ε0 > 0 is a certain number. Further, we define a differential ideal N (Rn) in thealgebra EM (Rn) as a set of elements f(ψ, x) ∈ E(Rn), such that for any compact K ⊂ Rn, and any α ∈ Nn

0

there exists N ∈ N0, such that for all ψ ∈ Aq , where q ≥ N , there holds the inequality

supx∈K

|∂αx f(ψε, x)| ≤ Cεq−N , for all ε ∈ (0, ε0) , (3.2)

where C > 0 and ε0 > 0. The Colombeau algebra is defined as a quotient algebra

G = EM/Nand its elements are called Colombeau new generalized functions.

2. The embeddings of the spaces of distributions D′(Rn) and smooth functionsC∞(Rn) into the Colombeaualgebra is realized by means of the Colombeau regularization

D′(Rn) f(x) −→ f(x) ∗ ψ(x) , ψ(x) ∈ A0(Rn) ,

where ∗ is the convolution. It can be easily proved that for f(x) ∗ ψε(x) estimate (3.1) holds, i.e., f(x) ∗ ψ(x)is a moderate function. Moreover, for a distribution f(x) its regularization f(x) ∗ ψ(x) belongs to the idealN if and only if f(x) = 0. Thus, the embedding D′ ⊂ G is injective and preserves operations of addition,multiplication by numbers, and derivation. The embedding of the space of smooth functionsC∞(Rn) ⊂ G is analgebra monomorphism, and the space C∞(Rn) is a differential subalgebra in the Colombeau algebra.

For example, the Colombeau regularization of δ-function and the Heaviside function H1(x) are given by theformulas

δ(ψε, x) =1εnψ(

x

ε

), H1(ψε, x) =

∫Rn

1+

1εnψ(

x − ξ

ε

)dξ ,

where H1(x) = 1, if x1 > 0 and H1(x) = 0, if x1 < 0, Rn1+ = x ∈ Rn : x1 > 0.

4 A version of the Colombeau algebra

1. Let us consider the space E(Rn) of all infinitely differentiable functions (sequences) F (x, ε) = fm(x, ε) :m = 0, 1, 2, . . ., ε > 0, x ∈ Rn. On E we introduce the structure of a differential algebra, defining theoperations componentwise:

(F 1 + F 2)m(x, ε) = f1m(x, ε) + f2

m(x, ε) ,

(F 1F 2)m(x, ε) = f1m(x, ε)f2

m(x, ε) ,(∂α

xF)1

m(x, ε) = ∂α

x f1m(x, ε) ,

(4.1)

where F j(x, ε) = f jm(x, ε) ∈ E , j = 1, 2, m = 0, 1, 2, . . . .

Let EM be a differential subalgebra of moderate elements F (x, ε) = fm(x, ε) ∈ E such that for anycompact K ⊂ Rn and any α ∈ Nn

0 there exists N ∈ N such that for all m ≥ N

supx∈K

|∂αx fm(x, ε)| ≤ Cε−N , for all ε ∈ (0, ε0) , (4.2)

where C > 0 is a constant, ε0 > 0 is a certain number. Note that in EM there are constant-sequences ofthe form F (x, ε) = fm(x, ε), where fm(x, ε) = f(x) ∈ C∞(Rn). Denote by N the set of all elementsF (x, ε) = fm(x, ε) from the algebra E such that for any compact K ⊂ Rn and any α ∈ Nn

0 and any p ∈ Nthere exists such N0 ∈ N that for all m ≥ N0

supx∈K

|∂αx fm(x, ε)| ≤ Cεp , for all ε ∈ (0, ε0) , (4.3)

where C > 0, ε0 > 0.

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Math. Nachr. 278, No. 11 (2005) / www.mn-journal.com 1323

Clearly, N ⊂ EM and N is a differential ideal in the algebra EM . The quotient algebra

G = EM/Nwill be called a Colombeau type algebra. Elements F (x) of this algebra are called generalized functions.An equivalence class of sequences which defines the element F (x) will be denoted by F (x) =

[F (x, ε)

]=[fm(x, ε)]. The algebra G of generalized functions is associative and commutative.

2. Define the embedding of the space of distributions and infinitely differentiated functions into the algebra G.Let f(x) ∈ D′(Rn) be a distribution. We define its regularization as the convolution

f(x, ε) = f(x) ∗K(x, ε) = 〈f(t),K(x− t, ε)〉 , ε > 0 , (4.4)

where K(x, ε) = 1εn ω

(xε

),∫ω(z) dz = 1, the mollifier ω(z) ∈ D(Rn), i.e., the kernel K(x, ε) is a regular-

ization of the δ-function. If f(x) ∈ S′(Rn) one can choose ω(z) ∈ S(Rn). Here ω(z) is not fixed. For allϕ(x) ∈ D(Rn)

limε→+0

〈f(x, ε), ϕ(x)〉 = 〈f(x), ϕ(x)〉 .

Definition 4.1 Let f(x, ε) be regularization (4.4) of the distribution f(x) ∈ D′(Rn). The function

f(m)(x, ε) =m∑

s=0

(−1)s ∂sεf(x, ε)s!

εs , m = 0 , 1 , . . . , (4.5)

will be called a canonicalm-regularization (representation) of the distribution f(x).Using the convolution properties, the m-regularization of the distribution f(x) can be written in the form

f(m)(x, ε) = f(x) ∗K(m)(x, ε) , (4.6)

where the kernel has the following form

K(m)(x, ε) =m∑

s=0

(−1)s ∂sεK(x, ε)s!

εs , m = 0 , 1 , . . . .

Theorem 4.2 The canonical m-regularization f(m)(x, ε) of a distribution f(x) ∈ D′(Rn) is a moderateelement, m = 0, 1, 2 . . . .

P r o o f. This theorem can be proved by analogy with [40, Proposition 9.1.]. In view of the representationtheorem [43, 6.26.], there exists a continuous function f0(x) and a partial derivative ∂k

x such that f(x) = ∂kxf

0(x)on K in distributional sense (in the weak sense). Then from (4.4), making the change of variables t = x − εξ,we obtain

∂αx f(x, ε) = ∂α

x

⟨∂k

t f0(t),

1εnω

„x − t

ε

«⟩=

1εα+k

∫f0(x− εξ)∂α+k

ξ ω(ξ) dξ .

So, we have supx∈K |∂αx f(x, ε)| ≤Mε−N , where N = α+ k.

From (4.4) and (4.5), making the change of variables t = x− εξ, we obtain

ε∂αx ∂

1εf(x, ε) = ε∂1

ε

⟨f0(t), 1

εα+k+n

`∂α+kω

´„x − t

ε

«⟩= − 1

εα+k

∫f0(x− εξ)

((∂α+kω

)(ξ) + ξ

(∂α+k+1ω

)(ξ)

)dξ .

Thus, since f(1)(x, ε) = f(x, ε) − ε∂1εf(x, ε), we have supx∈K |∂α

x f(1)(x, ε)| ≤ Cε−N , N = α+ k.In a similar way, from (4.4) and (4.5) for m = 2, 3 . . . , we obtain

supx∈K

|∂αx f(m)(x, ε)| ≤ Cε−N , N = α+ k .

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1324 Shelkovich : New versions of the Colombeau algebras

Denote by OD′(εk

)the collection of distributions f(x, ε) ∈ D′(Rn) such that

〈f(x, ε), ϕ(x)〉 = O(εk

),

for any ϕ(x) ∈ D(Rn), where O(εk

)is understood in the ordinary sense, i.e., there exists a right-neighborhood

U+0 of the point ε = 0, such that∣∣O(

εk)∣∣ ≤ const εk for any ε ∈ U+0 \ 0.

Theorem 4.3 Let f(m)(x, ε) be the canonical m-regularization (4.5) of f(x) ∈ D′(Rn). Then for any com-pact K ⊂ Rn, any α ∈ N0, and any m ∈ N0 we have

〈∂αx f(m)(x, ε), ϕ(x)〉 = 〈∂α

x f(x), ϕ(x)〉 +O(εm+1

), ε −→ +0 ,

for any function ϕ(x) ∈ D(Rn), i.e., in the weak sense

∂αx f(m)(x, ε) = ∂α

x f(x) +OD′(εm+1

), ε −→ +0 .

P r o o f. First, we prove this statement for α = 0. Consider the weak limit of the expression ∂sεf(x, ε) as

ε→ +0. From (4.4), using the convolution property, we obtain

〈f(x, ε), ϕ(x)〉 = 〈f(x), ϕε(x)〉for any function ϕ(x) ∈ D(Rn), where

ϕε(x) = K(x, ε) ∗ ϕ(x) =∫ω(t)ϕ(x + εt) dt ,

K(x, ε) = K(−x, ε).Thus for s = 0, 1, . . . we have

〈∂sεf(x, ε), ϕ(x)〉 = 〈f(x), ϕε;s(x)〉 ,

where

ϕε;s(x) = ∂sε ϕε(x) =

∫ω(t)∂s

εϕ(x + εt) dt

=∫ω(t)

(n∑

k=1

tk∂

∂xk

)s

ϕ(x + εt) dt =∑|β|=s

∫tβω(t)∂β

xϕ(x + εt) dt .

It follows from this,

〈∂sεf(x, ε), ϕ(x)〉 =

∑|β|=s

⟨(−1)β∂β

x f(x),∫tβω(t)ϕ(x+ εt) dt

⟩.

From this expression we find that

limε→+0

〈∂sεf(x, ε), ϕ(x)〉 =

∑|β|=s

⟨(−1)βΩβ∂

βx f(x), ϕ(x)

⟩, for all ϕ(x) ∈ D(Rn) ,

where Ωβ =∫ω(t)tβ dt. That is, weak limits limε→+0 ∂

sεf(x, ε) are distributions defined by the following

equalities

∂sεf(x, 0) = lim

ε→+0∂s

εf(x, ε) =∑|β|=s

(−1)βΩβ∂βxf(x) . (4.7)

In particular, f(x, 0) = f(x).Using formulae (4.7), we apply the Taylor formula to the function

Js(ε) =⟨εs∂s

εf(x, ε), ϕ(x)⟩

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for ε ∈ (0, ε0) and for s = 0, 1, . . . ,m:

Js(ε) =m∑

k=s

εk

(k − s)!〈∂k

ε f(x, 0), ϕ(x)〉 +εm+1

(m+ 1)!〈∂m+1

ε f(x, θsε), ϕ(x)〉 ,

where 0 < θs < 1. Then for m-regularization (4.5) of the distribution f(x) we obtain

〈f(m)(x, ε), ϕ(x)〉 =m∑

s=0

m∑k=s

(−1)s

s!(k − s)!⟨∂k

ε f(x, 0), ϕ(x)⟩εk

+1

(m+ 1)!

m∑s=0

(−1)s

s!⟨∂m+1

ε f(x, θsε), ϕ(x)⟩εm+1 .

Changing the summation order in the expression above and taking into account the expression

k∑s=0

(−1)s

s!(k − s)!=

1 , k = 0 ,

0 , k = 1 , 2 , . . . ,

we obtain

〈f(m)(x, ε), ϕ(x)〉 = 〈f(x), ϕ(x)〉 +εm+1

(m+ 1)!

m∑s=0

(−1)s

s!⟨∂m+1

ε f(x, θsε), ϕ(x)⟩. (4.8)

In view of the above remarks, it is clear that 〈∂m+1ε f(x, θsε), ϕ(x)〉 is a continuous function for ε ≥ 0.

Consequently, for ε ∈ (0, ε0) and for any ϕ(x) ∈ D(Rn)∣∣⟨∂m+1ε f(x, θsε), ϕ(x)

⟩∣∣ ≤ C ,

where C is a constant, and the statement of Theorem follows from (4.8). Analogously, one can obtain the prooffor the case of derivative ∂α

x f(m)(x, ε).

Corollary 4.4 Let f(m)(x, ε) be the canonical m-regularization (4.5) for f(x) ∈ C∞(Rn). Then for anycompact K ⊂ Rn, any α ∈ N0 and any m ∈ N0 we have

∂αx f(m)(x, ε) = ∂α

x f(x) +O(εm+1

).

This corollary can be proved by analogy with Theorem 4.3 applying the Taylor formula for each term of thecanonical m-regularization (4.5) of the function ∂α

x f(m)(x, ε).

Corollary 4.5 There exists an embedding C∞(Rn) ⊂ G which is an algebra monomorphism preserving theoperations of addition, multiplication and partial differentiation, i.e., C∞(Rn) is a differential subalgebra in theColombeau algebra.

P r o o f. The proof is essentially similar to the standard one [8, Ch. 1] and [40, III]. Note that the Colombeaualgebra G contains constant-sequences F (x, ε) = fm(x, ε), fm(x, ε) = f(x) ∈ C∞(Rn). In view of (4.3), ifF (x, ε) ∈ N , then f(x) = 0. Therefore, the mapping

C∞(Rn) f(x) −→ fm(x, ε) = f(x) + N ∈ Gis an algebra monomorphism which preserves the operations of addition, multiplication and partial differentiation.

This statement can be proved by the mapping

C∞(Rn) f(x) −→ f(m)(x, ε) + N ∈ G ,where f(m)(x, ε) is a canonical m-regularization (4.5) of smooth functions f(x), m = 0, 1, . . . . Indeed, in viewof Corollary 4.4 from the Theorem 4.3,

F (x, ε) = f(m)(x, ε) − f(x) ∈ N .

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1326 Shelkovich : New versions of the Colombeau algebras

Corollary 4.6 There exists an embedding D′(Rn) ⊂ G which preserves the operations of addition, multipli-cation by numbers, and partial differentiation.

P r o o f. Let us embed a distribution f ∈ D′(Rn) into G by the mapping

D′(Rn) f(x) −→ f(m)(x, ε) + N ∈ G ,where f(m)(x, ε) is the canonical m-regularization (4.5) of distribution f(x), m = 0, 1, . . . . It follows from

Theorem 4.2 that F (x, ε) = f(m)(x, ε) ∈ EM , and in view of Theorem 4.3, f(x) = 0 if and only if F (x, ε) =f(m)(x, ε) ∈ N , hence the mapping is injective.

3. Let

V =f(x, ε) ∈ C∞(Rn)(0,ε0)

be the space of regularizations of distributions from D′(Rn), i.e., for any f(x, ε) ∈ V there exists f ∈ D′(Rn)such that (in the weak sense) w − limε→+0 f(x, ε) = f(x). Let

V0 =f(x, ε) ∈ V : w − lim

ε→+0f(x, ε) = 0

.

It is clear (see, for example, [19, Ch. 1.2.1.]) that

D′(Rn) = V/V0 .

If f(x, ε) belongs to the equivalence class which defines the distribution f(x) ∈ D′(Rn) then f(x, ε) =f(m)(x, ε)+V0, where f(m)(x, ε) is the canonicalm-regularization (4.5) of the distribution f(x), m = 0, 1, . . . .

Let EM,D′ ⊂ E be the differential subalgebra of all elements F (x, ε) = f(x, ε) ∈ V. Denote by ND′ theset of all those elements from algebra EM,D′ for which relation (4.3) holds. Clearly, ND′ is a differential ideal.Now we introduce the Colombeau type algebra of generalized functions

GD′ = EM,D′/ND′ .

The spaces of distributions D′ and functions C∞ can be embedded into algebra GD′ using constructions fromCorollaries 4.5 and 4.6.

5 Colombeau’s algebra generated by harmonic representations of distributions con-nected with a half-space

5.1 Harmonic representations of distributions

Let Rn+1+ = (x, ε) ∈ Rn+1 : x ∈ Rn, ε > 0. By G = G

(Rn+1

+

)we denote the space of functions f(x, ε)

with the following properties:

1. f(x, ε) is harmonic in Rn+1+ ; i.e., ∆f(x, ε) = 0, where ∆ =

∑n+1k=1

∂2

∂x2k

is the Laplace operator andxn+1 = ε;

2. f(x, ε) has a finite order of growth, when approaching Rn(to the border of Rn+1

+

), i.e., if for any compact

set K ⊂ Rn there exist constants C > 0 and α > 0 such that

|f(x, ε)| ≤ Cε−α , ε ∈ (0, ε0) , x ∈ K , (5.1)

where ε0 > 0 is a certain number.

Let H0 be a class of functions f(x, ε) ∈ C∞(Rn+1

+ ∪ Rn)

harmonic in Rn+1+ and such that f(x, ε) = 0 in

Rn. In fact, it is a class of functions harmonic in Rn+1 and odd with respect to ε.Below we shall use the following theorem, which was first proved by A. M. Kytmanov in [30] (see also in [31])

and later by Li Bang-He and Li Ya-Qing in [2].

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Theorem 5.1 ([30] and [31, Corollary 11.9]) Every function f(x, ε) ∈ G(Rn+1

+

)determines a distribution

f(x) ∈ D′(Rn) on Rn via

limε→+0

〈f(x, ε), ϕ(x)〉 = 〈f(x), ϕ(x)〉 , for all ϕ(x) ∈ D(Rn) . (5.2)

For each distribution f(x) ∈ D′(Rn), there is a function f(x, ε) ∈ G(Rn+1

+

)for which (5.2) holds. Moreover,

f(x, ε) ∈ C∞(Rn+1

+ ∪ (Rn \ supp f(x)

)), and f(x, ε) = 0 on Rn \ supp f(x). Thus

D′(Rn) = G(Rn+1

+

)/H0

(Rn+1

+

).

The function f(x, ε) ∈ G(Rn+1

+

)is called a harmonic representation (regularization) of a distribution f(x) ∈

D′(Rn).

P r o o f. Here we give an outline of the proof. If f(x) ∈ E ′(Rn) ⊂ D′(Rn), where E ′(Rn) is the space ofdistributions with a compact supports (or f(x) ∈ O′

−n(Rn), where O′−n(Rn) is the space of distributions which

can be extended to the functions C∞(Rn), decreasing together with all their derivatives faster than (1 + |x|)−n

as |x| → ∞ (see [6, 13.5]), then any harmonic regularization of a distribution can be obtained by the Poissonintegral representation [30], [31, §11–12] and [54, §11]):

f(x, ε) = 〈f(t),P(x− t, ε)〉 , x ∈ Rn , ε > 0 , (5.3)

where

P(x, ε) =1εn

ω

„ |x|ε

«, ω(z) =

Γ((n+ 1)/2)π(n+1)/2

1(z2 + 1)(n+1)/2

, z ∈ R , (5.4)

is the Poisson kernel for the half-space, Γ(x) is the gamma-function,∫∞−∞ ω(z) dz = 1. Substituting the Poisson

kernel P(x, ε) in (4.4) for K(x, ε), we obtain (5.3). In the one-dimensional case (n = 1) the Poisson representa-tion turns into Cauchy representation with the Cauchy kernel K(x, ε) = 1

πε

x2+ε2 , x ∈ R.Suppose f(x) ∈ D′(Rn) is an arbitrary distribution. Now, in order to prove the Theorem 5.1 in [30] and [31,

Theorem 11.7] (for n = 1 see also [6, 5.9]) the standard construction is used. This technique is similar to theMittag–Leffler method. The essence of this method is as follows.

Let Ω be an arbitrary domain in Rn+1. We represent Ω in the form Ω =⋃∞

k=1 Ωk, where Ωk is an increasingsequence of domains Ωk ⊂ Ωk+1 such that the space H(Ω) is dense in H(Ωk). Here H(Ω) is the space ofharmonic functions in Ω. Let ηk(x) ∈ D(Rn) be a sequence of functions for which supp ηk ⊂ Rn ∩ Ωk+1

and ηk = 1 on Rn ∩ Ωk. Then fk(x) = f(x)ηk(x) ∈ E ′(Rn), and limk→∞ fk(x) = f(x) (in distributionalsense). Therefore, harmonic representations fk(x, ε) of a distribution fk(x), k = 1, 2, . . . , can be obtained bythe Poisson representation (5.3) and (5.4). But, in the general case, the function fk(x, ε) will not have a limit ask → ∞, so it is necessary to correct it.

Since fk+1(x) − fk(x) = 0 on Rn ∩ Ωk, the function fk+1(x, ε) − fk(x, ε) is harmonic in(Ω \ Rn

) ∪ Ωk.Consequently, there is a harmonic function hk(x, ε) ∈ H(Ω) such that |(fk+1(x, ε)−fk(x, ε)

)−hk(x, ε)| < 2−k

on Ωk, k = 1, 2, . . . .It is clear that the series

f(x, ε) =∞∑

k=0

(fk+1(x, ε) − fk(x, ε) − hk(x, ε)

), f0(x, ε) = h0(x, ε) = 0 ,

converges uniformly in Ω \ Rn. Indeed, the remainder∣∣∣∣∣∞∑

k=k0

(fk+1(x, ε) − fk(x, ε) − hk(x, ε)

)∣∣∣∣∣ ≤∞∑

k=k0

∣∣fk+1(x, ε) − fk(x, ε) − hk(x, ε)∣∣ < 21−k0

in Ωk0 , i.e., f(x, ε) is harmonic in Ω \ Rn. The partial sum of this series is

Fk0(x, ε) =k0−1∑k=0

(fk+1(x, ε) − fk(x, ε) − hk(x, ε)

)= fk0(x, ε) −

k0−1∑k=0

hk(x, ε) .

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1328 Shelkovich : New versions of the Colombeau algebras

Here fk0(x, ε) ∈ G(Rn+1

+

)and on Ωk0 ∩ Rn we have limε→+0 fk0(x, ε) = fk0(x) = f(x)ηk0 (x) (in distribu-

tional sense).The remainder f(x, ε)−Fk0 (x, ε) is a harmonic function in Ωk0 , so f(x, ε) ∈ G

(Rn+1

+

). Moreover, it is clear

that limε→+0 f(x, ε) = f(x) in distributional sense. Thus f(x, ε) is a harmonic representation of the distributionf(x).

Example 5.2 Now we give examples of harmonic regularizations (representations) of distributions δ(x) andP(xk |x|−n−1

):

δ(x, ε) =Γ((n+ 1)/2)π(n+1)/2

ε

(|x|2 + ε2)(n+1)/2, P

(xk |x|−n−1, ε

)=

xk

(|x|2 + ε2)(n+1)/2,

where P(xk |x|−n−1

)is the “principal value” of the function xk |x|−n−1, k = 1, . . . , n.

5.2 Polyharmonic representations of distributions

Let Gm = Gm

(Rn+1

+

), m = 1, 2, . . . , be a space of functions f(x, ε) with the following properties:

1. f(x, ε) is polyharmonic (m-harmonic) in Rn+1+ , i.e., ∆mf(x, ε) = 0;

2. f(x, ε) has a finite order of growth when approaching Rn, i.e., (5.1) holds.

Here G1

(Rn+1

+

)= G

(Rn+1

+

).

Theorem 5.3 A function f(x, ε) ∈ Gm

(Rn+1

+

)if and only if

f(x, ε) =m−1∑k=0

fk(x, ε)εk , (5.5)

where fk(x, ε) ∈ G(Rn+1

+

)is a harmonic function with finite order of growth, k = 0, 1, . . . ,m− 1.

P r o o f. The necessity of the formula (5.5) follows from [31, Lemma 12.8]. To prove the sufficiency of thisformula we suppose that the representation (5.5) holds for a function fk(x, ε) ∈ G

(Rn+1

+

), k = 0, 1, . . . ,m− 1.

Then, using the formula

∆(fg) = g∆f + f∆g + 2n∑

j=1

(∂f

∂xj

∂g

∂xj+∂f

∂ε

∂g

∂ε

),

we will have from (5.5)

∆f(x, ε) =m−3∑j=0

εj(j + 1)(

(j + 2)fj+2 + 2∂fj+1

∂ε

)+ 2εm−2(m− 1)

∂fm−1

∂ε.

Taking into account that ∂fk+1∂ε ∈ G

(Rn+1

+

)and applying the operator ∆ to this expression, we obtain

∆2f(x, ε). By similar calculation we have ∆mf(x, ε) = 0.From Theorems 5.1 and 5.3 it follows that the weak limit (5.2) of any m-harmonic function f(x, ε) ∈

Gm

(Rn+1

+

)defines a distribution f(x) ∈ D′(Rn) on Rn.

Definition 5.4 Let f(x) ∈ D′(Rn) be a distribution, and let f(x, ε) ∈ G(Rn+1

+

)be its harmonic represen-

tations. The function (4.5) will be called the canonical m + 1-harmonic regularization (representation) of thedistribution f(x), m = 0, 1, 2 . . . .

It follows from (4.5) and Theorem 5.1 that ∆m+1f(m)(x, ε) = 0.If f(x) ∈ E ′(Rn), from (5.3), (5.4) and (4.5) we obtain the m-Poisson integral representation

f(m)(x, ε) = 〈f(t),Pm(x − t, ε)〉 , x ∈ Rn , ε > 0 , (5.6)

where

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Pm(x, ε) =Γ((n+ 1)/2)π(n+1)/2

m∑s=0

(−1)s

s!εs∂s

ε

„ε

(|x|2 + ε2)(n+1)/2

«,

m = 0, 1, 2, . . . .Theorem 5.5 The canonicalm+ 1-harmonic regularization f(m)(x, ε) of a distribution f(x) ∈ D′(Rn) is a

moderate element, i.e., f(m)(x, ε) ∈ Gm+1

(Rn+1

+

), m = 0, 1, 2 . . . .

P r o o f. (See the proof of Theorem 4.2) Let f(x) ∈ E ′(Rn) be a distribution with a compact support K andlet f(x, ε) be its harmonic regularization. As is known, there exists k ∈ N0 such that,

|〈f(x), ϕ(x)〉| ≤ C supx∈K, |α|≤k

|∂αxϕ(x)|

for all ϕ(x) ∈ D(Rn). Then, by using the Poisson integral representation (5.3) and (5.4) for s = 0, 1, . . . , wecan obtain ∣∣εs∂s

εf(x, ε)∣∣ = εs

∣∣⟨f(t), ∂sεP(x− t, ε)

⟩∣∣ ≤ Cεs supx∈K, |α|≤k

∣∣∂αx ∂

sεP(x− t, ε)

∣∣ ≤ Mε−k .

Thus we have proved that the canonical m + 1-harmonic regularization g(m)(x, ε) of the distribution f(x) ∈E ′(Rn) has a finite order of growth, i.e., (5.1) holds.

In order to prove the Theorem for the general case (f(x) ∈ D′(Rn)) we use the Mittag–Leffler method [30]and [31, Theorem 11.7] (for n = 1 [6, 5.9]) in the above-cited algorithm.

Theorem 5.6 Let f(x) ∈ D′(Rn) be a distribution, and let f(m)(x, ε) be its canonical m + 1-harmonicregularization. Then for any compact K ⊂ Rn, any α ∈ Nn

0 , and any m = 0, 1, . . . ,m we have

∂αx f(m)(x, ε) = ∂α

x f(x) +OD′(εm+1

), m = 0 , 1 , 2 , . . . .

P r o o f. (See the proof of Theorem 4.3) First, we prove this statement for α = 0. Let f(x) ∈ E ′(Rn) be adistribution with a compact supportK and let f(x, ε) be its harmonic regularization. By using the Taylor formulain ε ∈ (0, ε0), we rewrite all terms in (4.5) for s = 0, 1, . . . ,m as follows:

εs∂sεf(x, ε) =

m−s∑k=0

∂k+sε f(x, 0)

k!εk+s +

εm+1

m+ 1!∂m

ε f(x, θsε) , 0 < θs < 1 . (5.7)

By using the Poisson integral representation (5.3) and (5.4), it is easy to see that ∂jεf(x, ε) ∈ G

(Rn+1

+

), j =

0, 1, . . . ,m+ 1 and ∂jεf(x, 0) ∈ D′(Rn) are distributions determined by

〈∂jεf(x, 0), ϕ(x)〉 = lim

ε→+0〈∂j

εf(x, ε), ϕ(x)〉 , for all ϕ(x) ∈ D , j = 0 , 1 , 2 , . . . , m ,

and f(x, 0) = f(x).Then, as in the proof of Theorem 4.3, by using (4.5) and (5.7), we obtain (4.8).The case ∂α

x f(m)(x, ε) can be proved similarly.If f(x) ∈ D′(Rn) we use the Mittag–Leffler method [30] and [31, Theorem 11.7] in the above-cited algorithm.

5.3 The Colombeau algebra

Let us consider the space H(Rn) of all sequences F (x, ε) = Fm(x, ε) : m ∈ N0, ε > 0, x ∈ Rn, whereFm(x, ε) = f(m)(x, ε) ∈ Gm+1

(Rn+1

+

)is the canonical m + 1-harmonic regularization of distribution f(x) ∈

D′(Rn), i.e., ∆m+1Fm(x, ε) = 0, m = 0, 1, 2, . . . .We introduce an associative and commutative differential algebra H∗(Rn) generated by the space H(Rn),

defining the operations componentwise as in (4.1). Elements F ∗(x, ε) of this algebra are finite sums of finiteproducts of functions (sequences) from H(Rn).

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1330 Shelkovich : New versions of the Colombeau algebras

It follows from Theorem 5.5 that H∗(Rn) is a differential algebra of moderate elements (sequences). Denoteby N ∗(Rn) the set of all elements (sequences) F ∗(x, ε) = Fm(x, ε) from H∗(Rn) for which the relation(4.3) holds. The quotient algebra

Gharm(Rn) = H∗(Rn)/N ∗(Rn)

will be called the Colombeau algebra of generalized functions generated by harmonic (or polyharmonic) regu-larizations of distributions.

Following Corollary 4.5 and Corollary 4.6, we embed the space C∞(Rn) and the space D′(Rn) into thealgebra Gharm(Rn) as a subalgebra and as a subspace, respectively.

Note that unlike the general scheme from Section 4.3, in this case the same Poisson kernel K(x, ε) = P(x, ε)(5.4) is used.

6 Colombeau’s algebra generated by analytic representations of distributions, connectedwith an octant

6.1 Analytic representations of distributions

Let

Γ+ = y = (y1, . . . , yn) ∈ Rn : yk > 0, k = 1, . . . , nbe the first octant in Rn. Denote by H(Γ+) the associative and commutative Vladimirov algebra [54] with theidentity and free of zero divisors consisting of all functions f(z), z = x + iy, which are holomorphic in thetubular domain T Γ+ = Rn + iΓ+ and satisfy the following condition of growth: for any cone Γ′ ⊆ Γ+ thereexists a numberK(Γ′, f, α, β) > 0 such that for certain α, β > 0 the following inequality holds

f(z) ≤ K(Γ′)(1 + |z|α) |y|−β , z ∈ T Γ+ . (6.1)

It is known that if f(x + iy) ∈ H(Γ+), then in the Schwartz space S′(Rn) there exists a boundary value (inthe weak sense)

f(x) = w − limy→0, y∈Γ+

f(x+ iy) ,

which is independent of the sequence y → 0, y ∈ Γ′ ⊆ Γ+, its support belonging to the closing of the octant Γ+.Let the vector σ = (σ1, . . . , σn) have components σk = ±1, k = 1, . . . , n. Denote by

Γσ = y = (y1, . . . , yn) ∈ Rn : σkyk > 0, k = 1, . . . , nthe octants of the space Rn. Let Hσ(Γσ) be the Vladimirov algebra of all functions fσ(x+ iσy)

(z = x+ iy ∈

T Γ+)

holomorphic in the tubular domain T Γσ = Rn + iΓσ with respect to the variables zσ = x + iσy, whereσy = (σ1y1, . . . , σnyn).

As is known [6, 13.8.], [51] and [54], each distribution from f(x) ∈ S′(Rn) can be represented as the weaklimit of the sum of 2n holomorphic functions

f(x, y) =∑

σ

fσ(x + iσy) , x ∈ Rn , y ∈ Γ+ , (6.2)

where

fσ(x+ iσy) = σ1 . . . σnf(x+ iσy) ∈ Hσ(Γσ) , f(x+ iy) ∈ H(Γ+) .

The function f(x, y) will be called the analytic representation of the distribution f(x). Denote by H the spaceof analytic representations of distributions from S′(Rn). According to [51],

S′(Rn) = H/HP ,

where HP is the space of pseudopolynomials. For example, the analytic representation of a distribution f(x) ∈S′(Rn) can be constructed either by the Bochner–Vladimirov type integral representation [24] and [25] or byn-dimensional Cauchy integral representation [7].

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Definition 6.1 Let f(x, y) be the analytic representation (6.2) of f(x) ∈ S′(Rn). The function

f(m)(x, y) =m∑

|s|=0

(−1)|s|∂s

yf(x, y)s!

ys , x ∈ Rn , y ∈ Γ+ , (6.3)

will be called a canonical m-analytical regularization (representation) of the distribution f(x), m = 0, 1, . . . .(Unlike Definitions 4.1 and 5.4, here s is not a scalar but a vector s = (s1, . . . , sn) ∈ Nn

0 .)Theorem 6.2 Let f(m)(x, y) be the canonical m-regularization (6.3) of f(x) ∈ S′(Rn). Then for any m ∈

N0 and any ∂αx we have in the weak sense as y → +0, y ∈ Γ′ ⊆ Γ+

∂αx f(m)(x, y) = ∂α

x f(x) +OD′(|y|m+1

).

The proof is analogous to those of Theorems 4.3 and 5.6.

6.2 The Colombeau algebra

Let us consider the space A(Rn) of all sequences F (x, y) = Fm(x, y) : m = 0, 1, 2, . . ., y ∈ Γ+, x ∈ Rn,where Fm(x, y) = f(m)(x, y) is the canonical m-analytical regularization of a distribution f(x) ∈ S′(Rn).

We introduce an associative and commutative differential algebra A∗(Rn) generated by the space A(Rn),defining the operations componentwise as in (4.1). Elements F ∗(x, y) of this algebra are finite sums of finiteproducts of functions (sequences) from A(Rn).

It follows from (6.1) that A∗(Rn) is a differential algebra of moderate elements (sequences), i.e., for anycompact K ⊂ Rn and any α ∈ Nn

0 there exists N ∈ N such that for all m ≥ N

supx∈K

|∂αx f(m)(x, y)| ≤ C |y|−N , for all y ∈ Γ′ ⊆ Γ+ , (6.4)

where C > 0. Denote by N ∗A(Rn) the set of all elements (sequences) F (x, y) = f(m(x, y) ∈ A∗(Rn), such

that for any compact K ⊂ Rn, any α ∈ Nn0 , and any p ∈ N there exists N0 ∈ N such that for all m ≥ N0

supx∈K

|∂αx f(m)(x, y)| ≤ C |y|p , for all y ∈ Γ′ ⊆ Γ+ , (6.5)

where C > 0.The quotient algebra

Ganalyt (Rn) = A∗(Rn)/N ∗A(Rn)

will be called the Colombeau algebra of generalized functions generated by analytic regularizations of distribu-tions.

As in Sections 4 and 5, it can be easily shown that the spaces of distributions S′(Rn) and functions C∞(Rn)can be embedded into the Colombeau algebra Ganalyt as a subalgebra and as a subspace respectively.

Remark 6.3 In [24], the theory of binary multiplication of distributions mentioned above was considered,using the analytic representation. In order to construct this theory the following way was used. Let W∗ bean associative and commutative differential algebra generated by the space H of analytic representations ofdistributions from S′(Rn). Elements f∗(x, y) of algebra W∗ are finite sums of finite products of functionsf(x, y) ∈ H . Denote by W0 the space of functions f∗(x, y) ∈ W∗ with the following properties: there exists adecomposition

f∗(x, y) =n∑

k=1

f∗k (x, y) , f∗

k (x, y) ∈ W∗ , k = 1 , . . . , n ,

such that

limyk→+0

∫f∗

k (x, y)ϕ(x) dx = 0 , for all ϕ(x) ∈ D(Rn) ,

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1332 Shelkovich : New versions of the Colombeau algebras

uniformly on Γ[k]+ = y ∈ Γ+ : yk = 0, k = 1, . . . , n. We can prove that HP ⊂ W0, H ∩ W0 = HP ,

H ·HP ⊂ W0.Next, we introduce a linear space

S∗ = W∗/W0

of so-called hyperdistributions. Let A∗ : W∗ → W∗/W0 be the canonical homomorphism. There exists aninjective embedding S′(Rn) ⊂ S∗.

The product of two distributions f1(x), f2(x) ∈ S′(Rn) is defined as the hyperdistribution

f∗(x) = f1(x) · f2(x) = A∗[(f1(x, y) +HP

)(f2(x, y) +HP

)] ∈ S∗,

where fj(x, y) is an analytic representation of distribution fj(x), j = 1, 2.

7 A subalgebra of asymptotic distributions (n = 1)

7.1 An algebra of asymptotic distributions

In [9] and [10] we constructed an associative algebra E∗ of asymptotic distributions with the identity and free ofzero divisors. E∗ is generated by the linear span of (one-dimensional) associated homogeneous distributions. Weshow that algebra E∗ is a subalgebra in the Colombeau algebra Gharm(R) from Section 5.

Here we outline the construction of the algebra E∗.

Definition 7.1 A distribution f(x) ∈ D′(R) is said to be associated homogeneous (in the broad sense) ofdegree λ and order k, k = 0, 1, 2, . . . , if for all a > 0 and for all ϕ(x) ∈ D(R) we have the relation

〈f(x), ϕ(x/a)〉 = aλ+1〈f(x), ϕ(x)〉 +k∑

r=1

aλ+1 logr a〈fk−r(x), ϕ(x)〉 ,

where fk−r(x) is an associated homogeneous (in the broad sense) distribution of degree λ and order k − r,r = 1, 2, . . . , k, i.e.,

f(ax) = aλf(x) +k∑

r=1

aλ logr afk−r(x)

(here we set f0(x) = 0 for k = 0).Associated distributions of zero order coincide with homogeneous distributions.One can prove that in the sense of Definition 7.1 for each λ there exist two linearly independent associ-

ated homogeneous (in the broad sense) distributions of degree λ and order k, k = 0, 1, 2, . . . . For example,(x ± i0)λ logk(x ± i0) are such distributions for any λ; xλ

± logk x± are such distributions for λ = −1,−2, . . . ;P(x−n± logk−1 x±

)are such distributions for λ = −n, n ∈ N. Here, according to [18, Ch. I, §4.2., (2) and (6)],

for Reλ > −n− 1, λ = −1,−2, . . . ,−n,

˙xλ

+ logk x+, ϕ(x)¸ def

=

Z 1

0

xλ logk x

ϕ(x) −

n−1Xj=0

xj

j!ϕ(j)(0)

!dx

+

Z ∞

1

xλ logk xϕ(x)dx +

n−1Xj=0

(−1)kk!

j!(λ + j + 1)k+1ϕ(j)(0) , ϕ(x) ∈ D(R) .

According to [18, Ch. I, §4.2., (4) and (7)],DP`x−n

+ logk x+

´, ϕ(x)

Edef=

Z ∞

0

x−n logk x

ϕ(x) −

n−2Xj=0

xj

j!ϕ(j)(0) − xn−1

(n − 1)!ϕ(n−1)(0)H(1 − x)

!dx , ϕ(x) ∈ D(R) ,

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where H(x) is the Heaviside function,⟨xλ− logk x−, ϕ(x)

⟩def=

⟨xλ

+ logk x+, ϕ(−x)⟩,⟨

P(x−n− logk x−

), ϕ(x)

⟩def=

⟨P(x−n

+ logk x+

), ϕ(−x)

⟩,

and [18, Ch. I, §4.4., 5.],

(x± i0)λ = xλ+ + e±iλπxλ

− ,

(x± i0)λ logk(x± i0) =∂k

∂λk(x± i0)λ .

Remark 7.2 In [18, Ch. I, §4.1.] the following definition is given: for any k, the distribution fk ∈ D′(R) iscalled an associated homogeneous distribution of order k and of degree λ if for any a > 0 and for any ϕ ∈ D(R)we have

〈fk, ϕ(x/a)〉 = aλ+1〈fk, ϕ〉 + aλ+1 log a〈fk−1, ϕ〉 , (7.1)

where fk−1 is an associated homogeneous distribution of order k − 1 and of degree λ, k = 1, 2, 3, . . . . In [52,Ch. X, 8.] a definition of an associated homogeneous distribution was introduced by analog of relation (7.1),where in the right-hand side of (7.1) log a was replaced by logk a. These definitions were introduced by analogywith the definition of associated eigenvectors.

Next, in [18, Ch.I, §4.2.] and [52, Ch. X, 8.] it is stated that xλ± logk x±, λ = −1,−2, . . . , is an associ-ated homogeneous distribution of degree λ and order k, and P

(x−n± logk−1 x±

)is an associated homogeneous

distribution of degree −n and order k.It is easily verified that these distributions are not associated homogeneous in the sense of the above defini-

tion [18, Ch. I, §4.1.] for k = 2, 3, . . . . For example, log2(ax±) = log2 x± + 2 log a log x± + log2 a. It isimpossible “to preserve” the definition [18, Ch. I, §4.1.] even if we suppose that a distribution fk−1 may dependon the variable a. Thus a direct extension of the notion of the associated eigenvector to the case of distributionsis impossible. This is connected with the fact that any homogeneous distribution is an eigenfunction of all thesimilitude operatorsUaf(x) = f(ax) (for all a > 0), while any distribution xλ± logk x±

(or P

(x−n± logk−1 x±

))is not an associated homogeneous function of all the similitude operators.

Thus we introduce a notion of an associated homogeneous distribution by Definition 7.1. The distributionsmentioned above are associated homogeneous in the sense of Definition 7.1.

7.2

Let us introduce a linear space E ⊂ D′(R) generated by linear combinations of associated homogeneous distri-butions: (x − ck ± i0)λ logp(x − ck ± i0), λ ∈ R, p = 0, 1, 2, . . . , ck ∈ R, k = 1, . . . , s. For λ > 0 we set(x − ck ± i0)λ ≡ (x− ck)λ.

Now we introduce the space of harmonic regularizations of distributions from E:

h = spann(x − ck ± iε)λ logp(x − ck ± iε) : λ ∈ R, p = 0, 1, 2, . . . ; ck ∈ R, k = 1, . . . , s; ε > 0

o,

where span denotes the linear span, z = x+ iε, z = x− iε, log z = log |z| + iargz, Imz = 0, argz ∈ [0, π).Elements of this space belong to the Vladimirov algebras [54].

We introduce an associative and commutative differential algebra h∗ with unity and without zero divisors gen-erated by the space h. Elements f∗(x, ε) of this algebra are finite sums of finite products of functions f(x, ε) ∈ h,ε > 0.

A product of distributions from the space of distributionsE will be defined as the weak asymptotic expansionof the product of regularizations of distributions as ε→ +0.

Theorem 7.3 Each element f∗(x, ε) of the algebra h∗ has a unique canonical representation in the form of aweak asymptotic series

f∗(x, ε) =∞∑

m=0

n0(m)∑n=0

fαm,n(x)εαm logn0(m)−nε , ε −→ +0 , (7.2)

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1334 Shelkovich : New versions of the Colombeau algebras

where αm ∈ R is an increasing sequence, n0(m) ∈ N0, the coefficients fαm,n(x) are distributions from thespace E. In particular, for any element f∗(x, ε) ∈ h ⊂ h∗ its canonical representation (7.2) is its Taylorexpansion.

The equality (7.2) is understood in the weak sense, i.e., for all ϕ(x) ∈ D(R) and all m ≥ 0, n =0, 1, . . . , n0(m):

〈fαm,n(x), ϕ(x)〉

= limε→+0

〈f∗(x, ε), ϕ(x)〉 −

m−1∑k=0

n−1∑j=0

〈fαk,j(x), ϕ(x)〉εαk logn0(k)−jε

ε−αm log−(n0(m)−n)ε .

This theorem was proved in [9, Corollary from the Theorem 4.3.] for the case ck = 0, k = 1, . . . , s. Theproof can be easily generalized for the general case by using the ideas presented in [46] and [48].

Definition 7.4 1) Either the weak limit f∗ε (x) = A∗[f∗(x, ε)] = limε→+0 f

∗(x, ε) ∈ E for f∗(x, ε) ∈ h;2) or the asymptotic expansion (7.2)

f∗ε (x) = A∗[f∗(x, ε)] =

∞∑m=0

n0(m)∑n=0

fαm,n(x)εαm logn0−nε , ε −→ +0 , (7.3)

given by Theorem 7.3, for f∗(x, ε) ∈ h∗ \ h is called the asymptotic distribution f∗ε (x) = A∗[f∗(x, ε)] [9].

(Here the subscript ε in f∗ε (x) means that the expansion depends on an asymptotic sequence εαm lognε.)

By E∗ = A∗[h∗] we denote the linear space of all asymptotic distributions f∗ε (x) = A∗[f∗(x, ε)], f∗(x, ε) ∈

h∗. Thus, according to (7.3), any asymptotic distribution is represented as a weak asymptotics series (1.1). It isproved in [9] that the mapping A∗ : h∗ → E∗ is an isomorphism and A∗[h] = E, i.e., the linear span of the setof associated homogeneous distributions E is a subspace in the space E∗.

Each asymptotic distribution f∗ε (x) fromE∗ \E can be identified with a vector-valued distribution (sequence)

finite from the left (1.2), where distributions f(αm,n)(x) ∈ A are coefficients in the weak asymptotic series (7.2)of the regularization f∗(x, ε) = A∗−1f∗

ε (x). Each distribution f(x) from the subspace E can be identifiedwith the vector-valued distribution f∗(x) =

(f(αm,n)(x)

), where f0,0(x) = f(x), while all the other compo-

nents fαm,n(x) are equal to zero. We denote by A′∞ the infinite-dimensional linear subspace of vector-valued

distributions (sequences) (1.2).Now let us introduce the operation of multiplication on the space of asymptotic distributions E∗ (or the space

of vector-valued distributions A′∞), and thus, for distributions from E.

Definition 7.5 The product of asymptotic distributions f∗ε,1(x), f

∗ε,2(x) from E∗ is defined as the asymptotic

distribution

f∗ε (x) = f∗

ε,1(x)f∗ε,2(x) = A∗

[f∗1 (x, ε)f∗

2 (x, ε)]∈ E∗ , (7.4)

where f∗k (x, ε) = A∗−1f∗

ε,k(x) ∈ h∗ is a regularization of asymptotic distribution f∗ε,k, k = 1, 2.

Since h∗ is an associative and commutative algebra of functions and A∗ : h∗ → E∗ is an isomorphism,multiplication (7.4) is associative and commutative. The multiplication (7.4) turns the space of asymptotic distri-butions E∗ (or the space of vector-valued distributions A′∞) into an algebra. It is a nontrivial multiplication ofasymptotic series (1.1).

Consider the space D∗ of vector-valued functions (sequences) finite from the right

ϕ∗(x) =(ϕβm,n(x)

), ϕβm,n(x) ∈ D(R) , (7.5)

where βm ∈ R is an increasing sequence, n ∈ N0.As in [48], [10], and [27]–[29], one can consider vector-valued distributions f∗(x) from A′∞ as continuous

linear vector-functionals on the space of test vector-valued functions (sequences) D∗. Here the action of a vector-valued distribution f∗(x) on a test vector-valued function ϕ∗(x) is defined as

〈f∗(x), ϕ∗(x)〉 =∞∑

m=0

∞∑n=0

〈fαm,n(x), ϕ−αm,−n(x)〉 ,

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where the last sum contains a finite number of terms. Obviously,

〈f∗(x), ϕ∗(x)〉 = P .f . limε→+0

〈f∗(x, ε), ϕ∗(x, ε)〉 ,where

ϕ∗(x, ε) =0∑

m=−∞

∞∑n=−∞

ϕαm,n(x)εαm log−nε , ε > 0 ,

P.f. is the finite part of the weak limit limε→+0 in the Hadamard sense.By analogy with the construction used in the papers [27]–[29], the space D∗ can be equipped with sequentially

complete topology.

Example 7.6 (See [9], [47] and [48])

(a)

(x + i0)−m (x− i0)−n

= A∗[(x+ iε)−m(x− iε)−n]

= (−1)m−1

(i

2

)m+n−2

πm+n−2∑

s=0

B+s

is

s!δ(s)(x)ε−m−n+s+1

+ (−1)n−1 iπ

(m+ n− 1)!

((−1)m + 2−m−n+2B+

m+n−1

)δ(m+n−1)(x) + P (x−m−n)

+ (−1)m−1

(i

2

)m+n−1 ∞∑s=m+n

is((−1)s(x+ i0)−s−1B+

s − (x− i0)−s−1B−s

)ε−m−n+s+1 , ε −→ +0 ,

where m,n = 1, 2, . . . ; B+s = Bs(m,n), B−

s = Bs(n,m), and the coefficients Bs(m,n) have the form

Bs(m,n) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

s∑k=0

Cn−1m+n−k−2C

ks (−2)k , 0 ≤ s ≤ m− 1 ,

m−1∑k=0

Cn−1m+n−k−2C

ks (−2)k , s ≥ m.

(b)

x δ(x) = A∗[12((x+ iε) + (x − iε)

) · −12πi

(1

x+ iε− 1x− iε

)]=

∞∑k=0

(−1)k

1πP

(1

x2k+1

)ε2k+1 +

1(2k + 1)!

δ(2k+1)(x)ε2k+2

, ε −→ +0 .

(c) (δ(x)

)2 = A∗[(−1

2πi

(1

x+ iε− 1x− iε

))2]

=1

2πεδ(x) +

12π2

∞∑k=1

(−1)k

π

(2k)!δ(2k)(x)ε2k−1 + P

(1

x2k+2

)ε2k

, ε −→ +0 .

Here we consider the element(δ(x)

)2 ∈ A′∞ in the form of an asymptotic distribution. On the other hand,

(δ(x)

)2 = (. . . , fk(x), fk+1(x), . . . )

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1336 Shelkovich : New versions of the Colombeau algebras

is a vector-valued distribution, where fk(x) = 0, k < −1, f−1(x) = 12π δ(x), f2k−1(x) = (−1)k

2π(2k)! δ(2k)(x),

f2k(x) = (−1)k

2π2 P (x−2k−2), k = 1, 2, . . . . For all test functions ϕ∗(x) ∈ D∗ we haveD`δ(x)

´2, ϕ∗(x)

E

=1

Dδ(x), ϕ1(x)

E+

1

2π2

∞Xk=1

(−1)k

(2k)!

Dδ(2k)(x), ϕ−2k+1(x)

E+

fiP

„1

x2k+2

«, ϕ−2k(x)

flff.

7.3 E∗ as a subalgebra of Gharm(R)

Let us consider the space H0(R) of all sequences F (x, ε) = f(m)(x, ε) : m = 0, 1, 2, . . ., where f(m)(x, ε)is given by formula (4.5) and f(x, ε) ∈ h (see Definition 5.4). Let H∗

0(R) be an associative and commutativedifferential algebra generated by the space H0(R). According to Subsection 5.3, we have H0(R) ⊂ H∗

0(R) ⊂H∗(R), H∗

0(R) ∩ N ∗(R) = 0. If f∗(x, ε) ∈ h∗ are finite sums of finite products of functions f(x, ε) ∈ hthen we denote by F ∗

m(x, ε) finite sums of finite products of functions f(m)(x, ε). In view of these facts andTheorem 5.6, the mappings

h f(x, ε) −→ f(m)(x, ε) + N ∗(R) ∈ Gharm(R) ,

h∗ f∗(x, ε) −→ F ∗m(x, ε) + N ∗(R) ∈ Gharm (R)

(7.6)

are one-to-one.Taking into account (7.6) and the fact that A∗ : h∗ → E∗ is an isomorphism, we can identify a product(

f(m),1(x, ε)+N ∗)(f(m),2(x, ε)+N ∗) from the Colombeau algebra Gharm (R) with the asymptotic distribu-

tion A∗[f1(x, ε)f2(x, ε)] from E∗, where fj(x, ε) ∈ h is a harmonic regularization of distribution fj ∈ E, andf(m),j(x, ε) is its canonical m + 1-harmonic regularization, respectively, j = 1, 2. Therefore, by (7.6) we candefine the mapping E∗ → Gharm(R) which is an algebra monomorphism. Thus the product (7.4) in the algebraE∗ coincides with the product in the Colombeau’s algebra Gharm (R).

8 Some applications

It is well-known that Colombeau’s algebras are applied in nonlinear problems of mathematical physics. Sincethese applications are described in many papers and books (see, for example, [19], [40] and [41]), below werestrict the discussion to the case of the algebra of asymptotic distributions E∗.

The fact that elements of the algebra E∗ are represented in the form of weak asymptotic series (1.1) allowsto consider products of asymptotic distributions as weak asymptotic series up to oD′

(εN logpε

), ε → +0 (for

details, see [9] and [27]–[29]). This representation is very useful for various applications.A. Using Example 7.6, (c), we obtain up to OD′(ε), the following relation

(δ(x)

)2 =1

2πεδ(x) , ε −→ +0 .

This relation can be used in the proof of the optical theorem in quantum scattering theory in the form(δ(E)

)2 =δ(E) · δ(0), where δ(0) = T/(2π), T → +∞, E is the energy, T is the time [4].

B. In [11]–[16], [49] and [50] a new approach (the weak asymptotics method) to solving the singular-frontproblem was developed. Recall that a singular-front problem is the following: for a system of conservation lawsto describe the propagation and interaction of singular fronts starting from the initial positions. The classicalsingular-front problem for the case of shocks was solved by A. Majda [37] and [38]. The key role in our methodis played by the definition of a weak asymptotic solution to the Cauchy problem, which admits passing to thelimit in the weak sense as ε→ +0, where ε > 0 is the regularization parameter. The background of our approachis constructing singular superpositions of distributions (multiplication of distributions).

As is well-known, the theory of nonlinear hyperbolic systems usually assumes systems to be strictly hyperbolicwith genuinely nonlinear or linear degenerate characteristic field and have the conservative form. On the otherhand, it is recognized that most of the physical systems do not fit into the standard theory of conservation laws.

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The Riemann problem in this “nonclassical” situation does not possess a weak L∞-solution except for someparticular initial data. In contrast to the standard cases, here the (linear) component of the solution may containDirac measures and must be sought in the space of measures, while the nonlinear component of the solution hasbounded variation (see below). This is the reason to introduce a new type of generalized solutions called δ-shocks.

Recently, the theory of δ-shock type solutions for systems of conservation laws has attracted intensive atten-tion. Several approaches to constructing δ-shock type solutions are known. An apparent difficulty in definingsuch solutions arises due to the fact that, to introduce a definition of the δ-shock type solution, we need to definethe singular superpositions of distributions (for example, the product of the Heaviside function and the deltafunction). We also need to define in which sense a distributional solution satisfies nonlinear systems.

Study the propagation of δ-shocks in two (one-dimensional) hyperbolic systems of conservation laws. Weconsider the system

L1[u, v] = ut +(f(u) − v

)x

= 0 , L2[u, v] = vt +(g(u)

)x

= 0 , (8.1)

where f(u) and g(u) are polynomials of degree n and n + 1, respectively, n is even. The well-known Keyfitz–Kranzer system [26]

ut + (u2 − v)x = 0 , vt + (u3/3 − u)x = 0 (8.2)

is a particular case of system (8.1). We also consider the system

L1[u, v] = ut +(F (u)

)x

= 0 , L2[u, v] = vt +(vG(u)

)x

= 0 , (8.3)

where F (u), G(u) are smooth functions. Both systems (8.1) and (8.3) are linear with respect to v, u =u(x, t), v = v(x, t) ∈ R, and x ∈ R.

In order to describe the propagation of δ-shock, we must solve the Cauchy problem for the systems (8.1) or(8.3) with the δ-shock front initial data

u0(x) = u00(x) + u0

1(x)H(−x) ,v0(x) = v0

0(x) + v01(x)H(−x) + e0δ(−x) , (8.4)

where u0k(x), v0

k(x), k = 0, 1, are given smooth functions, e0 is a given constant, H(ξ) is the Heaviside function,δ(ξ) is the Dirac delta function.

We shall seek a δ-shock wave type solution of the Cauchy problem (8.1), (8.4) or (8.3), (8.4) in the form

u(x, t) = u0(x, t) + u1(x, t)H(−x+ φ(t)) ,

v(x, t) = v0(x, t) + v1(x, t)H(−x + φ(t)) + e(t)δ(−x+ φ(t)) ,(8.5)

where u0(x, t), u1(x, t), v0(x, t), v1(x, t), e(t) and φ(t) are desired functions. Here (8.5) corresponds to thestructure of the initial data (8.4). A pair of distributions (8.5) will be called a δ-shock wave type solution of theCauchy problem if it satisfies special integral identities [15], [16], [49] and [50].

Systems (8.1) and (8.2) have a specific property. Namely, they have no balance of singularities. Let (u, v) be aδ-shock type solution of system (8.2). Hence, u contains the Heaviside functionH , while v contains the Heavisidefunction H and δ-function (see (8.5)). It is easily seen that the term (u2 − v)x contains the distributions H , δ,δ′, while the term ut contains only the distributions H and δ. Analogously, the term vt contains the distributionsH , δ, δ′, but the term (u3/3 − u)x contains only the distributions H , δ. Seemingly, it is impossible to obtainδ-shock type solutions for systems (8.1) and (8.2). Nevertheless, we proved the existence of exact δ-shock wavetype solutions for above systems [49] and [50]. Note that in [26], only an approximate solution of the Cauchyproblem for system (8.1) with piecewise constant initial data (8.4) was constructed. Moreover, in [26] it is notdefined in which sense a distributional solution satisfies the nonlinear system (8.1).

A δ-shock wave type solution (8.5) of the Cauchy problem is constructed as a weak limit of a weak asymptoticsolution. A pair of functions

(u(x, t, ε), v(x, t, ε)

)will be called a weak asymptotic solution of the Cauchy

problem (8.1), (8.4) or (8.3), (8.4) if the following estimates hold [15], [16], [49] and [50]:

L1[u(x, t, ε), v(x, t, ε)] = oD′(1) , L2[u(x, t, ε), v(x, t, ε)] = oD′(1) , ε −→ +0 .

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1338 Shelkovich : New versions of the Colombeau algebras

We shall seek a weak asymptotic solution in the form of the sum of the singular ansatz (8.5) regularized withrespect to singularities and corrections:

u(x, t, ε) = u0(x, t) + u1(x, t)Hu(−x+ φ(t), ε) +Ru(x, t, ε) , (8.6)

v(x, t, ε) = v0(x, t) + v1(x, t)Hv

(− x+ φ(t), ε)

+ e(t)δ(− x+ φ(t), ε

)+Rv(x, t, ε) , (8.7)

where, according to (4.4), δ(x, ε) = 1ε ωδ

(x/ε

)is a regularization of the delta function,Hj(x, ε

)=

∫ x/ε

−∞ ωj(η) dηis a regularization of the Heaviside function H(x), j = u, v, ωδ(η), ωu(η) and ωv(η) are mollifiers. Here wecould also use harmonic regularizations δ(x, ε) = ε

π(x2+ε2) andHu(x, ε)

= Hv(x, ε)

= 12

(1+ 2

π arctan(x/ε)).

Here the so-called corrections Ru(x, t, ε) and Rv(x, t, ε) are desired functions which must admit the esti-mates:

Rj(x, t, ε) = oD′(1) ,∂Rj(x, t, ε)

∂t= oD′(1) , ε −→ +0 , j = u , v .

Since the generalized δ-shock wave type solution (8.5) is defined as a weak limit of (8.6), in view of theabove estimates, the corrections Ru(x, t, ε) and Rv(x, t, ε) do not make a contribution to the generalized so-lution of the problem. However, these terms give contributions to the weak asymptotics of the superpositionf(u(x, t, ε)

)− v(x, t, ε), g(u(x, t, ε)

)or F

(u(x, t, ε)

), v(x, t, ε)G

(u(x, t, ε)

), and hence play an essential role

in the construction of the generalized solution to the problem.In the context of constructing of a δ-shock wave type solution (8.5) of the Cauchy problem (8.1), (8.4) we

obtain explicit formulas for the “right” singular superpositions [49] and [50]:

f(u(x, t)

)− v(x, t) def= limε→+0

(f(u(x, t, ε)

)− v(x, t, ε))

= f(u0) − v0 +[f(u) − v

]H(−x+ φ(t)) ,

(8.8)

g(u(x, t)

) def= limε→+0

(g(u(x, t, ε)

))= g(u0) +

[g(u)

]H(−x+ φ(t)) + e(t)

[f(u)][u]

δ(−x+ φ(t)) ,(8.9)

where the distributions u(x, t) and v(x, t) are defined in (8.5), the functions u(x, t, ε) and v(x, t, ε) are definedin (8.6), [h(u, v)] is a jump in the quantity h(u, v) across the discontinuity line x = φ(t), and the limits areunderstood in the weak sense.

In the context of constructing of a δ-shock wave type solution (8.5) of the Cauchy problem (8.3), (8.4) weobtain the following “right” singular superpositions [15] and [16]:

F(u(x, t)

) def= limε→+0

F(u(x, t, ε)

)= F (u0) +

[F (u)

]H(−x+ φ(t)) , (8.10)

v(x, t)G(u(x, t)

) def= limε→+0

v(x, t, ε)G(u(x, t, ε)

)= v0g(u0) +

[vG(u)

]H(−x+ φ(t)) + e(t)

[F (u)][u]

δ(−x+ φ(t)) .(8.11)

In fact, by formulas (8.10) and (8.11) we define the singular superposition (the product) of the Heaviside func-tion and the delta function. In contrast to system (8.3), in the case of specific systems (8.1) and (8.2) we do notdefine (!) the product of the Heaviside function and the δ-function. We stress that, although in the case of systems(8.1) and (8.2), u(x, t) does not depend (!) on the term e(t)δ(−x + φ(t)) (in view of (8.5)), the “right” singu-lar superposition g

(u(x, t)

)determined by (8.9), does depend (!) on this term. Thus one can say that the term

e(t)δ(−x + φ(t)) “appears from nothing”, and the “right” singular superposition g(u(x, t)

)is determined ac-

cording to the nonlinear system (8.2). Thus, the unique “right” singular superpositions (8.8)–(8.11) can beobtained only by constructing a weak asymptotic solution (8.6) of the Cauchy problems.

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Math. Nachr. 278, No. 11 (2005) / www.mn-journal.com 1339

Acknowledgements The research was supported by DFG Project 436 RUS 113/593/3 and by the Russian Federation forBasic Research, Grant 02-01-00483.

The author is greatly indebted to Michael Oberguggenberger who read the first manuscript version of the paper and drewthe author’s attention to some inaccuracy and misty ideas. The author is also grateful to the referee whose remarks helped inimproving the paper.

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