REND. SEM. MAT. UNI VERS. POLITECN. TORINO

Vol.43°, 1 (1985)

Agostino Prastaro

GEOMETRODYNAMICS OF NON-RELATIVISTIC CONTINUOUS MEDIA: II. DYNAMIC AND CONSTITUTIVE STRUCTURES/**

Summary: An intrinsic formulation of Continuum Mechanics on the affine Galilean space-time M is given, emphasizing the role of the dynamic equation as a geometric structure. In particular, a continous body is described as a geometric structure on M. Thus, the study of symmetry properties of this structure allows us to give useful classifications of continous bodies and to state generalized forms ofNoether's theo­rem. These considerations are applied to incompressible fluids. Existence and uniqueness theorems for regular solutions are obtained.

1. Introduction.

The purpose of this paper is to give an intrinsic space-time formulation, of Newtonian Continuum Mechanics, emphasizing the role of some geometri­cal structures which support continuum physics.

The theoretical interest of non-relativistic Continuum Mechanics is far from being exhausted; recent developments of General Relativity and gauge theoriesallowed to extend new suggestive points of view to Galilean Mechanics, (seerefs. [2, 13, 14, 15, 16, 22, 26]).

The aims of this paper are the following. First, we consider the flow structure on Galilean space-time and some

useful geometric objects related to a flow. We define a continous body with respect to a frame \p as a geometrical structure of Galilean space-time M + Classiftcaziorie per soggetto: AMS (1980): 55R05, 58G35, 58640, 58H99, 73B05, 73B10, 76A02.

90

by means of a fiber bundle C and a differential equation Ek CJ$k (C) on € . This geometrical picture allows us to treat Continuum Mechanics as a uni­fied field theory and to consider constitutive, dynamical and symmetry pro­perties in a more natural way. In the 3-plet (€, Ek, \p) all the information on a continous body is enclosed. So, any other geometrical formulation of Continuum Mechanics, e.g. lagrangian, symplectic or gauge type is essentially surmonted by the latter general structure.

To be noted that in our geometric formulation we systematically use de­rivative spaces (introduced by us in ref. [23]) which makes the substitution of jets of a map with their derivatives possible. Jet spaces have a natural embedding in derivative spaces which give a natural environment for derivatives. The rea­son for introducing such objects is essentially that in this way we can treat derivatives and any related object (as connections, Lie derivative etc.) as "ten­sor fields"Le! we can represent them by means of differential algebraic formulas. To this mathematical reason we add a physical meaning to use derivative spa­ces. In fact, by means of an observer we can induce on a vector bundle of geometric objects a splitting in time and space components. Then, by using the canonical embedding of jet-derivative spaces in derivative spaces we can induce the space and time analysis frome the derivative spaces to the jet-deri­vative spaces. These considerations are developed in part I [25].

Of particular interest are the following results:

1) a generalization of the concept of isotropy related to the introduction of a physical structure k on M (e.g. the metric field g) , which recovers the usual formulation of isotropy for continuous bodies when k =g .

2) some generalizations of Noether's theorem allowing conservation laws which are not derived from any symmetry of Lagrangian but rather from symmetries of the dynamic equation.

3) existence theorems of global Cw solutions for perfect incompressible fluids.

4) the proof of non existence of C°° solutions for the non isothermal Na-vier-Stokes equation, for generic generalized initial condition.

Notations.

(Af, J\4, a) = affine structure of Galilean space-time M .

(T, T, P) - affine structure of time space T.

91

<£ = {T:M-+ T}= fiber structure of Galilean space-time.

M =F fiber of r over t £ T.

5 = ker (DT ) C Af, g = Euclidean structure on S , g —field of geometric objects on M corresponding to ~g .

D = symbol of derivative, D = symbol of the Frechet derivation.

fj"=volume form on S corresponding to g;T? = field of geometric objects on M corresponding to rf.

/ = space-component of a field of geometric objects f-:M-*W with respect to a frame \jj [25] .

G = Galilean group; SG —special Galilean group; LG = linear Galilean group; SLG = special linear Galilean group. [25].

8 = symbol of infinitesimal variation [23 ].

a = canonical differential form on M .

(Ek)+h —h -th prolongation of a differential equation Ek CJ& (f)

[5].

gk symbol of the differential equation Ek CJ &k (£) .

2. Flow and related fields of geometric objects.

In this section we shall introduce a geometric structure that interprets the fundamental concept of flow in the framework of Galilean space-time. Furthermore, we shall emphasize some geometric objects that are directly re­lated to a flow.

Definition 2.1. A (global) flow on M is a map 0 : T X M -*M such that: (a) <j)x:M •+ M is a transformation of M ,V\ET y (b) (j)0=idM't

A local flow is a flow defined on open J XU in T X M .

A global flow <j> on M gives to M the structure of principal fiber bun­dle: (M, (j> /, M/T; T) , where 4> I is the canonical projection M-+M/T.

A very strong relation exists between flow and frame.

92

Proposition 2.1. A flow is equivalent to a frame. More precisely one has the following implications:

(a) frame \p = > flow 0 = \p °(0X irfM ) ° p ° (Z'^XT, TT2) : T X M ->M , being ir2 the projection TXM-+M and p the bijection TXTXAf-> -+TXTXM;

(b) flow ^ = > frame i//0 = 0 o (jS'Xi^) o (z^Xr , n2) :. T X M -> M , being ft' :T XT->T','f? (t,t') = t - t' . We call \//0 the co-moving frame of the flow 0 .

Proposition 2.2. For any flow 0 we get: r(0 (p, X)) = r(p) + X . Thus any map 0Xf =0X IAff is a diffeomorphism 0Xf : M't0-+Mt+K . In other words, (j>K is a fiber bundle diffeomorphism of & . The corresponding diffeomor­phism 0 on the basis T is an affine map such that D 0 = zdT .

In the following table we list some important fields of geometric objects associated with a flow.

Tab. 1 - Important fields of geometric objects associated with a flow 0

name definition particular characterizations rigid isochoric

velocity 0 = 3 0 : M ^ ^ ( ^ ) «

acceleration $= Vjfi'.M-*vTM inf. strain e = ( l / 2 ) J C ^ e = 0

spin s = (1/2) rot 0 •

deformation gradient F = D 2 0 Fxeo(S) det F = ± 1 strain qr*fc<+> ^ = |

volume strain VK=€* V=I? inf. volume strain v=^u v = 0

displacement vector a(X,p) = 0(X,p)-p

(-f) The distance d of two points p,p'GM{ is changed under the folio-

93

wing equation:

d(^(p'),<t>x(p)) = (Ux(p)(v>v)+ \\w\\)1/2

with v=p'-p and w G 5 , w = 4>x (pf)~4>x (p)~D <j>x (p) (v) .

3. Galilean continuum bodies.

Let us now give the geometric structure defining a continous body.

Definition 3.1. 1. A Galilean continuum system (of order k ) is a couple CS = ((E , Ek) where-, (i) (C, M, ir^; C) is a bundle of geometric objects on M (configuration bundle) a. section c of ire is called a configuration; (ii) Ek CJ$k ((E) is a differential equation of order k on € (dynamical equation), J$k ((E) is called the state space of CS . 2. An elastic constitutive map of CS is a differential operator on €, K : /#* (C) ->• iPC ; an anelastic constitutive map of CS is a functional fiber dif­ferential operator on (EK: C°° (J$k "(C))-> C°° (̂ C) (see ref. [27,11]); JK is a bundle of geometric objects on M . We say that an elastic constitutive map K is reducible to order k', 0< k' < k , if a differential operator of order k' exists on (E, K1, such that K = K' ° 7rfe ,̂ . We say K trivial if a field of geometric objects exists on M K' such that K = K' ° irk .We call K^ntity, corresponding to the configuration c , the field K . c = K o Dk c . Similar de­finitions can be given for anelastic constitutive maps. 3. An internal constraint is a subbundle C' of € over M.

Example. An important example is the Galilean electromagnetic system ES having (E = Aj M and Ex = ker (d) = Maxwell equation (d is the exterior differential). A section of A^ M is denoted by F. With respect to a frame \jj one can associate to ES two constitutive maps: (a) Electric map E.F^ = i//JF ; (b) Magnetic map H.F = * (F) . Then we can see that F can be written in the usual matrix form by means of E. and H. . Moreover, with respect to the frame \p , the Maxwell equation can be written in the follo­wing way: rotE=£. H, (div H) . = 0 .

94

Definition 3.2 A continuum system CS has a flow structure if the configura­tion bundle C has the following structure:

where €, is a bundle of geometric objects on M .

In the following table we list some important constitutive maps of a con­tinuum system with flow structure. In that table grad 0 is defined by

grad^ 6 = 'g*j.vT.M o J T # M W)oDd:M^vTM

being yu r*^ : (T*Af) J ->z;T*M the isomorphism given by j v T * M :«»->• - + ( 7 r r , A , ( a ) , / 3 s a | S ) ( + > . • "

Tab. 2 - Important constitutive maps

name K ^.-symbol Kc-symbol

rheological vS\M = P St. i

0. *' = P body force vTM # . ^ . c = £

mass density T%M = R A

R. i . c = p temperature TjjAf = 0 0. @.c = d

power force T%M W. W.c = w=pg(B,v^)

thermal conductivity T°0M = A A. A. c = X

heat flux vTM c Q. c=q^~\ grad^0

momentuum flux SIM 7: T. c = p v® v — P

interior energy. T°0M = E, £. E. c = e

energy flux TM * tc= vpVe+teiv^v^

+ q-'glv^ )JP )]

mass flux / * ( # ) Jf. ^ c = pz>

*

(+) 'g is the canonical isomorphism vT*M=vTM .

95

Definition 3.3 A Galilean continuous body is a triplet CB = (C;Ek>\lj), where: (i) i// is a frame; (zV) (0, £^) is a continuum system with flow-structure and such that Ek CJ&k ((C) is defined by means of the following exact sequence:

0-+Ek*-*J$k (C) m=&'»>*\ 1K=J&($)®J&2 (^)e'TgiW ,

where. JD is called dynamical operator and defined by means of the follo­wing differential operators on C:

(a) 2t. c — '^t. (v, c) = div(pv) : continuity operator;

(b) &. c = &. (v , c) = div ( r c)- pB : motion operator;

(c) A.'C = A.(V,C) = div (f. c) — W.c : energy operator.

By a direct computation we can prove that a dynamical configuration c = (v , c) must satisfy the following system of partial differential equations in adapted coordinates:

(1) continuity equation: p(G'iQ + G*-k vk) + (&v0- P) + ^ V (P^1) = 0 J

(2) motion equation: p [G^00 + 2G;'fe z>* 4- G\k vlvk + (<bc0.V) + v* (dx{. tf)] +

(3) energy equation: p [(dx0. e) 4- vk (bx . e)] 4- \GJ.{q% + (bxi.q

1)]- [Gjk (vk-

Remark 3.1 Given a configuration space € and a frame \jj , a. Galilean con­tinuous body can be identified as a point of the constitutive configuration space *& (€, K (M) , namely the space of constitutive mapping of € into K (M) = vTM © £© 0 © A ©JP © £ . #f (Af) is called canonical constitutive space. We call a point of ^(C, IK (M)), a constitutive configuration and we write (.^., #., 0., A., $., E.) . Since € and £^ are two fundamental elements of CB we can obtain classifications of continuous bodies specifying some restriction on these structures, or on the constitutive maps which concur to determine Ek , i.e. giving some restrictions on the constitutive configura­tions. So we bbtain the definitions of simple materials, viscoelastic materials, perfect fluids, elastic materials, etc..

96

4. Symmetries and isotropy

Other important categories of materials are identified in relation to some symmetry properties of CB. Therefore let us now clarify the concept of sym­metry properties for continous bodies and for constitutive maps.

Definition 4.1 A super-symmetry f of CB is a transformation / of J$k ((£) such that it preserves CB, i.e.:

(i) f is a fiber bundle transformation over a physical symmetry of TT̂ , that is a fiber bundle transformation / of <E over a transformation fM^Q;

(ii) f is a fiber bundle transformation of Ek . ' /

The set of super-symmetries of CB forms a group G (CB) called su­per-symmetry group of CB . We set H = HomM (TT , TTC) . Let <ge denote the physical group of <L , that is, the set of physical symmetries of n^ . One has the canonical isomorphism; ^e— GXH. Furthermore, we have the canonical homomorphism <p : G (CB) -> # c = G X H , given by <£(/) = ( / M ' ft ° C (/_1)) • W e s e t - ^c (CB) = Im0C^ c . In the following table some important subgroups of G (CB) are listed.

Tab. 3 - Important subgroups of G (CB)

name symbol definition canonical isomorphism

symmetry g.

super-gauge g.

dynamical g.

gauge g.

CG (CB)

SGau(CB)

DG (CB)

G (CB)

{feG(CB)\f=j&%)} ~cy€<pB)cy c

{f^G(CB)\fc=idt,fM=id} aHom e (p M ,p M ) (CS)CH ( + )

{f<=G(CB)\f=J&(fc), SDG(CB)CG fe=c(fM)}

{f€G(CB)\f=Jdk(fe), =H(CB)CH /eejf(t)}

G (CB) acts naturally way on the configuration space C°° ((E) and on the

state space C°° (/#* ((E)). Thus having fixed a configuration c and a state

(+) H S H o m j ^ y . ( t) i .e . /M = « V

97

s we denote by G (CB) and G (CB) the corresponding isotropy groups respectively.

Proposition 4.1 1. Let c be any dynamical configuration (that is Dkc.M^> -+Ek) Ai c( =f~g°c°fM with f€ £ ^ , then c' is also a dynamical confi­guration if fc ec <s€ (CB) , 2. CGc(CB) = CGDkc(CB).

3. Let "0 be a 1-dimensional subgroup of G (CB) . Then:

(i) <l>CGc(CB) < = >£xc = 0 with X = 3 0 ;

(») <l>CGDkc(CB) < = >£xDk c = 0.

Definition 4.2 An infinitesimal super-symmetry of CB is a Killing vector field X on /#* (C) with respect to G (CB) .

Proposition 4.2 Let X be a vector field on J$k ((C) . Then the following propositions are equivalent: 1) X is an infinitesimal super-symmetry of CB ; 2) (a) X\ Ek is a vector field on Ek ;

(b) X projects on an infinitesimal physical symmetry of <C , that is a vector field X on C which projects on an infinitesimal symmetry XM

of the space-time M .

Remark 4.2 If f. IR XJ$k (<£)-+J$k (<E) is a 1-dimensional subgroup of G (CB), then X = df is an infinitesimal super-symmetry of CB . We also have XM = dfM .and X€

= 3 / c , where fM and f€ are the correspon­ding 1-parameter group of transformations on M and <E respectively.

Proposition 4.3 Let G be an r-dimensional Lie group. Let {Xlt-...,Xf} be the infinitesimal generators of the action of G on J$k (C) . Let Ek

be locally characterized by equations F* = 0 , 1 <i <s . Then G CG (CB) if and only if

*

XyFr=0 ,Kk<r , l< /<s = w-4 = dira/#*(0-dim^ .

98

Proof; Let 0 be the action map of G on J$k ((E). Let us put f =Fo</) . Let us assume G C G (CB). Then, since DVu(e) = 0J(idQ, F) ° D<j>u (e) , Vw €J&k (€) , e = unit of G , we get in particular that if u E£^ 0 ' = 0 so D$l (e) = 0 . On the other hand if {Zl} ..., Z } is a basis of the Lie al­gebra A (G) of G , corresponding to {X1}'..., Xy} we get 0 = <D<^(<?), Z p > = ®(lWG >^)(A^(K)) = (Zp .F')(«) , VuGEk.

The viceversa is trivial. O

Remark 4.3 Under the hypothesis of the last proposition we have that X I Ek

are the infinitesimal generators of the restricted action of G on Ek . Fur­thermore, Xlf..., X generate an involutive r-distribution JEQ on Ek . So Ek is foliated with invariant submanifolds of dimension r locally cha­racterized by equations <£*= const., i = 1,..., q — r, q = dimEk, being {&} a fundamental set of s —r invariant of the distribution JEQ , (i.e. for any vector field X belonging to 2EG is X.<b,= 0).

For a workable criterion to get the infinitesimal generators correspon­ding to the symmetry group CG (CB) we emphasize the following Proposi­tion (see ref. [19]).

Proposition 4.3/1 Let X : Jd* (€).-• TJ&* ((C) be trje k-th prolongation of an infinitesimal physical symmetry of € . Then, X is an infinitesimal symmetry of CB if and only if

X .Fi= 2X( |) Fs

s s

where X^ are smooth numerical functions on J$k (€) .

Let us now consider the relation between the symmetry properties and the constitutive maps of a continous body. This analysis will lead us to give a rigorous interpretation of the concept of isotropy.

Definition 4.3 Let Jf = (JK,M, n^ K) be a vector bundle of geometric objects; so K is a linear functor. Let K. : C°° ((C) -* C°°(IK) be a constitu­tive map for CB . Let ^m=GX HomM (TT^, irK) be the physical group of JC, i.e. the group of fiber bundle transformations \p of IK over tran­sformations \pM €(1 . A K-symmetry is an element of the K-group K (CB)

99

of CB defined by K (CB) = {(/, \jj) G G (CB) X <SK \ K°f= \jj o K, \JJM =

Definition 4.4 A Galilean physical structure is a couple (M , k) , where:

(a) M is the Galilean space-time; (b) k is a field of geometric objects of a suitable bundle of geometric objects. The set of all \jj ^^m

s u c n that k is related to itself is called the general k-isotropy group on M : 0k , i.e.

Examples (1) O as G X O (S). (2) 0 =SG X SO (S) .

Lemma 4.1 Let & be a field of geometric objects of J^and let / be a transformation of M . Then for any numerical function we get fi : M^-IR :

Proof. /•Qi.k)=K(f)oQi.k)ofi=K{f)oXfLof^).{kof1) . By using the condition that If is a linear functor we get:

Lemma 4.2 Let k be a field of geometric objects of j f and let ($, f M ) ^ € Horn (flV, T^ , , then, for any numerical function we get ju : M -* tt :

Proof. Let ^ = h ° K(fM) , then we get:

100

Theorem 4.1 If k' and k are two fields of geometric objects of j f then the following conditions are equivalent:

1. k'= h.k , with h : M -»• 1R numerical function invariant for f EO^ and Qk, ; (Ok is the canonical image of Ok into G). 2- <v = 0*

Proof. 1. Let k'-h.k then if (*P,fM)^Ok we get:

k'=h.k = >j±°k'of~1=h.^okof-1=>^ok,of-i=h.k=kr/ .

Viceversa, if (\p, fM) £Ok,, from k>:=h.k we get:

^ok'of-1 =h. \ljokof~1 =>k'=h. il/ofcof-i =>iJmk=hm [hokof-1'=>k = \ljoko

°/MJ . ie- L W e 0 * • 2. Let Ok,

= Ok . Let {k,k1, ..., km] be a basis of C™ (K) . So we can write: (4) k' = a.k +OL\ k. ,

with a,a1 numerical functions. Let ($,fM)^Okt=Ok. Then we get: i//o£oM

= (a°f$ . i> o k°'fi[+(s*<>f$ . ± o *,.o/J*»(5) *'=(a»/^1).*+(a'»^1). *» °^i°f~M • By comparing (4) with (5) we impose that it must be:

(6) ( a - a ° / - 1 ) . H a i l - ( a , ' o / - 1 ) J o ^ f i = 0 :

But eq. can be in general verified if and only if

( a. -.a © f~j = 0 [ k' = a.k 1 1

a1 = 0 a invariant for /*.. E 0, . • \ ' Ai — k

Definition 4.5 The general k-isotropy group on M with weight a belonging to a sub-group A of IR is: 0Ak = {W,fM)e <gm\-ak=^o k o fj£ ,

Example k =rj , A = {- 1, 4- 1} . Then we get: 0 { _ 1 + 1 j = G X Unim (S). Of course we have 0 C O C O ; 1 x 1 i .

101

Definition 4.6 1. A dynamical configuration c is isotropic if G (CB) = 0 . 2. Let us put: Kc(CB)={(f,\p)eK(CB)\kx = \lj°K.cof-j}CK(CB). Let GKC (CB) and Horn* (irm, n^) (CB) be the canonical images of KC(CB) into G and HomM (n , ̂ ) respectively. Let us put: <$KC(CB)= = G/Cc (CB) X Horn* ' (7^ ,7^) (CB) . Then, the dynamical configuration c is (fe,/Q-isotropic if <$KC(CB) = Ok-

Thus we have the following

Proposition 4.4 1. If c is a dynamical configuration (k,K)-isotropic, then we get K.c = h.k , where h is a ^-invariant numerical function on M .

2. In particular, if c is a dynamical configuration (g, ^-isotropic then B—^t.c —~p.g , where p is a numerical function G-invariant called pressure.

Proof. 1. A direct consequence of Theorem 4.1.

2. A direct consequence of point 1 of Proposition 4.4 and the canonical iso­morphism & P=G X GL (S) .

Definition 4.7 1. A continuous body is isotropic if it has a (k , /0-isotropic dynamical configuration.

2. A continuous body is (k , K)-isotropic if it has a (& , K")-isotropic dyna­mical configuration.

3. A solid material is a continuous body such that <g$(CB) CO , being y& (CB) the canonical image of 0t(CB) into <$p .

4. A /7«id material is a continuous body such that ^^(CB) D O tx, x v .

Proposition 4.5 If CB is a solid continuous body such that there is a (g ,<g) isotropic dynamical configuration c then must c§0t(CB) = O .

Example. In ref. [22] it is proven that perfect incompressible fluids and New­tonian incompressible fluids are ufluid materials" in the sense referred to in Definition 4.7/4.

102

5. Constitutive maps and Noether theorems.

We now consider the properties of invariance of a constitutive map K under a 1-parameter group of /^-symmetries of CB . So we shall obtain gene­ralized forms of Noether's theorem.

Theorem 5.1 Let K: J&k (([)•+ K be a constitutive map of CB = ((E, Ek, \p) and if, \jjt) a 1-parameter group of /C-symmetries for CB . Then we get the following equations:

(7) dK = 0 , with Kt=^toKo^ -

(8) XJDK=j , with X=df and j = X ° K , X = b\p .

We call (7) and (8) Noether equations for K , and /. the Noether sour­ce generated by \p for K.

Proof, Let us consider the "deformation" of K by means of (f \jjt) , that is K = \jj_ o (idJR X A> (idm Xf) o i -JR X J$k ((C) -> K , being ""/; = £ * and i the canonical inclusion IR XJ&k (€)-* 1R XIR X J$k (€). So K0 = = K . On the other hand, being (f.,\p)EK(CB) for any £G2R, it must also be K^==K . Therefore, 3 A - 0 . By the explicitly computation of dK we get a r = X o K - T ( / C ) o X . n

Corollary 5.1 in particular if K is a real valued function / on J&k ((C) eq. (8) gives

(9) XJD/=0".

Corollary 5.2 1. Suppose Ek =kerK . For any dynamical configuration c we get:

(10) (XJDK-)o£ )* r =x 0 ,

being X0 : M-* TIK the vector field X computated along the zero section o f «m •

103

2. In particular if \pt =MC (tyM t) we have:

(11) (XJDK) o Dk c = 0 .

Proof. 1.(10) follows from (8).

2. As K is a linear functor, \pt is a linear map V £ G ZR , so i/^(0) = 0 . D

In coordinates eq. (11) is as follows:

(12) K> oDk c = 0 ; [Xa(3fa. K?)]oDk c = 0 ,

being {#a, z ;} a fibered coordinate system on IK, {fa } a coordinate sy­stem on J$k (€) and # W ° t f .

Remark 5.1 Corollary 5.1 interprets Noether's theorem. In fact if L :J& ((C) -> -*ZR is a first order Lagrangian then eq. (9) gives XJDL o D c = 0 , for any configuration c . Then by a direct computation, and assuming / =•' J& (f^ t) we get in coordinates and for any dynamical configuration c:

(bx . YfX) = 0 with Y» = {(L ° Dc)X^+ ( (9y L) o Dc) [X^o c -

( 1 3 ) - ( a ^ . ^ ' ) ^ ] } ^ , ^ ^ ^ .

Eq. (13) represents Noether's Theorem in its classical form (see ref. [9]). Let us now give a conservation theorem for physical entities represented

by differential forms.

Theorem 5.2 Let K = A°M and the constitutive map K given by K.c = = D* c* X , where 2 is a p-form on J& ((C) . Let us consider a deforma­tion ut of a section u0=Dkc of J$k ((C)->M and a 1-parameter sub­group ^ of G .'Then put (K.c)t =$* (K o« t) . If ( K c j ( = i ( . c 4 ^ , where af is a 1-parameter of (p - l)-forms on M , then we get:

(14) dfi=j,

with: (a) j = -Dhc* (XJdZ) - (XJDk c* d2) = No ether source generated by tf;x=.a«,x = a£;

104

(b) j3 = (XJD*c*2) 4- Dkc* (XJX)-da = Noether current genera­ted by K.

Proof. From eq. (7) we get:

dda = d(Kx) = d(ji*(u*X.) = d(XJDkc*?;) +

+dDkc*(XJX)+Dkc*(XJdX)+(XJDkc*dX) . •

Remark 5.2. If 2 is the Cartan form associated with a first order Lagran-gian L (see ref. [9]) and c is a dynamical configuration then (X_\Dc*dX) = = 0z=Dc*(X_\d2) (Hamilton form of Euler-Lagrangfe equation) and DC*'LF=

= (L ° Dc) T / + V. Thus, eq. (14) becomes d(3 = 0 and we recover the usual expression of Noether's theorem (see ref. [9]). Let us now generalize the abo-be results.

Theorem 5.3 1. Let K (CB) be an m-dimensional Lie group. Then we have m indipendent equations like (14) -with a = 0 and

(15) X = Xj = bu=(Fj)^ e'u?ea , ; = (1, . . . » ,

being (F.^f the representation of ;' = (1, ...,m) infinitesimal generators of K(CB) into the vector space C°°(/#*(£)), {ea } a basis of. C°° (/#*.(€)) and e1 real parameters/*"^ 2. Furthermore, if we consider the sub-group CK (CB) of K (CB) charac­terized by the condition (f,\p)£CK (CB) *>f=J$k (f€) , then

(16) X. = du = (A.)a el cP v ,

with (A.)" representing the Lie algebra of CK(CB) into C°°(C) and

(+) Note that on Galilean space-time there is a canonical volume form 7] given by 77 = = aAr? .

(•+ + ) Of course in this theorem it is assumed that (E can be identified with a vector bun­dle. For example, if (E is given by €=jd(q) ^ <T, where <L is a vector bundle, then with respect to an inertial frame \jj we have the isomorphism jd(q) = vTM. So, with respect to \p we get the indentification of €. with the vector bundle C=vTM X. C

105

{va } a basis of C°° ((E) ;c is a section of 7Tff such that u0 =Dk c .

Proof. Proof can be obtained by a direct but long computation. D

Finally by using Remark 4.3 and Corollary 5.1 we can recognize conser­

vation laws which are essentially derived from the symmetry properties of Ek .

Theorem 5 4 Let HQG (CB) be r-dimensional Lie group. Let <£r , r = = (1, ..., q - r) be (q - r)-independent differential invariants of the involutive distribution JEH . Then for any function $T we get Noether's theorem. For k = 1 we get

with FM = [(^>r0 D c ) x^ + Qyt $ T j 0 Dc (aayj h, ^ = y/'MJij ,

for any dynamical configuration c which should also be extremal foi $T, being Z the deformation of c, by means of a 1-parameter subgroup f of H and XM = dfM .

Proof. In fact 4>T can be considered an invariant Lagrangian under H. •

Remark 5.3 The effective construction of such conservation laws (dynamic conservation laws) requires the development of a variational calculus for La-grangians of higher order than 1. This is well given in ref. [14, 26, 27, 32] by using a completely intrinsic and global framework.

6. Incompressible fluids.

In this section we shall apply the former geometric considerations to a concrete example. Thus, we will obtain some interesting results on the exi­stence of solutions for incompressible fluids. In particular, we shall prove that existence and uniqueness of Cw global solutions for perfect incompressible fluid can be checked by using the formal theory of partial differential equa-

106

tions [5, 6, 7, 8, 9, 10, 11, 17, 19, 20, 21, 27,; 29, 30, 31] beside some consi­derations on the symmetry properties of dynamic equations.

Furthermore, by considering the structure properties of the non-iso-thermic Navier-Stokes equation, we can prove a theorem of non-existence of C°° solutions for viscous incompressible fluids for any choice of initial condi­tions.

Definition 6.1 A perfect incompressible fluid is a continuous body such that

(i) <L = / # (< )̂ e n ® 0 with IT = T[J M = pressure space (thus any configu­ration c is a 3-plet c = (v,p,d);

(ii) R is a trivial constant constitutive map; 0

{Hi) 3H.c = pv®v-\-pg , (p EIR = mass density); /

(Hit) £. c=pv(e + i g(%' ^ p - ^ g r a ^ e-p'g(P^)Jg.

We are now interested in studying the isothermal case in detail. Then, we must take into account an internal constraint (isothermal constraint): (EQ = = i ^ ( ^ ) ® n C f . The corresponding dynamical equation is a first order equation called Euler equation andjdenoted by (E) . (E) in adapted coordi­nates is given by the following system of partial differential equations of first order.:

v'

(17) continuity equation: (GiQ+ Gik vk) + (dx{ vl) = 0 ;

(18) motion equation: p [G>0 4- 2 G ^ vk + G\k vl vk + (dxQ. vj) + v1 (bx^i)]-

-(dx.. p)gii-pBj=0 .

The most important structure property of (E) is given by the follo­wing

Theorem 6.1 The Euler equation (E) is formally integrable with respect to an inertial frame.

Proof. Note that the Euler operator E> = (3fy #) is an epimorphism so

(E) =ker E>0 is just a sub-fiber bundle of J # ((E0) over M . Similarly, one

can see that the first prolongation E>(^ : j&2 (C0)-*J& (1K0) of D0

and its symbol a2 (E)^1^) are morphisms of constant rank. So (E)+1 =

107

=ker(£)|)1)) and g2 =ker a2 ( D ^ ) are a sub-manifold of J&2(C0) and a

vector fiber bundle over (E) respectively. Furthermore, we can see that the symbol gt of (E) is involutive [5, 21, 28]. So, from Theorem 8.2 of ref. [5], in order to prove that (E) is formally integrable it remains to be proven that the map (E)++(E) is surjective. Let us now consider a family Fx of vector spaces over (E) defined by means of the following exact sequence:

0 -*gi -* S%M ® vT(EQ^ T*M 9 F0 -+ Fx -* 0

where F0 is the natural bundle of the embedding of (E) into J&(€0) . One can see that Fx is a vector bundle over (E) as g2 is a vector bundle over (E) (see ref. [25] pg. 66) and that dim Fx = 0 . Under these condi­tions we know [8] that a section K (E) : (E)-* Fx exists such that the follo­wing sequence p2 t : (2s)+1-* (E)K^ Fx = 0 is exact, i.e. the map p21 is surjective. The proof is now completed. •

We now come to the local existence theorem.

Theorem 6.2 Let i// be an inertial frame. Then, given q E(ii)+Jb, there is a Cw local dynamical configuration c of a perfect incompressible fluid such that: (i) D1 + h c (x) = q , with # = p 1 + A> (g) GAf , where p1+h is the canonical map (E), h -+M .

Proof. As Atf has a natural paracompact topological structure, M admits a Cw structure (see ref. [41]). Furthermore, we can consider (E) as an ana­lytic p.d.e.-l. So, from Theorem 6.1 we can state the existence of a Cw lo­cal solution c = (y,p) over a neighborhood A of x=p1 + h(q)E;M , such that D1 + /7 c (*)=<?• D

We shall now check theorems of existence of solutions for (E) related to the symmetry group of (E).

The infinitesimal generators of the symmetry group G of (E) are vector fields X : C0 -> T€0 on the configuration space C0 such that:

(a) X is 7rc -related with a vector field X on M which belong to the infinitesimal generators of the natural action of the Galilei group on the space-time M ;

(b) The first prolongation X : Jd (<E0) -> TJ& (<C0) of X , when it is restricted to (E) , Is a vector field on (E) .

108

Then by using Proposition 4.3/1 we get the following

Theorem 6.2 The algebra of infinitesimal symmetry of the perfect incompres­sible fluid is generated by vector fields X = Xa dx 4- X 3 x. 4- Y 3f : <CQ -+ -* T€0 , with respect to a coordinate system {x01, x\ J} on €0 , such that X = Xa dx is a Galilean infinitesimal symmetry and X1 and y are nu­merical functions on €Q given by:

Xl= X (dx .Xl)xr

K K 3 r Y

being C^f-^iR any numerical function on M such that (3 # . £ ) = -- 2 p g.5 (£x By . With respect to the symmetry group G we recognize the following submanifold of ) # (C0): J$ (<C0)

G = {u ejd (C0)l 3 G-invariant sections c E.C°°(£0) ; Dc (irl (u)) = u }C J& (£0) . Then, there is a natural fiber bundle morphism 7TG .J&(C0)

G -*J&(C0/G) over 7rG • €0-^€0/G given by ir^ . u=Dc (x)>-> Dc/G ([x]G) where c/G is the unique section of C0/G^M/G corresponding to the G-invariant section c , and [x]G is the image of x EAf under the canonical projection M-+M/G . Further­more, one can see that given a section c of TT£

-+M/G exists such that the following diagram a c/G section of C0/G

<£o

M

7T, •> <T0/G

•c/G

" G

^Af/G

is commutative if and only if Dc : M-+J& (C0)G

Then, one has the following important.

Theorem 6.3 A section c of n is a G-invariant solution to (E) if and only if c/G is a solution of the ''reduced" differential equation (£) = = TTG ( ^ j , where G £ = (£) H J& (C0)G .

By using Theorem 6.2 and Theorem 6.3 we get some important theorems on the existence of global regular solutions to (E).

109

Theorem 6.4 With respect to an inertial frame, the Euler equation (E) admits global time-translation invariant solutions of class Cw if B is con­servative.

Proof. As £~B = 0 , from Theorem 6.2 we get that the Euler equation admits the infinitesimal symmetry X = dx0 : (E0 ->• T€0 . This corresponds to the following finite transformation of C0 :

GT : (*°, xl, it, s)»(x° + fx, x\ $, f) , VM em .

The manifold M/GT is an affine Euclidean 3-dimensional manifold with space of free vector identified with S and with metric identified with g . Furthermore, one has the global (non-canonical) isomorphisms: M/GT= iR3 , and €0/GT^ 1R1 Therefore, equation {E)GT is written:

div u = Gjk uk+ (££.. u*) = 0 (E)GT

pVuu^rgtU(p-p1) = p[Gikutuk +« ,(3$,. u0] + (^rp)g^-pB^0

where u is a vector field on M/GT , p is a numerical function on M/GT

and 7 is the potential of B . Then, putting w = gradT , where T is a numerical function on M/GT the above equation becomes:

AT = 0

p = j Pg (« , u) + P7 +.C , C = constant.

So, if we consider that the fluid fills all the space, the only global time-translation invariant solution is the trivial v = \p , p = py , in fact this case 1 is a constant function. On the other hand, for limited domains with Dirichlet or Neumann conditions, one has the existence and uniquennes of the solution. This is, in fact, a direct consequence of a classic result on the C^ harmonic functions on JRn . •

Theorem 6.5' Let \p be an inertial frame and B conservative. Then the Euler equation (jE) admits global invariant solutions of class Cw corre-

110

sponding to waves travelling in a fixed direction if Lg B = 0 , with X = in­finitesimal generator of the action on M of the 1-dimensional sub-group G of G given by:

Gc : <j> : T X M ~>M , <)> (JU , x) = \jj (p , x) 4- JJLC W

being w a fixed unit vector belonging to _5 and c a constant (=wave1 ve­locity).

Proof. Now we shall consider the finite transformation of (U0 :

Gc : (x°, x1, x2, x3, x\ f) -+(x° 4- JU , x1 4- JJLC, x2, x3, x\ f) , Vp G2R #

the corresponding infinitesimal generator is X = cdx1 4- 3A;0 . From Theo­rem 4.8 we know that X is a fine infinitesimal symmetry for (E) . Fur­thermore , M/G is an Euclidean affine 3-dimensional manifold with space of free vectors identified with S and metric g . So we get the following glo­bal (non-canonical) isomorphisms M/G = ]R3 and C0/G = R1 . The equation (E)-c is given by

Gjk uk + (3£;. «•') = 0 ( £ ) - c

p [G/ t ««' «* - c (d{, «0 + a' O ? . »>')] + (3g.. p)i' - pB = 0

where w is a vector field on M/G and p is a numerical function on — c £ •

(CQ/G .. Let us put u — grad T , where T is a numerical function on M/G , then the above equation can be written

AT = 0 gradp =X(u) ,

1 where X (u) = ~ grad ( - pg (u , u) 4- py) 4- (9£i . «*) p3{. is a vector field

on M/G . For the first equation one has the considerations developed above. Fr;om the second equation, by using the metric isomorphism we get

( • ) d p - a ( « ) , where a (u) is the differential form corresponding to X (u). As Hk (M/Gc ; IR) = 0 , for k > 0 , and da («) = 0 , we get that p is globally defined by means of equation (•) . •

I l l

Let us consider the viscous case. As a consequence of the internal fric­tion in the fluid we shall take into account processes of energy dissipation. Then, the configuration space of a non-isotermic viscous incompressible fluid is € = €0XM 0 being 0=M X ]R the temperature bundle. Furthermore, the corresponding dynamic operator is ID- (2tt &, S) : J&2 {€) -*IK = = !K0XM TgM with

2t. c = &.(Vip, d) = div(pv)

&.c = $.(v,p, d) = div[pv®v+pg-2x(£vg)]-pB

S. c = Z(v,p,d) = div {pv [e + ~ g 0A ̂ ,v ̂ )] - ugradfl -

-'g ( ^ )J[-pg+2x£vg]-pg (B,v^ )

for any configuration c = (v, p, 6) G C°° ((C) , with 6 = temperature field, e = interior energy, v = thermal conductivity and x =.viscosity. Taking into account the thermodynamic constitutive equations for e, v and X'e~ = e (9), v = v (6), X = X(0) and assuming that v and x a r e constants and C = (90. e) = constant, we get that the non-isothermic Navier-Stokes equa­tion (NS) can be written in coordinates adapted to an inertial frame as fol­lows: (A/5) =04) + (#) + (C)

(A) Gjjkv

k + (dxi.vi) = 0

(B) plG^tfv* + (dx0.vi)+v\dxivi)]+giJ(dx..p)+

- 2 X {Gikvs(bxs.g

kr)+Gjkiv

s(bxs.gki)+vs(bx. bxsf')+

-G\kgHbxs.vk)-Gikg

k\dxs^)-2G^

-gs^(dxibxs.vi)-(dxidxs.vi)gi9}--pBj=0

(C) pCp[(dx0.dHvk(dxkd)]-[vGji(dxs.d)gsi^v(bxs.d)(^^

-2x[^^3*p.^^ vk (dx^) +

+ (bx/)vP(bxp.gs%ks-(bxi.v

k){bx.^^

There is an important structural difference between the equations (E) and (NS) . In fact we have the following

112

Theorem 6.16 (NS) is not a formally integrable differential equation.

Proof. In fact one can see that even if the symbol of (NS) is involutive and the maps JD, 2D(1) , a3 (E>(1)) have constant rank, the map (NS)+1 -• (NS) is not surjective. In fact dim g3 = 84 and dim (NS)+1~ dim (NS) = 80 . •

From the above theorem we also prove the following

Theorem 6.17 For any choice of point q E (NS) , (generalized initial con­ditions) we can not find C°° solutions c for (NS) such that D2 + bc(x) = = 4 , with x=ir2 + h (q)GM(i)

Proof. In fact if C°° solutions existed such that the condition (k) would be satisfied for any q€(NS)+h , this would mean that the map p2+h 2

:(NS)+h^ -+(NS) is surjective for any h > 0 . But, this is not true from Theorem 6.16.

' •

Remark 6.1 From the above considerations it follows that the set of initial conditions such that equation (NS)' admits solutions is characterized by the kernel of the curvature map k (NS) -. (NS) -> Fx . T o this set belong, for example, initial conditions corresponding to small Reynolds numbers.

Thanks. I would like to thank the Referee for its criticism.

APPENDIX A - Fibered morphisms of J$k (W) induced from fibered mor-phisms of W .' '

Let IT : W -• V be a fibered manifold and (/,/) a fibered isomorphism of TT . Then 9k (Ifi^T*^)®!^-1 (f,f) is a transformation of J&(W)C C0* (W) . This allows us to define the functor J&k (f) =®k (J,f) \j&k (W) . In fact,we have, for any sEC°°(W), 9k,(f}f)oDksof-1=Dk(foso^1) = >

9*(f,f)oDk s = tf (fosof-i)of=(D*'s)of , being 7 = / o s *f\ Thus ®k(f> f) transforms a point u = Dk7(x) E J{)k (W) in a point u = = (Pks)(f(x)) with 7=fosof~1 .

Similarly, one can prove that if / is a fibered morphism over V bet­ween the fiber bundles W and W' over V , then J&k (f) =

113

=®k (idv,f) \j$k (W) is a fibered morphism J&k (W)->JVk (W) over V

APPENDIX B - First prolongation of (£) in an inertial frame coordinate sy­stem.

{bxa.Gik)vk + Gijk(bxa.v

k) + (itfx)iaJ) = ^ ;

APPENDIX C - Galilean pseudogroup.

We can characterize G and 5G, as Lie pseudogroups on the Galilean space-time. (For the theory of Lie pseudogroups see refs.[l, 6, 7, 8, 11, 17, 20, 21, 27]). More precisely. The Galilean group G (resp. SG) is a Lie pseu­dogroup characterized in adapded coordinates by the following system of non linear partial differential equations of first order:

{txvnbXs.r)gimof=gis

(dx0.f<>)=l

(bxk.f°) = 0 ( SG

(det(3*../>)=l) | ,fx=xa°f:M-+R.

In fact an affine transformation on Af, / : M -+M will belong to G if and only if it satisfies the following non-linear differential equation of first

A ^. A

order: Gt (Af)=ker(G) , where G is the differential operator G: C°(MXM)""* -ZC°°(vS0

2M®T*M) given by G.s = G (idM,f) = (f*g~g) + (f*o- a). Thus G.s = 0 splits into /*g =g and /*a = a . Thus we have the following equi­valences:

(a) f*o = a < = > fT = idT ; Dfr = 0 , where /T = T O / and D is the symbol of vertical derivative (see ref. [23]).

*) f*g=g< = >fi\seo(S).

114

(The proof for the pseudogroup SG is similar). Furthermore, LQ (resp. SLG) is characterized by the following system of linear partial differential equations:

j (dx..gh)X>+(dx..X>)gjs+(dxs.Xm)gmi=:0

LG \ (dx . X°) = 0 }SLG

«dx..X> = 0)

[10

[11

[12

REFERENCES

0

/

S.S. Chern, Pseudogroupes continus infinis, Geometrie Diff. Coll. Int. du CNRS, Strasbourg (1953), 119-135.

C. Duval and H.P. Kunzle, Rep. Math. Phys., 13 (1978), 351-368.

D.G. Ebin and J. Marsden, Ann. Math., 92 (1970), 102-163.

X. Ehrenpreis, V.W. Guillemin and S. Sternberg, Ann. Math., 82 (1965), 128-138.

H. Goldschmidt, J. Differential Geom., 1 (1967), 269-307.

H. Goldschmidt, J. Differential Geom., I., 6 (1972), 357-373; II., 7 (1972), 67-95; III., 11(1976), 167-223.

H.. Goldschmidt, Bull. Am. Math. Soc , 84 (4) (1978), 531-546.

H. Goldschmidt and D.G. Spencer, I., II., Acta Math., 1 (36) (1976), 103-239. III., IV., J. Differential Geom., 13 (1978), 409-526.

H. Goldschmidt and S. Sternberg, Ann. Inst. Fourier, Grenoble, 23 (1) (1973), 203-267.

V. Guillemin and S. Sternberg, Bull. Am. Math. Soc, 70 (1964), 16-47.

A.*K. Kumpera and D.G. Spencer, Lie equations, h General Theory, Annals of Math. Studies No 73, Princeton University Press, 1972.

H.P. Kunzle, Ann. Inst. Henri Poincare, 17 (A-4) (1972), 337-362.

115

[13] H.P. Kunzle, GRG, 7 (1976).

[14] B.A.Kupershmidt, Lect. Notes, Math. 775 (1980), 168-218

[15] A. Lichnerowicz, Theories relativistes de la gravitation et de Uelectromagnetisme, Masson, Paris 1973.

[16] A. Lichnerowicz, Relativistic hydrodynamics and magnetohydrodinamics, Benjamen, N.Y., 1967.

[17] E.B. Malgrange,./. Differential Geom., I., 6 (1972), 503-522; II., 7 (1972), 117-141.

[18] E. Massa, GRG, I., II., 5 (5) (1974), 55-591; III., 5 (6) (1974), 715-736.

[19] P.J. Olver, J. Differential Geom., 14 (1979), 497-542.

[20] J.F. Pommaret, Ann. Inst. Henri Poincare, A18 (4) (1973), 285-352.

[21] J.F. Pommaret, Systems of partial differential equations and Lie pseud ogroups, Gor­don and Breach, N.Y., 1978.

[22] A. Prastaro, Stochastica, 3 (2) (1979), 15-31.

[23] A. Prastaro, I., Boll. Un. Mat. Ital., (5) 17-B (1980), 704-726; II., III., Boll. Un Mat. Ital. (5) S.-FM (1981), 69-129.

[24] A. Prastaro, Boll. Un. Mat. Ital., (5) 18-A (1981), 411-416.

[25] A. Prastaro, I., Rend Sem. Mat. Torino, 4 (2) (1982), 89-117.

[26] A. Prastaro, Riv. Nuovo Cimento, 5 (4) (1982), 1-122.

[27] A. Prastaro, Dynamic conservation laws, Geometrodynamics Proceedings (1985),283-420. World Scientific P.C. Singapore.

[28] I.M. Singer and S. Sternberg, J. Anal. Math., 15 (1965), 1-114.

[29] D.G.Spencer, Bull. Am. Math. Soc, 75 (1965), 1-114.

[ 30] W.J. Sweeney, Pacific J. Math., 20 (1967), 559-570.

[31] W.J.Sweeney, Acta Math., 120(1968), 223-277.

[32] F. Takens, J. Differential Geom., 14 (1979), 543-562.

116

[33] E. Whitney, Geometric integration theory, Princeton, 1957.

AGOSTINO PRASTARO. Dipartimento di Matematica - Universita della Calabria - 87036 Arcavacata di Rende (Cosenza) Italy.

Lavoro pervenuto in redaztone il 28/10/1983 e in versione definitiva il 29/6/1984.