ДОКЛАДЫ, ПОЛУЧЕННЫЕ ПОСЛЕ 1 ЯНВАРЯ 1967 г. REPORTS RECEIVED AFTER JANUARY 1, 1967 NON-LINEAR RELATIVISTIC PARTIAL DIFFERENTIAL EQUATIONS

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1 ДОКЛАДЫ, ПОЛУЧЕННЫЕ ПОСЛЕ 1 ЯНВАРЯ 1967 г. REPORTS RECEIVED AFTER JANUARY 1, 1967 Секция 5 Section 5 NON-LINEAR RELATIVISTIC PARTIAL DIFFERENTIAL EQUATIONS IRVING E. SEGAL Introduction As is well known, evolutionary partial differential equtions involving non-linear local interactions are of importance in many diverse connections. The theory of the Navier-Stokes (and related) equations has had great impact both in pure and applied mathematics; these equations are typical of non-linear parabolic equations with a local interaction. In relativity and quantum field theory one has primarily to* deal with hyperbolic equations, with similarly local interactions. We present here an account of recent developments concerning fairly typical such equations, with emphasis on global aspects relevant to the cited applications; these are, more specifically, the temporal asymptotics and the phase space structure on the solution manifold. The Cauchy problem The basic partial differential equations of quantum mechanics are conveniently taken, for purposes of global analysis as well as for generality, in the evolutionary form (1) u' = Au + K(u,t), where и =u(t) has values in a Banach space L, A is a given unbounded operator in L, and К («, t) is a given non-linear function of и ancf t, quite commonly not everywhere defined on L. For many theoretical purposes, the corresponding integral equation (Y) t u(t) = W(t-t 0 )u(t 0 )+lw(t-s)k(u(s),s)ds, where W (s) denotes the one-parameter semi-group generated by A r is more fundamental. From this abstract standpoint the basic distinction between the parabolic and hyperbolic cases is that in the former to

2 682 ДОКЛАДЫ, ПОЛУЧЕННЫЕ ПОСЛЕ 1 ЯНВАРЯ 1967 г. A has typically a real semibounded spectrum, while in the latter A has a pure imaginary spectrum; in the hyperbolic case, W (s) is a full group, not merely a semi-group. An important and relatively typical hyperbolic equation is the time-independent (i.e. autonomous) second-order equation <2) <&"(t) + B*0(t)=--J(Q)), where В is a given non-negative self-adjoint operator in a Hilbert space H, and J is a given non-linear operator in H. This equation is readily subsumed under equation (Г) by taking L as the set of all pairs [/, g] with f 6 H a and g g H 0 " 1, where a is an adjustable parameter, and H a denotes the Hilbert space completion of the domain of B a relative to the norm \\f \\ a = \\B a f\\, \\-\\ denoting the norm in H; W (t) as the unitary operator on L whose matrix decomposition in the representation L = H a + H 0-1 is / cos(*b) Ш \ В sin (tb) cos (tb) (these operators being extended from H to the H b for arbitrary b in theobvimis manner); and K(u, t) as the mapping [f, g] ->- [0, J(f)\- The choice of a is influenced by physical considerations, and strongly affects the existence of energy-inequalities and the continuity of К as an operator in L. The "scalar relativistic equation with local interaction", Пф= m\ + F (ф), where F is a given function of a complex variable such that F(0) = F' (0) =0, is readily subsumed, for example, the value a = 1 corresponding to the conventional energy norm, but in more than two space dimensions, the mapping К will not in general be'continuous when F is a polynomial, unless a larger value of a is taken. Developing classical differential equation methods in functional analytic form, the basic theory of such equations may be developed, a representative result being (with a taken as 1). Theorem I. If J is locally lipschitzian from H 1 to H, then the Cauchy problem for equation (2), in its integrated form (Г), has a unique beat solution. If in addition there exists a non-positive function E on H whose Fre chet differential exists at every point Ф 6 H 1 and has the form: ^ - Re ( Ф, X P), then the solution exists and is unique globally in time. If J is Frechet différentiable, then the differential equation is satisfied in the strict form ( 1) // ana only if the initial data [f, g] are in the domain of the generator of the one-parameter group W, i.e. if f H *and g H 1.

3 REPORTS RECEIVED AFTER JANUARY 1, For example, the equation (3) p = 2 ф т ф + gy* (g>0, p odd and > 0) has unique global solutions in one and two space dimensions, and also in three dimensions in case /7 = 3, to the Cauchy problem, as first shown by K. Jörgens [4] by a classical treatment, inasmuch as E (Ф) may be taken as const«\ ФН -1. The partial conflict between the non-negativity and regularity requirements on the functional J (Ф) is, illustrated by the circumstnace that the mapping Ф -^ Ф р is not continuous if the number of space dimensions n exceeds 3 and p > 3, or even \in =3 and p > 3, from H 1 to H; with a change in the value of a, these mappings may be made coiitinuous, and a local existence theorem obtained, but the non-negativity feature is lost, and thereby the global existence. Nevertheless, in this case, Theorem 1 may be applied to u cut-off" equations, involving modified functionals /, which are spatially non-local, and combined with compactness arguments to show Theorem 2. The Cauchy problem for equation (3) with arbitrary finite-energy initial data has a global weak solution, for arbitrary n and (odd) p. It is difficult to determine whether these solutions are unique, or whether they are globally regular if initially so. This is however the case as regards the solutions primarily relevant to dispersion theory, In the case n = 3, as indicated below. Dispersion of solutions Dispersion theory concerns the temporal asymptotics of two different evolutionary equations, having the same Cauchy data spaces, in relation to one another. To illustrate the concepts in a representative and fairly general case, the forward wave operator for equation (Г) is the transformation u 0 -+ ù from a solution u 0 (t) of the "free equation" u 0 (t) =W(t t 0 )u (t 0 ) to a solution и (t) of the given equation (Г), which is asymptotic as t-* oo to u 0 in the sense that (4) u(t) = u Q (t)+ J W(t-~s)K(u(s),s)ds. oo In general, especially for the time-independent case in which K(v, f) is independent of t, the infinite integral in equation (4) will fail to exist; although the wave operator is widely used in theoretical applied work, it is only with material restrictions on К that it has mathematical existence. Closely related to the wave operator, and still more impor-

4 684 ДОКЛАДЫ, ПОЛУЧЕННЫЕ ПОСЛЕ 1 ЯНВАРЯ 1967 г. tant in theoretical applied work, is the "dispersion" (also called "scattering" or "collision") operator; this is the transformation u Q -* u iy where щ is the solution of the free equation which is asymptotic as t ->- + oo to the same solution of equation (Г) to which u 0 (t) is asymptotic as t -> oo. Fairly representative of results on the wave operator is the Theorem 3. The forward wave operator exists for the equation Ф = т 2 ф + / ;, (ф) (m>0, F of class C<*>) and is unique within a specified regularity class, provided: (a) the free solution ф 0 is sufficiently regular (e.g. has Cauchy data which are infinitely différentiable and of compact support) ; (b) FM(%) = 0 (\X\ r ) for somer>2 + ^ j and j = 0, 1,..., j n, near % = 0. Corollary 3.1. The wave operator exists in the indicated sense for equation (3) provided: (a) n > 3, p > 3, and m > 0; (b) n = 1 or 2, p > 5, and m > 0. A considerably more general result may be given, but the statement is quite long. In the case m = 0 there are similar results provided n > 3, but the requirements on F are more stringent. The proof depends on the use of auxiliary norms in the Cauchy data space, and of suitable estimates of the temporal decay of the solutions of the free equation relative to these norms, as indicated below. The question of the existence of an asymptotic free solution ф 4 to the solution ф of the non-linear equation is of a different nature; unlike Theorem 3, which is essentially a local existence theorem (in a neighborhood of the time t = oo) and does not depend on energy boundedness, such bounds are essential in deriving results such as Theorem 4. Any finite-energy weak solution of the integrated form of equation (3) is asymptotic weakly, as tf-> + oo, to a uniquefiniteenergy solution of the free equation, provided n > 3 and m > 0. This shows the existence of dispersion when n > 3 and m > 0, for a suitable given solution of the non-linear equation, but the univalence, regularity, etc. properties of the dispersion operator are difficult to treat in the case of weak asymptotics. Strong asymptotics depend on estimates of temporal decay of auxiliary norms of solutions of the non-linear equation. In the case of the important equation Пф = = Ч )3 (g>0, n 3), a good estimate was obtained oy W. Strauss [9] by refined energy integral methods, which however are not readily generalized. Results for more general equtions, of a somewhat different nature, may be obtained relatively systematically by combining

5 REPORTS RECEIVED AFTER JANUARY 1, sharp estimates of- the decay of auxiliary norms of solutions of the associated linear relativistic equations (these are obtained by a method due essentially to A. R. Brodsky [1]) with a series of applications of conventional inequalities. In general, larger values of n, p and m lead to more rapid decay and stronger dispersion results, although the situation is somewhat complex; a representative result, for the case of greatest quantum-mechanical interest is Theorem 5. If n =3 and m > 0, the dispersion operator exists strongly for equation (3), for any given free solution щ (f) whose Cauchy data have a sufficient number of integrable derivatives, if g is sufficiently small. More specifically, there exists a solution и (t) of the equation t u(t) = u 0 (t) + J W(t-s)K(u(s))ds, oo and a free solution u t (f) such that DO u(t) = Ui(t)- J W(t-s)K(u(s))ds, t both integrals being absolutely convergent in the energy norm. The general method may be sketched briefly as follows. Setting ф (t) = ф (x, t) as a function of x, and defining Ф 0 (t) similarly, equation (4), with К (и, s) as in equation (3), is equivalent to the equation t Ф(t) = ф 0 (t)+ J sin ((t-s)в)в' 1 (go(s) p ) ds, B=(m 2 I-A) 1/a. oo Taking L r norms, writing sin (tb) B' iy as (sin (tb) B' b ) (B b ~ l T), and noting that if b is sufficiently large, sin (tb) B~ h is convolution with a function G tib (x), it follows, using the Hausdorff-Young inequality, that Ф(0,.<: 1Фо(*) 1г+ \ oo \\G t. s>b \\ q \\B^o(sr\Wd S for certain q and q'. Now Ф (t) \\ T and \\G t ^8^\\q are norms of free solutions, and may be relatively sharply majorized (explicit boqnds are given in [7]; another variant of Brodsky's approach is indicated in [8d, Remark 1]); inequalities of Sobolev type will bound В Ь ~ ] Ф (s) p \\ q» in terms of В Ф (s) p \\ q * for an integer с; \\ В С Ф (s) v \\ q» may now be bounded in terms of the norms of derivatives of Ф (s) p, e.g. ВФ (s) p g «<const ( Ф (s) p g. + Ф! (S) Ф (s)*' 1 Hg.); Holder's inequality now bounds the last expression by E (s) d \\ Ф (s) \\{, where E (s) is the energy

6 686 ДОКЛАДЫ, ПОЛУЧЕННЫЕ ПОСЛЕ 1 ЯНВАРЯ 1967 г. at times, and d, e, f are certain constants. The free parameters may be adjusted in a number of ways to arrive at an inequality of the form t Ф(011г<ЛГ(ф 0 )(1НЧ* ГЧ (ф</с \ (1 + *- Я )-, Ф( в ) {, where N is a certain norm on the free solution at time со, Я(ф 0 ) is the energy of the free solution, and С is a constant. An estimate adequate for the proof of the indicated theorem may be obtained by taking r == oo, b =2, and q > 4; with these values, e" < < 1 and / > 1, corresponding to a bound which is in itself insufficient, but for sufficiently small g, and in combination with related estimates used in the proof of Theorem 3, it follows that ф (x, t) = = 0( \t\~ 1+e ) uniformly in x, for every e > 0, implying the convergence of the integrals in question. It is likely that further application of the Q same method will show that ф(*, t) = 0(\t\ -^ + 8 )> which is virtually best possible. oo Phase space structure of the solution manifold The solution manifold of a hyperbolic equation may be regarded physically as a phase space which differs from conventional phase spaces: fundamentally only in being infinite-dimensional. This analogy may be made mathematically explicit through the introduction in the solution manifolds of suitable equations of the structures which are well known in the finite-dimensional case. Fundamental among these is the symplectic structure in the solution manifold of a second-order hyperbolic equation; this is relatively simply described in the case of equation (2). Theorem 6. The manifold of all finite-energy solutions of equation (2) is of class C (as a Frechet-Banach manifold) provided J is of class C, and its tangent vectors correspond naturally to solutions of the first-order variational equation, V (t) + B 2 4? (t) = (d 0(t) J) Y (t). On defining, for the tangent vectors represented by the solutions Ч^-} and 4 r 2 ( ) of the foregoing equation, the form Q<D<.) by the equation О ф (. ) = 0 М 0, *;(*)>-<*;('). T»W>. Q is a closed, non-degenerate, differential form of class C which is invariant under the one-parameter group of transformations on the solution manifold defined by the differential equation. In the case of a relativistic equation (e.g. that of Theorem 3), Q is invariant under the Poincarê group.

7 REPORTS RECEIVED AFTER JANUARY 1, In general the form of Q is more complicated, although similar results are valid. This is illustrated in the treatment by A. Lichnerowicz of the quantization of. linear relativistic local equations in curved space-time manifolds; the field commutator distribution D (x, x') is closely related to the form Q, and is the'difference of the advanced and retarded elementary solutions provided by the general theory of Leray. The relation to the given expression for Q is visible from the alternative, non-relativistic, definition of D (x, x') as the solution of the differential equation in question such that relative to a particular Lorentz frame, it has Cauchy data: D (x, x') \ t==i > = 0, d t D (x, x') = ô (* *') (x = (x, t)). The theory may in part be extended to non-linear equations, as may be illustrated by the case of the scalar equation Пф = F (ф) on a Lorentzian space-time manifold M, where F is a given C function. A tangent vector to the solution manifold at the solution ф may be identified with a solution X of the first-ordervariational equation, A, = F* (ф) Я. According to a result of Y. Fourës-Bruhat [2], every suitably regular such function % is of the form X (x) = jdq, (x, x')x X f(x, )dx'' for some infinitely différentiable function f of compact support on M, where о ф (x, x') is the commutator distribution for the indicated tangential equation. The form 2 ф (X, V) = J J Dq, (*, x') f (x) f (x') dx dx' depends only on %, %*, and ф, and not at all on the choices of / and f. In part extending and rigorizing developments indicated in [8gl, it follows from a study of the Frechet differential ôcpd v (x, x'), which may be expressed in terms of D^ (x, x'), that Theorem 7. Q is a closed, non-degenerate, C, second-order differential form on the manifold of all local solutions of the equation Пф = = F (ф), in the vicinity of any fixed point on the given Lorentzian manifold. The vector field X/ on the solution manifold which assigns at the solution ф the tangent vector % given above represents an integral, over the times involved in the support of /, of the vector fieldscorresponding to infinitésimal vector displacements of the Cauchy date at each time. Relatively explicit expressions involving the Xf, such as the commutator [Xf, X g ] for any two functions / and g of the indicated type, may be given in terms of thed (x, x') functions above; these provide an algebraic interpretation for domains of influence, etc.; e.g. two points P and Q are outside each others' domains of influence if and

8 88 ДОКЛАДЫ, ПОЛУЧЕННЫЕ ПОСЛЕ 1 ЯНВАРЯ 1967 г. only if [Xf, Xg\ = 0 whenever the supports of f and g are contained in sufficiently small neighborhoods of P and Q respectively. Another important structure in a phase space is an invariant integral. In a phase space of finite dimension 2n, Q n provides such an integral; when n =oo, Q n has no meaning; however, for an important -class of linear hyperbolic equations (e.g. relativistic wave equations in Minkowski space) there is a natural canonical measure, of a generalized nature (a "weak distribution" as defined in [8j]); this measure is temporally invariant, and the corresponding flow is ergodic and in fact mixing. In the case of a non-linear equation, no temporally invariant measure in the solution manifold can be given explicity, but if the wave operator exists and is regular in a sufficiently strong sense, it will induce according to results of L. Gross [3] a transformation of an invariant weak (probability) distribution on the free solution manifold into an invariant weak distribution in the solution manifold of the non-linear equation. This amounts to the construction of a (generalized) stationary stochastic solution to the non-linear equation which is asymptotic at time oo to a given stationary stochastic solution of the associated linear equation. Stochastic and quantized equations Stochastic equations are those for which the unknown function «Ф (x, t) is for each time t a (possibly generalized) random-variablevalued function on space; quantized equations are those in which ф (л:, t) is similarly (possibly generalized) operator-valued, and indeed it is commonly postulated that the commutator between the Cauchy data at two points x and x' oh a space-like surface is id (x, x'), and so can not vanish. Such equations are in general, when non-linear, not a priori mathematically meaningful, since they involve non-linear functions of generalized (i.e. weak) functions; indeed this is always the case for quantized equations satisfying commutation relations of the indicated nature. Weak solutions of stochastic non-linear equations may be obtained by the method just indicated, when the non-linear term is sufficiently regular, but the question remains open (even in the case of linear equations, in part) of whether these solutions are strong, in the sense that with probability one suitably différentiable versions of the generalized solutions obtained exist satisfying the given differential equation in the classical sense. When the non-linear term is irregular relative to the initial probability distribution in function space, e.g. if ф (л:, 0) is not well-defined at almost all points x, but a local non-linear function such as ф (x) p is involved, one is in the indicated mathematically

9 REPORTS RECEIVED AFTER JANUARY 1, 1967 ambiguous situation; however, there exist natural definitions, connected with the Wick products arising in the case of quantized fields. Two basically different interpretations of such quantized equations have been treated mathematically (exclusive of conventional perturbation theory as practiced in theoretical physics, which there is reason to doubt can be given comprehensive mathematical meaning). The first of these interprets polynomials in ф (x, t) at a fixed time t in terms of so-called Wick products, either in their conventional form applicable to the "free field",or in a generalized form based on an intrinsic characterization ([8e, 8k]). The Cauchy problem, for the quantized equation can then be made mathematically well-defined. There are, analytically speaking, two major questions in addition to the solution of this problem: (a) the existence (and uniqueness) of a regular positive linear functional E on the operator algebra A generated by the (bounded functions of) the fieldoperators, such that for any element A 6 6 A, (A*A*) has a non-negative frequency spectrum, as a function of the time t, where A 1 denotes the temporal displacement of the operator A through the time t (such a functional E is called a physical vacuum); (b) the existence, and nature (especially, relation to symmetric or skew-symmetric tensor algebras over the Hilbert space of free (classical) solutions, i.e. solutions of the first-order variational equation at the solution ф =0), of asymptotic linear fields as <-* -+ ± oo. While a substantial variety of results in these directions now exist, especially in the case of two space-time dimensions, in which case the free-field Wick products at a sharp time are strictly operatorvalued, crucial aspects of these problems remain unresolved. The second interpretation is concerned primarily with the propaga.- tion of the quantized field from time oo to + oo, rather from one finite time to another, i.e. with the dispersion of the field.the (linear) quantum-theoretic dispersion operator is taken as the induced action of the (non-linear) dispersion operator S c previously considered, acting on the classical solution manifold of the given equation, on a certain implicitly-defined S c -invariant class of functionals over the solution manifold. In the simplest case, this class consists of the holomorphic functionals relative to a complex structure on the solution manifold which is determined by the condition that it be invariant under S c and extend the symplectic structure described earlier to a Kählerian one. For a quantized field interacting with an external source or potential, this formalism gives results equivalent to the more conventional, first-indicated formalism; unlike this conventional formalism, it is adaptable to the association of a quantum field with a suitably structured infinite manifold which is not necessarily defined by a partial differential equation.' Massachusetts Institute of Technology, USA

10 690 ДОКЛАДЫ, ПОЛУЧЕННЫЕ ПОСЛЕ 1 ЯНВАРЯ 1967 г. REFER ENCES [1] В г о d s к у A. R., Asymptotic decay of solutions to the relativistic wave equation and the existence of scattering for certain non-linear hyperbolic equations, Ph. D. Thesis, M.I.T., Cambridge, Mass., [2] F о u r è s - В r u h a t Y., Propagateurs et solutions d'équations homogènes hyperboliques, C. R. Acad. Sei. Paris, 251 (1960), [3] Q r о s s L., Integration and non-linear transformations in Hilbert space, Trans. Amer. Math. Soc, 105 (1960), [4] Jörg en s K., Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen, Math. Zeits., 77 (1961), [5] L e r a y J., Hyperbolic partial differential equations, Institute for Adv. Study, Princeton, N.J., U.S.A., [6] L i с h n e r о w i с z A., (a) Theorie quantique des champs sur un espacetemps courbe. Cours de l'ecole d'eté de Physique théorique des Houches (France), 1963; (b) Propagateurs et commutateurs en relativité generale, Jnst. Hautes Études Sei. Publ. Math., 10 (1961), 1-56; (c) Champs spinoriels et propagateurs en relativité générale, Bull. Soc. Math. France. [7] N e 1 s о n S., Asymptotic behavior of certain fundamental solutions to the Klein-Gordon equation, Ph. D. Thesis, M.I.T., Cambridge, Mass., [8] Segal I., (a) Non-linear semi-groups, Ann. Math., 78 (1963), ; (b) The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France, 91 (1963), ; (c) Differential operators in the manifold of solutions of a non-linear differential equation, Jour. Math. pur. appl., 44 (1965), ; (d) Quantization and dispersion for non-linear relativistic equations. Proc. Conf. on Math. Th. El. Particles, M. I. T. Press, Cambridge, Mass., 1966, ; (e) Interpretation et solution d'équations non linéaires quantifiées,' С. R. Acad. Sei. Paris, 259 (1964), , sec. 1-3 (heuristic). (f) Conjugacy to unitary groups within the infinite-dimensional symplectic group, Argonne Nat. Lab. report ANL-7216, 1966, 1-11; (g) Quantization of non-linear systems, Jour. Math. Phys., 1 (1960), , sec. 4A-B (heuristic); (h) Explicit formal construction of nonlinear quantum fields, Jour. Math. Phys., 5 (1964), ; (\) La variété des solutions d'une équation hyperbolique, non linéaire, d'ordre 2, C.I.M.E. lectures on non-linear partial differential equations at Varenna, 1964; (j) Abstract probability spaces and a theorem of Kolmogoroff, Amer. Jour. Math,, 76 (1954), ; (k) Non-linear functions of random distributions and generalized normal products. Proc. Conf. on Funct. Integration and Constr. Quant. Fid. Theory, M.I.T., Cambridge, Mass., April, [9] S t r a u s"s W., (a) Les operateurs d'onde pour les equations d'onde nonlinéaires indépendantes du temps, CR. Acad. Sei. Paris, 256 (1963), ; (b) La deccroissance asymptotique des solutions des equations d'onde non-lineaires, CR. Acad. Sei. Paris, 256 (1963),