The uniform order convergence structure on ML(X)

The Uniform Order C on v ergence Structure o n ML ( X ) J H v an der W alt Departmen t of Mathematics a nd Applied Mathematics Univ ersit y of Pretoria Pretoria 0002 South Africa Octob er 24, 2018 Abstract The aim of this pap er is to set up appropriate uniform conv ergence spaces in which to reform ulate and enric h the Order Completion Method [25] for nonlinear PDEs. In this regard, w e consider an appropriate space ML ( X ) of normal low er semi-conti nuous functions. The space ML ( X ) appears in the ring theory of C ( X ) and its v arious extensions [15], as well as in the theory of nonlinear, PDEs [25] and [28]. W e define a uniform conv ergence structure, in the sense of [11], on ML ( X ) such that the induced conv ergence structure is the order conv ergence structu re, as introduced in [7] and [32]. The u niform conv ergence sp ace completion of ML ( X ) is constructed as the space all normal low er semi-contin uous functions on X . It is then show n how these ideas may b e app lied to solv e nonlinear PDEs. In particular, w e construct generalized solutions t o the Navier-Stokes Equations in three spatial dimensions, sub ject to an initial condition. Keywords Gener a l T opolo gy , Uniform Conv er gence Str uctures, F unction Spaces, Ordered Spaces 2000 Mathematics Sub ject Classification 54A2 0, 46 E05, 0 6F30 1 In tro duc tion It is widely held that, in contradistinction to ODEs, ther e can b e no general, type indep endent theory for the e x istence a nd reg ularity of the solutions to PDEs [8], [14]. As seen in the s equel, this is in fact a misunderstanding which is often attributed to the mo r e co mplex geo metry of R n , with n ≥ 2, as app osed to that of R which is relev a n t to ODEs alone, see [8]. Indeed, the difficulties that are typically encountered when solving PDEs by the usual function a nalytic metho ds, which a re per ceived to arise form the complica ted geometry of R n , ar e rather due to the inherent limita tio ns of the function analytic metho ds themselves, and are therefo r e technical o bstacles, rather than conceptual ones. The above is exemplified by the app earance of not only one, but t wo general, t yp e indep endent theories fo r the so lutions of nonlinear P DEs. The Central Theory of PDEs, developed by Neuber ger [23], se e a lso [2 4], is bas ed on a genera lized metho d of steep est descent in suitably cons tructed Hilber t Spaces. The Order Completion Metho d, as develop ed b y Ob erg uggenberge r a nd Rosinger [25], is based on the Dedekind co mpletion of suitable spaces of equiv alence classe s of functions . 1.1 The Order Completion Metho d The metho d of Order Completion r esults in the exis tence of generalized so lutions to arbitrar y , contin uous nonlinear PDEs of the form T ( x, D ) u ( x ) = f ( x ) , x ∈ Ω (1) 1 with the right hand term f a contin uous function of x ∈ Ω, and the partial differe ntial op era tor T ( x, D ) defined thr ough a joint ly contin uous function F : Ω × R M → R by T ( x, D ) u : x 7→ F ( x, u ( x ) , ..., D α u ( x ) , ... ) (2) With the PDE (1) o ne asso ciates a mapping T : M m (Ω) → M 0 (Ω) where M m (Ω) is the space of eq uiv alence classes of functions which are con tinuously differe ntiable up to order m e verywhere except o n some clo sed nowhere dense set [25], under the equiv alence relation u ∼ v ⇔   ∃ Γ ⊆ Ω closed nowhere dense : 1) u, v ∈ C m (Ω) 2) x ∈ Ω \ Γ ⇒ v ( x ) = u ( x )   (3) The mapping T induces an equiv ale nce relation ∼ T on M m (Ω) through ∀ u, v ∈ M m (Ω) : u ∼ T v ⇔ T u = T v (4) With the mapping T one asso ciates in a canonica l wa y an injective mapping b T : M m T (Ω) → M 0 (Ω) where M m T (Ω) denotes the quotient s pace M m (Ω) / ∼ T . The space M m T (Ω) is order ed throug h ∀ U, V ∈ M m T (Ω) : U ≤ T V ⇔ b T U ≤ b T V (5) so that b T is an order isomorphic embedding. The mapping b T extends uniquely to an order iso - morphic embedding e T ♯ : M m T (Ω) ♯ → M 0 (Ω) ♯ (6) where M m T (Ω) ♯ and M 0 (Ω) ♯ denote the Dedekind order co mpletions of M m T (Ω) a nd M 0 (Ω) ♯ , resp ectively . This is summarized in the following c o mm utative diag ram: M m T (Ω) ✲ M 0 (Ω) b T ❄ M m T (Ω) ♯ ✲ b T ♯ M 0 (Ω) ♯ ❄ Sub ject to a mild assumption on the PDE (1), one ha s ∀ f ∈ C 0 (Ω) : ∃ ! U ♯ ∈ M m T (Ω) ♯ : b T ♯ U ♯ = f 2 where U ♯ ∈ M m T (Ω) ♯ is the uniq ue gener alized so lution to (1). T he unique g eneralized solution should be interpreted as the t otality of all super solutions, sub so lutions a nd exact so lutio ns to (1). Recently [6] it was shown that the genera lized so lutions to a PDE of the fo r m (1) may b e assimila ted with usual Hausdo rff co n tinuous functions, in the s ense that there is an order isomo rphism betw een M m T (Ω) ♯ and the space H nf (Ω) of all nearly finite Hausdo rff cont inuous functions. T aking into account the universality of the e x istence and regular it y result just describ ed, o ne may notice that there is a la rge sco pe for further enrichmen t of the bas ic theory of Orde r Completion [25]. In particular, the following may serve as guidelines for such a n enrichmen t. (A) The space of gener a lized so lutions to (1) may dep end on the PDE op erator T ( x, D ) (B) Ther e is no differential structure on the spa ce of generalize d solutions In order to accommo date (A), one ma y do aw ay with the equiv alence r elation (4) on M m (Ω) a nd consider a par tial o rder other than (5), which do es not depend o n the partial differential op erator T ( X , D ). Indeed, somewhat in the spirit of Sob olev, one may consider the partial order ∀ u, v ∈ M m (Ω) : u ≤ D v ⇔  ∀ | α | ≤ m : D α u ≤ D α v  (7) which could also s olve (B). Ho wever, such an appr oach pres ent s several difficulties. In particular, the existence of gener alized solutio ns in the Dedekind co mpletio n of the partially ordered set ( M m (Ω) , ≤ D ) is not clear. In fact, the pos sibly nonlinear ma pping T asso ciated with the PDE (1 ) cannot be extended to the Dedekind completion in a unique and meaningful wa y , unless T sa tisfies some a dditional a nd ra ther restrictive conditions. W e mention that the use of pa r tial order s other than (5) was inv estig ated in [25, Section 13 ], but the partial orders that are considere d ar e still tied to the PDE op erato r T ( X D ). Rega r ding (B), w e ma y reca ll that there is in general no connection betw een the usual order on M m (Ω) and the deriv atives of the functions that are its elements. 1.2 The Order Conv ergence Structure One p os sible wa y o f going be yond the basic theo ry of Order Co mpletion is motiv ated by the fact that the pro cess of taking the supremum of a subset A of a par tially order ed set X is essentially a pro cess of approximation. Indeed, x 0 = s up A means that the set A approximates x 0 arbitrar ily close from below. Approximation, how ever, is essentially a top olog ical pro ce ss. Hence a top olog ical type mo del for the pro c ess of Dedekind completion of M 0 (Ω) may serve as a starting p oint fo r the enrichmen t of the Order Completion Metho d. In this reg a rd, we recall that there are s everal useful mo des of c onv ergence on a par tially ordered s et, defined in ter ms of the par tial order, see for instance [12], [20] a nd [27]. In particula r, we consider the or der con vergence o f sequences defined on a partially or de r ed set X as ( x n ) order converges to x ∈ X ⇔ ⇔   ∃ ( λ n ) , ( µ n ) ⊂ X : 1) n ∈ N ⇒ λ n ≤ λ n +1 ≤ x n ≤ µ n +1 ≤ µ n 2) sup { λ n : n ∈ N } = x = inf { µ n : n ∈ N }   (8) It is well kno wn that the order conv erg e nce of sequences is in general not top olog ic al, as is demon- strated in [31 ]. That is, for a par tially ordere d set X there is no top ology τ on X such that the τ -con vergent sequences a re exac tly the order co nvergen t sequences. How ever, see [7] and [3 2], for a σ -distributiv e lattice X there exists a conv er gence structure λ o , in the sense of [11], o n X that induces the order conv er gence of se q uences through ∀ x ∈ X : ∀ ( x n ) ⊂ X : ( x n ) or der co n verges to x ⇔ [ {{ x n : n ≥ k } : k ∈ N } ] ∈ λ o ( x ) (9) 3 In particular, the order conv ergence structur e, defined a nd studied in [7 ] and [32] induces the order conv ergence of sequences through (9), and is defined as ∀ x ∈ X : ∀ F a filter o n X : F ∈ λ o ( x ) ⇔     ∃ ( λ n ) , ( µ n ) ⊂ X : 1) n ∈ N ⇒ λ n ≤ λ n +1 ≤ x ≤ µ n +1 ≤ µ n 2) sup { λ n : n ∈ N } = x = inf { µ n : n ∈ N } 3) [ { [ λ n , µ n ] : n ∈ N } ] ⊆ F     (10) and is Hausdorff, regula r a nd fir st countable, se e [3 2]. A par ticular case of the ab ove o ccur s when X is an Archimedean vector la ttice. In this cas e the conv erg e nc e structure λ o is a vector space conv erg ence structure, and as such it is induced by a uniform co n vergence str ucture, in the sense of [17]. Indeed, the Ca uch y filters are characterized as ∀ F a filter o n X : F is Cauch y ⇔ F − F ∈ λ o ( x ) The conv erg ence vector space completion of an Archimedean vector lattice X , equipp ed with the order conv ergence str ucture λ o may b e co nstructed as the Dedek ind σ -completion X ♯ of X , equipp e d with the or der conv er g ence structure, see [3 2]. If X is order sepa rable, then the completion of X is in fact its Dedekind completion. In the particular case when X = C ( Y ), with Y a metric s pa ce, then the conv er gence vector space completion is the set H f t ( X ) of finite Hausdor ff contin uous functions on Y , whic h is the Dedekind completion of C ( Y ). Let us now consider the p ossibility of applying the ab ov e results to the proble m of so lving nonlinear PDEs . In this rega rd, conside r a nonlinea r PDE of the for m (1 ), and the asso ciated mapping T : M m (Ω) → M 0 (Ω) The Order Completion Metho d is based on the abundance of appr oximate solutions to (1), whic h are elements of M m (Ω), and in g eneral one cannot exp ect these a pproximations to b e con tinuous, let alone sufficiently smo oth , o n the whole of Ω. Mor eov er, the space H f t (Ω) do es not contain the space M 0 (Ω). On the other hand, the space M 0 (Ω) is an o r der separable Archimedean vector lattice [25], and therefore one ma y equip it with the order conv ergence structure. T he completion of this spa c e will be its Dedekind completion M 0 (Ω) ♯ , a s desired. Ho wev er, there are sev er al obstacles. If one equips M m (Ω) with the subs pa ce conv erge nce structure, then the no nlinear mapping T is not necessar ily co nt inuous. Moreov er, the quotient space M m T (Ω) is not a linear spa ce, so that the completion pro ces s for c o nv ergence vector spaces do es not apply . It is therefo r e neces sary to develop a nonline ar topolo gical model for the Dedekind completion of M (Ω). 2 Spaces of Lo w er Semi-Con tin uous F unctions The notio n of a norma l lo wer semi-contin uous function, resp ectively normal upper semi-contin uous function, was int r o duced b y Dilw or th [13] in connection with the Dedekind co mpletion of spaces of cont inuous functions. Dilw orth introduced the concept for b ounde d , real v alued functions. Sub- sequently the definition w a s extended to lo c al ly b oun de d functions [3 ]. The de finitio n e x tends in a straight forward way to extended real v alued functions. In pa rticular, a function u : X → R , with X a top olog ical s pace, is normal low er semi-contin uous whenever ( I ◦ S ) ( u ) ( x ) = u ( x ) , x ∈ X (11) where I a nd S a re the Low er- and Upper B a ire Op erator s, see [2], [9] and [3 0], defined as I ( u ) ( x ) = sup { inf { u ( y ) : y ∈ V } : V ∈ V x } , x ∈ X (12) 4 S ( u ) ( x ) = inf { s up { u ( y ) : y ∈ V } : V ∈ V x } , x ∈ X (13) where u is any extended real v alued function on X . A nor mal lower semi-contin uous function u is called ne arly finite if the set { x ∈ X : u ( x ) ∈ R } is op en and dense in X . W e denote the spa ce of a ll nearly finite norma l low er semi-contin uous functions by N L ( X ). The space N L ( X ) is ordere d in a p oint wise wa y through ∀ u, v ∈ N L ( X ) : u ≤ v ⇔  ∀ x ∈ X : u ( x ) ≤ v ( x )  (14) The space N L ( X ) satisfies the following pr op erties. Theorem 1 The sp ac e N L ( X ) is De dekind c omplete. Mor e over, if A ⊆ N L ( X ) is b oun de d fr om ab ove, and B ⊆ N L ( X ) is b ounde d fr om b elow, then sup A = ( I ◦ S ) ( φ ) inf B = ( I ◦ S ◦ I ) ( ϕ ) wher e φ : X ∋ x 7→ sup { u ( x ) : u ∈ A } and ϕ : X ∋ x 7→ inf { u ( x ) : u ∈ B } Pro of. O ne may prov e the result directly . Howev er, it is straight forw ar d to show that N L ( X ) is order is omorphic to the set H nf (Ω) o f nearly finite Haus dorff contin uous functions [ ? ]. The res ult follows immediately from the resp ective result in [3]. Applying similar ar guments, w e obtain the following useful result. Prop ositio n 2 Consider any u ∈ N L ( X ) . Then ther e is a set U ⊆ X such that X \ U is of First Bair e Cate gory and u ∈ C ( X \ U ) . What is mor e, if v ∈ N L ( X ) and D ⊆ X is dense in X , t hen  ∀ x ∈ D : u ( x ) ≤ v ( x )  ⇒ u ≤ v Pro of. Again a direc t pro of of this pro po sition is av aila ble. How ever the result follows easily by considering the order iso morphism I : H nf ( X ) → N L ( X ) Prop ositio n 3 The sp ac e N L ( X ) is ful ly distributive. Pro of. Consider a set A ⊂ N L ( X ) such that sup A = u 0 F or v ∈ N L ( X ) w e must show u 0 ∧ v = sup { u ∧ v : u ∈ A } (15) 5 Suppo se that (15) fails for some A ⊂ N L ( X ) and some v ∈ N L ( X ). That is, ∃ w ∈ N L ( X ) : u ∈ A ⇒ u ∧ v ≤ w < u 0 ∧ v (16) Clearly , u 0 , v ≥ w so that there is s ome u ∈ A such that w is not la rger than u . In view of Prop ositio n 2 ∃ V ⊆ X nonempty , op en : x ∈ V ⇒ w ( x ) < u ( x ) (17) Upo n application of Pro po sition 1 we find ( v ∧ u ) ( x ) > u ( x ) , x ∈ V since the o pe rators I and S are monotone and idempotent [2 , Section 2 ]. Hence (16) cannot hold. This completes the pro o f. The set C nd ( X ) of all functions u : X → R that a re contin uous everywhere except on some closed nowhere dens e subse t of X , that is, u ∈ C nd ( X ) ⇔  ∃ Γ u ⊂ X close d nowhere dens e : u ∈ C ( X \ Γ u )  plays a fundamental role in the theory of Order Completion [25], as discussed in the introductio n. In particular, one consider s the quotient spa ce M ( X ) = C nd ( X ) / ∼ , where the equiv alence rela tion ∼ on C nd ( X ) is defined by u ∼ v ⇔  ∃ Γ ⊂ X clo sed nowhere dense : x ∈ X \ Γ ⇒ u ( x ) = v ( x )  (18) An or der is omorphic repre sentation of the spa c e M ( X ), consisting o f normal lower semi-contin uous functions, is obtained by co nsidering the set ML ( X ) =  u ∈ N L ( X ) ∃ Γ ⊂ X clo sed nowhere dense : u ∈ C ( X \ Γ)  (19) The adv antage of consider ing the space M L ( X ) in s tead o f M ( X ) is that the e le men ts of M L ( X ) are actual p oint v a lued functions on X , as app osed to the elements of M ( X ) which are equiv alence classes of functions. Hence the v alue u ( x ) of u ∈ M L ( X ) is completely determined for ev ery x ∈ X . Prop ositio n 4 The mapping I S : M ( X ) ∋ U 7→ ( I ◦ S ) ( u ) ∈ ML ( X ) (20) is a wel l define d or der isomorphism. Pro of. First we show that the mapping I S is well defined. In this regar d, consider some U ∈ M ( X ) and a ny u, v ∈ U . Let Γ ⊂ X be the clo sed nowhere dense set asso cia ted with u and v through (18). Since Γ is closed, it follows by (12) and (13) that ( I ◦ S ) ( u ) ( x ) = ( I ◦ S ) ( v ) ( x ) , x ∈ X \ Γ (21) Since X \ Γ is dense in X , it follows that ∀ x ∈ X : ∀ V 1 , V 2 ∈ V x : ∃ x 0 ∈ X : x 0 ∈ ( X \ Γ) ∩ ( V 1 ∩ V 2 ) 6 F or a ny x ∈ X we hav e inf { ( I ◦ S ) ( u ) ( y ) : y ∈ V 1 } ≤ ( I ◦ S ) ( u ) ( x 0 ) and ( I ◦ S ) ( v ) ( x 0 ) ≤ sup { ( I ◦ S ) ( v ) ( y ) : y ∈ V 2 } Hence it follows by (21) that inf { ( I ◦ S ) ( u ) ( y ) : y ∈ V 1 } ≤ sup { ( I ◦ S ) ( v ) ( y ) : y ∈ V 2 } so that (12) and (13) yields I (( I ◦ S ) ( u )) ≤ S (( I ◦ S ) ( v )) (22) It now follows form the idempotency and monotonicity of the o per ator I [2, Section 2 ] that ( I ◦ S ) ( u ) ≤ ( I ◦ S ) (( I ◦ S ) ( v )) Since the op erato r ( I ◦ S ) is also idempo tent , s ee [6, Section ], one o bta ins ( I ◦ S ) ( u ) ≤ ( I ◦ S ) ( v ) By similar arguments it follows that ( I ◦ S ) ( v ) ≤ ( I ◦ S ) ( u ) so that ( I ◦ S ) ( u ) = ( I ◦ S ) ( v ). It is obvious that the mapping I S is sur jective. T o see that it is injective, consider any U, V ∈ M ( X ). Then w e may assume that ∃ A ⊆ X nonempty , o pen : ∃ ǫ > 0 : ∀ u ∈ U , v ∈ V : 1) x ∈ A ⇒ u ( x ) < v ( x ) − ǫ 2) u, v ∈ C ( A ) (23) so that I S ( U ) ( x ) < I S ( V ) ( x ) − ǫ , x ∈ A It remains to verify ∀ U, V ∈ M ( X ) : U ≤ V ⇔ I S ( U ) ≤ I S ( V ) The implication ‘ U ≤ V ⇒ I S ( U ) ≤ I S ( V ) ’ fo llows b y similar ar guments a s tho s e employed to show that I S is well defined. Conv ers ely , suppo se that I S U ≤ I S V for some U, V ∈ M ( X ). The result now follows in the same wa y a s the injectivit y of I S . This completes the pro of. The following is now immediate. Corollary 5 The sp ac e ML ( X ) is a ful ly distributive lattic e. 3 The Uniform Ord er Con v ergence Struc ture on ML ( X ) As a conseq uence of Prop osition 3 one may define the or de r co nv ergence structure λ o on the space ML ( X ). The order conv erg ence structure induces the order convergence of se q uences o n ML ( X ) and is Hausdor ff, r egular and first countable. In order to define a uniform conv erg e nce structur e, in the sense of [1 1], we intro duce the following nota tion. F or any op en subset U of X , and any subset F of ML ( X ), we deno te b y F | U the restriction of F to U . That is, F | U =  v ∈ ML ( U ) | ∃ w ∈ F : x ∈ U ⇒ w ( x ) = v ( x )  7 Definition 6 L et τ b e the top olo gy on X , and let Σ c onsist of al l nonempty or der intervals in ML ( X ) . L et J o denote the family of filters on ML ( X ) × ML ( X ) that satisfy the fol lowing: Ther e ex ists k ∈ N such that ∀ i = 1 , ..., k : ∃ Σ i =  I i n  ⊆ Σ : 1) I i n +1 ⊆ I i n , n ∈ N 2) ([Σ 1 ] × [Σ 1 ]) ∩ ... ∩ ([Σ k ] × [Σ k ]) ⊆ U (24) wher e [Σ i ] = [ { F : F ∈ Σ i } ] . Mor e over, for every i = 1 , ..., k and V ∈ τ one has ∃ u i ∈ ML ( X ) : ∩ n ∈ N I i | V n = { u i } | V or ∩ n ∈ N I i | V n = ∅ (25) Theorem 7 The family J o of filters on ML ( X ) × ML ( X ) c onstit u tes a u niform c onver genc e structur e. Pro of. The fir s four axioms [11, Definition 2.1 .2] a r e clearly fulfilled, so it remains to verify ∀ U , V ∈ J o : U ◦ V exists ⇒ U ◦ V ∈ J o (26) So take any U , V ∈ J o such that U ◦ V exis ts, and let Σ 1 , ..., Σ k and Σ ′ 1 , ..., Σ ′ l be the collectio n of order int er v als a sso ciated with U and V , resp ectively , through Definition 6. Set Φ = { ( i, j ) : [Σ i ] ◦ [Σ ′ j ] exists } Then U ◦ V ⊇ \ { ([Σ i ] × [Σ i ]) ◦ ([Σ j ] × [Σ j ]) : ( i , j ) ∈ Φ } (27) by [11, Pro po sition 2 .1.1 (i)]. Now, ( i, j ) ∈ Φ exis ts if and only if ∀ m, n ∈ N : I i m ∩ I j n 6 = ∅ F or a ny ( i, j ) ∈ Φ, set Σ i,j =  I i,j n  where, for each n ∈ N I i,j n = [inf  I i n  ∧ inf  I j n  , sup  I i n  ∨ sup  I j n  ] Now, using (3), we find U ◦ V ⊇ \ { [Σ i ] × [Σ j ] : ( i, j ) ∈ Φ } ⊇ \ { [Σ i,j ] × [Σ i,j ] : ( i, j ) ∈ Φ } Clearly each Σ i,j satisfies 1) of (24). Since ML ( X ) is fully distributiv e, see Cor ollary 5, (25) also holds. This co mpletes the pro of. An imp orta nt fac t to note is that the uniform order conv erg ence structure J o is defined solely in terms o f the or de r on ML ( X ), and the to po logy on X . This is unusual for a unifor m conv er- gence structure on a function space. Indeed, for a space of functions F ( X , Y ), defined on some set X , and taking v a lues in Y , one defines the unifor m co n vergence structur e either in ter ms of the uniform co nv ergence structure on Y , or in terms o f a conv ergence structure on F ( X, Y ) which is suitably compatible with the algebr aic s tructure of the s pace. Indeed, a co n vergence vector space carries a natural uniform co nv ergence structure, wher e the Cauch y filters are deter mined by the linear structure. That is, F a Cauch y filter ⇔ F − F → 0 (28) This is also the case for the or der conv erg ence str ucture studied in [7] and [3 2]. The motiv ation for introducing a unifor m co n vergence s tr ucture that do es not dep end on the algebra ic structure of the set M L ( X ) co mes from nonlinear P DEs, and in par ticula r the Order Completion Metho d [25], as explained in the Intro ductio n. The conv ergence structure λ J o induced on ML ( X ) by the uniform convergence structure J o may b e characterized as follows. 8 Theorem 8 A filter F on ML ( X ) b elongs to λ J o ( u ) , for some u ∈ ML ( X ) , if and only if t her e exists a family Σ F = ( I n ) of nonempty or der intervals on ML ( X ) such that 1) I n +1 ⊆ I n , n ∈ N 2) ∀ V ∈ τ : ∩ n ∈ N I n | V = { u } | V (29) and [Σ F ] ⊆ F . Pro of. Let the filter F converge to u ∈ M L ( X ). Then, by [11, Definition 2.1.3], [ u ] × F ∈ J o . Hence by Definition 6 there exist k ∈ N and Σ i ⊆ Σ for i = 1 , ..., k such that (24) through (25) are satisfied. Set Ψ = { i : [Σ i ] ⊂ [ u ] } . W e claim F ⊃ \ i ∈ Ψ I i (30) T a ke a set A ∈ ∩ i ∈ Ψ I i . Then fo r each i ∈ Ψ there is a set A i ∈ I i such that A ⊃ ∪ i ∈ Ψ A i . F o r each i ∈ { 1 , ..., k } \ Ψ choose a set A i ∈ I i with u ∈ ML ( X ) \ A i . Then ( A 1 × A 1 ) ∪ ... ∪ ( A k × A k ) ∈ ( I 1 × I 1 ) ∩ ... ∩ ( I k × I k ) ⊂ F × [ u ] and so there is a set B ∈ F s uc h that B × { u } ⊂ ( A 1 × A 1 ) ∪ ... ∪ ( A k × A k ) If w ∈ B then ( u, w ) ∈ A i × A i for some i . Since u ∈ A i , w e get i ∈ Ψ and so w ∈ ∪ i ∈ Ψ A i . This gives B ⊆ ∪ i ∈ Ψ A i ⊆ A and so A ∈ F s o that (30) holds. Clearly , for each i ∈ Ψ, we have ∀ V ∈ τ : ∩ n ∈ N I i n | V = { u } | V (31) W riting each I i n ∈ Σ i in the form I i n = [ λ i n , µ i n ], we claim sup { λ i n : n ∈ N } = u = inf { µ i n : n ∈ N } (32) Suppo se this were not the c ase. Then ther e exists v , w ∈ ML ( X ) such that λ n ≤ v < w ≤ µ n , n ∈ N Then, in view of Pr op osition 2, ther e is some nonempty V ∈ τ suc h that v ( x ) < w ( x ) , x ∈ V which contradicts (25). Since ML ( X ) is fully distributive, the result follows up on setting Σ F =  [ λ n , µ n ] : 1) λ n = inf { λ i n : i ∈ Ψ } 2) µ n = sup { µ i n : i ∈ Ψ }  (33) The conv ers e is trivial. The following is now immediate Corollary 9 Consider a filter F on M L ( X ) . Then F ∈ λ J o ( u ) if and only if F ∈ λ o ( u ) . Ther efor e M L ( X ) is a uniformly Hausdorff uniform c onver genc e sp ac e. In p articular, a se quenc e ( u n ) on ML ( X ) c onver ges to u if and only if ( u n ) or der c onver ges to u . 9 4 The Completion of ML ( X ) This sec tion is concerned with constructing the completion of the uniform conv er gence spa c e ML ( X ). In this re g ard, r ecall that the co mpletio n of the co n vergence vector s pace C ( X ), eq uipp ed with the order conv er g ence str ucture, is the set of finite Hausdorff contin uous functions on X [7]. This space is order isomor phic to the set a all fin ite no rmal lower semi-c ontin uous functions . Note, how ever, that functions u ∈ ML ( X ) need not be finite everywhere, but may , in contradistinction to functions in C ( X ), a s sume the v alues ±∞ on a ny clo s ed nowhere dense subset of X . Hence we consider the space N L ( X ) o f near ly finite nor mal low er semi-contin uous functions on X . F ollowing the results in Section 3 , w e introduce the following uniform conv ergence structure on N L ( X ). Definition 10 L et τ b e t he top olo gy on X , and let Σ c onsist of al l nonempty or der int ervals in N L ( X ) . L et J ♯ o denote the family of filters on N L ( X ) × N L ( X ) that satisfy the fol lowing: Ther e exists k ∈ N su ch that ∀ i = 1 , ..., k : ∃ Σ i =  I i n  ⊆ Σ : 1) I i n +1 ⊆ I i n , n ∈ N 2) ([Σ 1 ] × [Σ 1 ]) ∩ ... ∩ ([Σ k ] × [Σ k ]) ⊆ U (34) wher e [Σ i ] = [ { F : F ∈ Σ i } ] . Mor e over, for every i = 1 , ..., k and V ∈ τ one has ∃ u i ∈ N L ( X ) : ∩ n ∈ N I i | V n = { u i } | V or ∩ n ∈ N I i | V n = ∅ (35) The following now follows by similar ar g ument s as those employed in Section 3. Theorem 11 The family J ♯ o of filt ers on N L ( X ) × N L ( X ) is a Hausdorff uniform c onver genc e structur e. Theorem 12 A filter F on ML ( X ) b elongs t o λ J o if and only if F ∈ λ o ( u ) . W e now pro ceed to s how that N L ( X ) is the completion of ML ( X ). That is, we show that the following three co nditions are satisfied: • The uniform convergence space N L ( X ) is complete • ML ( X ) is uniformly isomorphic to a dense s ubs pa ce o f N L ( X ) • Any uniformly co n tinuous mapping ϕ on ML ( X ) into a co mplete, Hausdorff uniform con- vergence spac e Y extends uniquely to a uniformly contin uous mapping ϕ ♯ from N L ( X ) into Y . Prop ositio n 13 The uniform c onver genc e sp ac e N L ( X ) is c omplete. Pro of. Let F b e a Cauch y filter on N L ( X ), so that F × F ∈ J ♯ o . Let Σ 1 , ..., Σ k be the families of order int er v als asso cia ted with F × F throug h Definition 10. Since N L ( X ) is Dedekind complete it follows by (3 5) that, for each i = 1 , ..., k sup { λ i n : n ∈ N } = u i = inf { µ i n : n ∈ N } (36) for some u i ∈ N L ( X ), where I i n = [ λ i n , µ i n ] for each n ∈ N . By Theor e m 12 each of the filters F i = [Σ i ] conv erg es to u i . Let G ⊇ F be an ultrafilter. Since F ⊇ F 1 ∩ ... ∩ F k it follows that G ⊇ F i for at least o ne i = 1 , ..., k , so that G conv erg es to u i . Therefore [1 1, Prop ositio n 2.3.2 (iii)] the filter F conv erg es to u i . This co mpletes the pro of. 10 Theorem 14 L et X b e a metric sp ac e. Then the sp ac e N L ( X ) is the uniform c onver genc e sp ac e c ompletion of M L ( X ) . Pro of. Fir st we show that the identit y mapping ι : ML ( X ) → N L ( X ) is a unifor mly cont inuous embedding. In this reg ard, it is sufficient to c o nsider a filter [Σ F ] where Σ F is a fa mily of nonempty order int er v als in ML ( X ) that satisfies 1) o f (2 4) and (25). Clea rly ∀ I n = [ λ n , µ n ] ∈ Σ F : ι ( I n ) ⊆ [ ι ( λ n ) , ι ( µ n )] (37) The family Σ ι ( F ) = ( I ′ n ) = { [ ι ( λ n ) , ι ( µ n )] : n ∈ N } (38) satisfies 1) of (34). T o see that (35) holds , we pro ceed by co n tra diction. Assume that for so me W ∈ τ ∃ u, v ∈ N L ( X ) : ∩ n ∈ N I ′ n | W ⊇ { u, v } | W (39) where u | W 6 = v | W . W e may assume that u ( x ) < v ( x ), x ∈ W . Clear ly , λ n ( x ) ≤ ϕ ( x ) ≤ u ( x ) < v ( x ) ≤ µ n , x ∈ W (40) for every n ∈ N , wher e ϕ ( x ) = sup { λ n ( x ) : n ∈ N } which is upp er semi- c o nt inuous. Applying Hahn’s Theo r em twice we find ∃ φ, ψ ∈ C ( W ) : { φ, ψ } ⊆ ∩ n ∈ N I n | W which contradicts (25) s o that (35) must hold. That ι − 1 is uniformly contin uous is trivial. T o see that ι ( ML ( X )) is dense in N L ( X ), consider any u ∈ N L ( X ), and set D u = { x ∈ X : u ( x ) ∈ R } Since D u is o pen, it follows tha t u restricted to D u is no rmal low er semi-contin uous. Since u is a lso finite on D u it follows, see [7, Pr o of o f The o rem 26] that ther e exis ts a sequence ( u n ) of cont inuous functions on D u such that u ( x ) = sup { u n ( x ) : n ∈ N } , x ∈ D u (41) Consider now the se quence ( v n ) =  ( I ◦ S )  u 0 n  where u 0 n ( x ) =  u n ( x ) if x ∈ D u 0 if x / ∈ D u (42) Clearly v n ( x ) = u n ( x ) for every x ∈ D u . W e cla im u = sup { v n : n ∈ N } (43) If (43) do es not hold, then ∃ v ∈ N L ( X ) : n ∈ N ⇒ v n ≤ v < u But then, in view of Pr op osition 2, and the fa c t that D u is o pen and dense, there exis ts an op en and nonempty set W ⊆ D u such that ∀ x ∈ W : n ∈ N ⇒ u n ( x ) ≤ v ( x ) < u ( x ) 11 which con tra dicts (41). Therefor e (43) must hold. The sequence ( v n ) is clearly a Cauch y sequence in ML ( X ) so that M L ( X ) is dense in N L ( X ). The extension pr op erty for unifor mly contin uous mapping s on ML ( X ) fo llows in the standa rd wa y . Note tha t in the ab ove pr o of, we actua lly show ed that N L ( X ) is the Dedekind completion of ML ( X ). Hence the uniform order conv erg e nc e structure provides a nonlinear to po logical mo del for the pr o cess o f taking the Dedekind completion o f ML ( X ). In view o f Pr op osition 4, this extends a previous result o f Anguelov [2] on the Dedekind co mpletio n o f M ( X ). 5 An A pplication to Nonlinear PDEs As an illustration of how the results developed in this pap er may be applied to the problem of obtaining genera lized solutions to no nlinear PDEs, we consider the Navier-Stokes e quations in three spatial dimensions given b y ∂ ∂ t u i ( x, t ) + P 3 j =1 u j ( x, t ) ∂ ∂ x i u j ( x, t ) − ν P 3 j =1 ∂ 2 ∂ x 2 j u i ( x, t ) + ∂ p ∂ x i ( x, t ) = f ( x, t ) P 3 i =1 ∂ ∂ x i u i ( x, t ) = 0 (44) where ( x, t ) ∈ Ω = R 3 × [0 , ∞ ), and f ∈ C 0  Ω , R 3  . W e also req uire the unknown function u = ( u 1 , u 2 , u 3 ) to satisfy the initial v alue u ( x, 0) = u 0 ( x ) , x ∈ R 3 (45) where u 0 ∈ C 2  R 3 , R 3  is a given, divergence free v ecto r field. The equations (44) are suppo sed to mo del the motio n of a fluid thr o ugh three dimensional space, where u sp ecifies the v elo c ity , and p the pressure in the fluid. W e write the equation (44) in the compact form T ( x, t, D ) v ( x, t ) = g ( x, t ) , ( x, t ) ∈ Ω where v = ( u, p ), g = ( f , 0) and the nonlinear P DE op erato r T ( x, t, D ) is defined thro ugh a contin uous mapping F : Ω × R K → R 4 by T ( x, t, D ) v ( x, t ) = F ( x, t, v ( x, t ) , ..., D α v ( x, t ) , ... ) , | α | ≤ 2 (46) With the system of P DEs (44 ) we can asso cia te a mapping T : C 2 (Ω) 4 ∋ u 7→ ( T 1 u, T 2 u, T 3 u, T 4 u ) ∈ C 0 (Ω) 4 (47) In view of (46), one may extend the mappings T uniquely to T : C 2 nd (Ω) 4 → C 0 nd (Ω) Then, for i = 1 , ..., 3 T i : X ∋ v 7→ ( I ◦ S )   ∂ ∂ t u + 3 X j =1 u j ∂ ∂ x i u j − ν 3 X j =1 ∂ 2 ∂ x 2 j u i + ∂ p ∂ x i   ∈ Y (48) and T 4 : X ∋ v 7→ ( I ◦ S ) 4 X i =1 ∂ ∂ x i u ! ∈ Y (49) define unique extensions o f the comp onents of T to X , where X = ML 2 0 (Ω) 4 , 12 Y = ML 0 (Ω) 4 where, for m ∈ N , ML m 0 (Ω) =    u ∈ ML 0 (Ω) 1) u ( · , 0) ∈ C m  R 3  2) ∃ Γ ⊂ Ω closed nowhere dense : u ∈ C m (Ω \ Γ)    (50) With the initial v alue problem (45) we asso ciate the mapping R 0 : X ∋ u 7→ u | t =0 ∈ Z (51) where Z = C 2  R 3 , R 3  That is, R 0 assigns to u ∈ X the restriction of u to the hyperplane R 3 × { 0 } . Note that this amounts to a sep ar ation of the problem of so lv ing the system of PDEs (44), a nd the problem of satisfying the initial v alue. This is a characteris tic featur e of the Order Completion Metho d [25], a nd the pseudo -top ologica l version of the theo r y developed here and in [33]. What is mor e, and a s will b e seen in the seq uel, this a llows for the rather straight forward and ea s y treatment of b oundary and / o r bounda r y v a lue problems, when compa red to the usual functional ana lytic metho ds. Define the mapping T 0 as T 0 : X ∋ v = ( u, p ) 7→ ( T v , R 0 u ) ∈ Y × Z (52) The mapping T 0 induces an equiv alence relation ∼ T 0 on X through ∀ v , w ∈ X : v ∼ T 0 w ⇔ T 0 v = T 0 w (53) The quotient space X/ ∼ T 0 is denotes X T 0 . Ther e is then an inje ctive mapping b T 0 : X T 0 ∋ V 7→ ( T 0 v , R 0 u ) ∈ Y × Z (54) where v = ( u, p ) is any mem b er of the equiv alence class V , such tha t the diagram X ✲ Y × Z T 0 ❄ ✲ X T 0 Y × Z q T 0 i ❄ b T 0 commutes, with q T 0 the quotient ma pping . W e equip the space ML 0 (Ω) with the uniform order conv er gence s tructure J o , and Y ca rries the pro duct uniform conv er gence structur e. The space Z carr ies the uniform con vergence s tructure J λ , see [11], asso cia ted with the co n vergence s tr ucture ∀ u ∈ Z : λ ( u ) = [ u ] (55) 13 That is, ∀ U a filter o n Z × Z : U ∈ J λ ⇔  ∃ u 1 , ..., u k ∈ Z : ([ u 1 ] × [ u 1 ]) ∩ ... ∩ ([ u k ] × [ u k ]) ⊆ U  (56) Note that J λ induces the c o nv ergence structure λ , and is unifor mly Hausdorff a nd complete [1 1]. In particular, the seq ue nce s whic h co n verge with r esp ect to J λ are exactly the constant sequence s . The pro duct space Y × Z ca r ries the pr o duct uniform conv ergence structure, which we deno te by J P . In view of Theor em 14 and [34, Theo rem 3.1 ] the completion ( Y × Z ) ♯ of Y × Z is N L (Ω) 4 × Z , equipp e d with the pro duct uniform conv er g ence structure with resp ect to the uniform conv er gence structure J ♯ o and the unifor m co nv ergence structure J λ . W e equip X T 0 with the initial unifor m conv ergence structure J T 0 with resp ect to the mapping b T 0 . Tha t is, ∀ U a filter o n X T 0 × X T 0 : U ∈ J T 0 ⇔  b T 0 × b T 0  ( U ) ∈ J P (57) Since b T 0 is injective, it is a uniformly co n tinuous embedding so that X T 0 is unifor mly iso mo rphic to a subspace of Y × Z . Therefore, see [34], the the ma pping b T 0 extends to a uniformly c o nt inuous embedding b T ♯ 0 : X ♯ t 0 → ( Y × Z ) ♯ (58) so that X ♯ t 0 is unifor mly iso morphic to a subspace of ( Y × Z ) ♯ . This is summarized in the following commutativ e diagram. X T 0 ✲ Y × Z b T 0 ❄ ✲ X ♯ T 0 ( Y × Z ) ♯ ❄ b T ♯ 0 A generalize d solution to (44) throug h (45) is any V ♯ ∈ X ♯ T 0 that satisfies the equatio n b T ♯ 0 V ♯ = g (59) The ma in re sult of this s ection, concer ning the existence of ge ne r alized so lutions to (44) through (45), is based on the existence of a pproximate solutions , which follows form the following [33]. W e include the pro of to illustra te the tec hnique. Lemma 15 Consider any g = ( f , 0) ∈ C 0 (Ω) and any ǫ > 0 . Then ∀ ( x 0 , t 0 ) ∈ Ω : ∃ v = ( u, p ) ∈ C 2 (Ω) : ∃ δ > 0 : ∀ ( x, t ) ∈ Ω :  k x 0 − x k < δ | t 0 − t | < δ  ⇒ g ( x, t ) − ǫ < T ( x, t, D ) v ( x, t ) < g ( x, t ) (60) wher e the or der ab ove is c o or dinatewise, and ǫ r epr esen t s the 4 dimensional ve ctor that c orr esp onds to the re al n umb er ǫ . 14 Pro of. Note tha t, for every ( x, t ) ∈ Ω, the function F satisfies { F ( x, t, ξ ) : ξ = ( ξ α ) | α |≤ 2 ∈ R K } = R 4 so that, for every ( x, t ) a nd ǫ > 0, there is s ome ξ ǫ ∈ R K such tha t F ( x, t, ξ ǫ ) = g ( x, t ). Let v = ( u, p ) b e the C 2 -smo oth function such tha t D α u ( x, t ) = ξ ǫ α The result now follows from the contin uit y of v , F and g . The following is essentially a version o f Lemma 15 above which incor po rates the initial condi- tion (45). Lemma 16 L et g and ǫ b e as in L emma 15 ab ove. Consider any u 0 ∈ C 2  R 3 , R 3  . Then ∀ x 0 ∈ R 3 : ∃ v = ( u, p ) ∈ C 2 (Ω) : ∃ δ > 0 : 1) ∀ ( x, t ) ∈ Ω :  k x 0 − x k < δ | t | < δ  ⇒ g ( x, t ) − ǫ < T ( x, t, D ) v ( x, t ) < g ( x, t ) 2) x ∈ R 3 ⇒ u ( x, 0 ) = u 0 ( x ) (61) Pro of. The pro of follows similar a r guments a s those employ ed in the pro of of Lemma 15 when one sets u ( x, t ) = u 0 ( x ) + ϕ ( t ) where ϕ ∈ C 2 ([0 , ∞ )) is an appro priate function such that ϕ (0) = 0. The main result of this section is no w the following. Theorem 17 F or any g = ( f , 0) ∈ Y and any u 0 ∈ Z , ther e exists a u nique V ♯ ∈ X ♯ T 0 such that b T ♯ 0 V ♯ = g (62) Pro of. Let Ω = [ ν ∈ N C ν (63) where, for ν ∈ N , the compact sets C ν are 4-dimensional interv als C ν = [ a ν , b ν ] (64) with a ν = ( a ν, 1 , ..., a ν,n ), b ν = ( b ν, 1 , ..., b ν,n ) ∈ R n and a ν,i ≤ b ν,i for every i = 1 , ..., n . W e also assume that C ν , with ν ∈ N are lo c ally finite, that is, ∀ ( x, t ) ∈ Ω : ∃ V x ⊆ Ω a neighborho o d of x : { ν ∈ N : C ν ∩ V x 6 = ∅} is finite (65) W e also assume that the in terio rs of C ν , with ν ∈ N , are pairwise disjoint. W e no te that such C ν exist, see [16]. Select ν ∈ N and ǫ > 0 ar bitrary but fixed. F or any ( x, t ) ∈ C ν , let δ ( x,t ) > 0 b e the po sitive nu mber and v ǫ ( x,t ) the function asso ciated with ( x, t ) throug h Lemma 15, if t > 0, and Le mma 16 if t = 0. Since C ν is compact, it follows that ∃ δ > 0 : ∀ ( x 0 , t 0 ) ∈ C ν : ∃ v = ( u, p ) ∈ C 2  R 4 , R 4  : 1)  k x − x 0 k ≤ δ | t − t 0 | ≤ δ  ⇒ g ( x, t ) − ǫ ≤ T ( x, t, D ) v ( x, t ) ≤ g ( x, t ) , ( x, t ) ∈ Ω 2) t 0 = 0 ⇒ u ( x ) = u 0 ( x ) , x ∈ R 3 (66) 15 Subdivide C ν int o n - dimensional in terv als I ν, 1 , ..., I ν,µ ν such that their interiors are pairwise disjoint and ∀ ( x 0 , t 0 ) , ( x, t ) ∈ I ν,i : 1) k x 0 − x k < δ 2) | t 0 − t | < δ If I ν,i ∩  R 3 × { 0 }  = ∅ , take a i to b e the center of the interv a l I ν,j . Then by (66) there exists v ν,i = ( u , p ) ∈ C 2  R 4 × R 4  such that g ( x, t ) − ǫ ≤ T ( x, t, D ) v ν,i ( x, t ) ≤ g ( x, t ) , ( x, t ) ∈ I ν,i (67) If, on the o ther hand, I ν,i ∩  R 3 × { 0 }  6 = ∅ , let a i denote the pro jection of the midp oint of I ν,i on the hype rplane R 3 × { 0 } . Then by (66) there ex ists v ν,i = ( u , p ) ∈ C 2  R 4 × R 4  such that (6 7) holds and u ( x, 0) = u 0 ( x ) , ( x, 0) ∈  R 3 × { 0 }  ∩ I ν,i (68) Now set v ǫ = ( u ǫ 1 , u ǫ 2 , u ǫ 3 , p ǫ ) = X ν ∈ N µ ν X i =1 v ν,i χ I nu,i ! (69) where χ I ν,i is the characteristic function of I ν,i . Clearly , v ǫ = ( u ǫ , p ǫ ) is C 2 -smo oth everywhere except on a clo sed nowhere dense set, w hich has measure 0, and u ǫ ( x, 0) = u 0 ( x ) everywhere except on a closed nowhere dense subset of R 3 × { 0 } . Now set w ǫ = ( u ǫ ∗ 1 , u ǫ ∗ 2 , u ǫ ∗ 3 , p ǫ ∗ ) where, for j = 1 , ..., 3 u ǫ ∗ j = ( I ◦ S )  u ǫ j  and p ǫ ∗ = ( I ◦ S ) ( p ǫ ) (70) Clearly the function w ǫ belo ngs to X . What is more,in view of (67) through (68), it follows that g − ǫ ≤ T w ǫ ≤ g and R 0 w ǫ = u 0 so that the sequence ( T 0 w n ) =  T 0 w 1 n  conv erges to  g , u 0  in Y × Z . F o r each n ∈ N , let W n denote the ∼ T 0 -equiv alence class genera ted by the function w 1 n . The sequence ( W n ) is Cauch y in X T 0 , and since b T 0 is unifor mly cont inuous, there exists V ♯ ∈ X ♯ T 0 that satis fie s (62). More ov er, V ♯ is unique, since the mapping b T ♯ 0 is a uniformly contin uous embedding. The uniquenes s of the gene r alized so lution should not b e misint er preted. Note tha t the completion of X T 0 consists of equiv alence classe s of Cauch y filters on X T 0 , under the equiv alence relation F ∼ C G ⇔  ∃ H a Cauch y filter : H ⊆ F ∩ G  In vie w of this, the so lution V ♯ is actually the equiv alence class of filters F on X T 0 such that b T 0 ( F ) c onv erges to  g , u 0  in Y × Z . What is more, V ♯ contains also all classic al , o r smo o th, solutions to (44) throug h (45), as well a s a ll nonclassic al solutio ns v = ( u, p ) ∈ C 2 nd (Ω) 4 , since eac h such a solution g enerates a Cauch y sequence in X T 0 . Hence our notion of a gener a lized solution is c onsistent with the usual c la ssical and noncla ssical solutions in C 2 nd (Ω) 4 to (44) through (45). Note that the metho d presented her e for the three dimens ional Navier-Stokes eq uations applies equally well to a ny dimension n ≥ 2. 16 6 Conclusion W e have co nstructed a n o rder isomo rphic re presentation M L ( X ) of the quo tient space M ( X ) consisting of norma l lower semi-contin uous functions on X . A nontrivial uniform conv ergence structure on ML ( X ), which induces the or de r convergence structure was co nstructed solely in terms of the order on ML ( X ). The completion of the uniform conv erg ence space ML (Ω) is obtained as the set N L ( X ) of nearly finite normal low er semi-c o nt inuous functions on X . This result essentially relies on the fact that N L ( X ) is the Dedekind c o mpletion o f ML ( X ). Hence we hav e established a topo logical type mo del for the Dedekind completion o f the space ML ( X ). 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