The Order Completion Method for Systems of Nonlinear PDEs: Solutions of Initial Value Problems

We present an existence result for generalized solutions of initial value problems obtained through the order completion method. The solutions we obtain satisfy the initial condition in a suitable extended sense, and each such solution may be represented in a canonical way through its generalized partial derivatives as nearly finite normal lower semicontinuous function.


Introduction
It is a virtual consensus among mathematicians specializing in nonlinear partial differential equations (PDEs) that a general and type independent theory for the existence and basic regularity of generalized solutions of such equations is not possible [1]; see also [2]. Within the setting of the usual linear topological spaces of generalized functions that are customary in the study of PDEs, this may perhaps turn out to be the case. Here we may point out two possible reasons for the failure of the mentioned customary spaces of generalized functions to contain solutions of large classes of linear and nonlinear PDEs.
Firstly, these spaces typically fail to contain sufficiently singular objects. Indeed, the Sobolev spaces have been so successful in the study of PDEs exactly because, in some cases, they lead to rather regular, in fact even smooth, generalized solutions of PDEs. On the other hand, singularities which may occur in the solutions of nonlinear PDEs may be rather arbitrary. We may recall that even in the case of analytic nonlinear PDEs, the Cauchy-Kovalevskaia Theorem [3] guarantees the existence of an analytic solution only on a neighborhood of a given noncharacteristic analytic hypersurface. As such, any solution which is defined on the whole domain of definition of a given analytic system of nonlinear PDEs, see, for instance [4], will in general admit singularities. In the simplest case of an analytic function with an essential singularity at a single point, the Great Picard Theorem states that the function will attain every complex value, with possibly one exception, in every neighborhood of the singularity.
This brings us to the second reason for the failure of usual spaces of generalized functions to contain solutions of large classes of systems of nonlinear PDEs. Namely, the generalized functions that are the elements of these spaces are typically defined in terms of certain growth conditions. This is true, for instance, of the Sobolev spaces, the elements of which must be locally integrable, and the Colombeau algebras of generalized functions [5], where the generalized functions are required to satisfy certain polynomial type growth conditions near singularities. In view of our remarks above concerning the existence of solutions of analytic systems of nonlinear PDEs, the deficiency of such growth conditions is clear. Indeed, an analytic function which has an essential singularity at just one point may grow faster than any polynomial near the singularity and may therefore also fail to be locally integrable on any neighborhood of that singularity.
However, and in contradistinction with the perviously mentioned insufficiency of the customary functional analytic methods, the order completion method, published in the 1994 monograph [6], delivers generalized solutions of a large class of systems of continuous nonlinear PDEs. These solutions are constructed as the elements of the Dedekind order completion of suitable spaces of piecewise smooth 2 Abstract and Applied Analysis functions. Furthermore, the solutions obtained in this way have been shown to satisfy a basic blanket regularity in the sense that the solutions may be assimilated with usual Hausdorff continuous interval valued functions [7].
Recently, see [8][9][10], the mentioned order completion method was reformulated and enriched by introducing suitable uniform convergence spaces. This has led to a significant improvement in the regularity of the generalized solutions obtained, as well as significant new insight into the structure of the solutions.
In this paper, we show how the techniques developed in [10] may be adapted in order to also incorporate initial and/or boundary values that may be associated with a given system of nonlinear PDEs. As it turns out, in order to incorporate such addition conditions into the theory, the methods that apply to the free problem need only be modified slightly. This state of affairs should be compared with the usual linear functional analytic techniques for solving linear and nonlinear PDEs, where the presence of initial and/or boundary values often leads to significant complications, which typically require entirely new methods. In this way we come to appreciate yet another advantage of solving nonlinear PDEs by the methods introduced in [10]. Namely, initial and boundary value problems are solved by essentially the same techniques that apply to the free problem.
The paper is organized as follows. In Section 2 we recall some basic concepts relating to the spaces of normal lower semicontinuous functions upon which the spaces of generalized functions are constructed in Section 3. The existence of generalized solutions of a large class of initial value problems is presented in Section 4, where we also discuss the structure and regularity of the solutions.

Normal Lower Semicontinuous Functions
In this section we recall some basic facts concerning spaces of normal lower semicontinuous functions upon which the spaces of generalized functions are constructed. In particular, the spaces of generalized functions are constructed as the completions of suitable uniform convergence spaces, the elements of which are normal lower semicontinuous functions. In order to make the exposition as self-contained as possible, we also include a brief account of the spaces introduced in [8][9][10].
In this regard, let Ω be an open subset of R , and denote by A(Ω) the set of extended real valued functions on Ω.

(5)
A function ∈ A(Ω) is normal lower semicontinuous at ∈ Ω whenever while is normal lower semicontinuous on Ω provided it is normal lower semicontinuous at every point ∈ Ω; see [12,13]. A normal lower semicontinuous function is called nearly finite whenever { ∈ Ω : ( ) ∈ R} is open and dense in Ω.
The set of nearly finite normal lower semicontinuous functions on Ω is denoted by NL(Ω). Clearly, every continuous, real valued function on Ω is nearly finite and normal lower semicontinuous, so that we have the inclusion Conversely, each function ∈ NL(Ω) is continuous on a residual set. That is, ∃ ⊂ Ω of first Baire category : ∈ Ω \ ⇒ is continuous at .
For ∈ N ∪ {0}, we consider the set Each of the spaces ML (Ω) is a sublattice of NL(Ω); see [14]. In particular, ML 0 (Ω) is -order dense in NL(Ω). That is, for each ∈ NL(Ω) we have ∃ ( ) , ( ) ⊂ ML 0 (Ω) : The spaces of generalized functions introduced in [10] are constructed as the completions of suitable uniform convergence spaces. In this regard, a uniform convergence structure is defined on ML 0 (Ω) in the following way. Definition 1. Let Σ consist of all nonempty order intervals in ML 0 (Ω). Let J denote the family of filters on ML 0 (Ω) × ML 0 (Ω) that satisfy the following. There exists ∈ N such that ∀ = 1, . . . , : (1) +1 ⊆ , ∈ N, The uniform convergence structure J is first countable and uniformly Hausdorff. Furthermore, a filter F on ML 0 (Ω) converges to ∈ ML 0 (Ω) with respect to J if and only if The completion of the space ML 0 (Ω) with respect to the uniform convergence structure J may be represented as the set NL(Ω), equipped with the appropriate uniform convergence structure. This completion result follows essentially as an application of the order completeness of NL(Ω) and the approximation property (18). The correct uniform convergence structure on NL(Ω) is defined as follows.
For ≥ 1 the usual linear partial differential operators : C (Ω) → C 0 (Ω), | | ≤ extend uniquely to mappings Abstract and Applied Analysis which may be defined as The space ML (Ω) is equipped with the initial uniform convergence structure J with respect to the family of mappings (22). That is, Clearly J makes each of the mappings (22) uniformly continuous. In fact, it is the coarsest uniform convergence structure with respect to which each of the mappings (22) is uniformly continuous. Since the family of mappings (22) is countable, it follows from the first countability of J that the uniform convergence structure J is also first countable. Furthermore, the family of mappings (22) separates the points of ML (Ω), that is: so that the uniform convergence structure J is uniformly Hausdorff. As such, we may construct its completion, which we denote by NL (Ω). This notation is due to the fact that, as we will shortly see, we may identify the completion of ML (Ω) in a canonical way with a subspace of NL(Ω) , for a suitable integer . Indeed, see [14], the mapping with ML 0 (Ω) equipped with the product uniform convergence structure, is a uniformly continuous embedding. As such, it may be extended in a unique way to an injective uniformly continuous mapping Here the mappings are the unique uniformly continuous extensions of the mappings (22).

Spaces of Generalized Functions
The space of generalized functions NL (Ω) was shown in [10] to contain generalized solutions of a large class of systems of nonlinear PDEs. However, as mentioned in Section 1, this existence result does not take into account any initial and/or boundary values that may be associated with a given system of nonlinear PDEs. In order to also incorporate such additional conditions, we need only modify the construction of the space NL (Ω) slightly.
In this regard, consider a system of nonlinear PDEs: with ∈ R, ∈ R −1 , ≥ 1, 0 ≤ < , ∈ N −1 , | |+ ≤ and with the Cauchy data We assume that the initial data (30) satisfies It follows immediately from the results presented in [10] that the system of nonlinear PDEs (29) admits a generalized solution in NL (R ) . However, such a solution may fail to satisfy the initial condition (30) in any suitable extended sense.
In order to incorporate the initial condition (30) into our solution method, we introduce the following spaces of functions. Denote by ML g (Ω) the set where Ω = R −1 × R. For each = 1, . . . , , every 0 ≤ < and each ∈ N −1 such that 0 ≤ | | + ≤ , we consider the space ML 0 , , (Ω), which is defined through Clearly, for every 0 ≤ < , and ∈ N −1 such that 0 ≤ | | + ≤ , and each = 1, . . . , we may define the partial differential operators as in Section 2 through The partial differential operator D , is defined in a similar way, namely, as The method for constructing generalized solutions of the initial value problem (29) to (30) presented here is essentially the same as that used in the case of arbitrary systems of nonlinear PDEs, which is developed in [10]. In particular, generalized solutions are constructed as elements of the completion of the space ML g (Ω), equipped with a suitable uniform convergence structure. In this regard, we introduce the following uniform convergence structure on ML 0 , , (Ω). (1) +1 ⊆ , ∈ N, Proof. The first four axioms of the definition of a uniform convergence structure [15] are clearly fulfilled, so it remains to verify In this regard, take any U, V ∈ J such that U ∘ V exists, and let Σ 1 , . . . , Σ and Σ 1 , . . . , Σ be the collections of order intervals associated with U and V, respectively, through Since ML 0 (Ω) is fully distributive, the conditions in (38) follow by Lemma 5. The second part of the proposition follows by the same arguments used in the proof of [8,Theorem 8].
The completion of ML 0 , , (Ω) may be represented as a suitable space of nearly finite normal lower semicontinuous functions. In particular, consider the space Note that ML 0 , , (Ω) ⊂ NL , , (Ω). As such, in order to show that NL , , (Ω) is the completion of ML 0 , , (Ω), we must introduce a Hausdorff uniform convergence structure J ♯ , , on NL , , (Ω) in such a way that the following conditions are satisfied.
where each ∈ NL , , (Ω) satisfies = sup{ : ∈ N} = inf{ : ∈ N}, and Σ = { : ∈ N} with That the family of filters J ♯ , , does indeed constitute a Hausdorff uniform convergence structure on NL , , (Ω) can easily be seen. Indeed, J ♯ , , is nothing but the uniform convergence structure associated with the following Hausdorff convergence structure through [  Proof. That NL , , (Ω) is complete follows immediately by the above remarks. Furthermore, it is clear that the subspace uniform convergence structure on ML 0 , , (Ω) is equal to J , , . To see that ML 0 , , (Ω) is dense in NL , , (Ω), consider any ∈ NL , , (Ω). We claim where Suppose that this were not the case. Then, since NL(Ω) is Dedekind order complete, it follows that there is some 0 ∈ NL(Ω) so that 0 = sup A. Since (55) does not hold, it follows that 0 must satisfy > 0 . Furthermore, it is clear from (51) that 0 ∈ NL , , (Ω). In view of (10) and the semicontinuity of and 0 we have From (51) it follows that ∩ R −1 × { 0 } = 0. As such, it follows that + ( /2) ∈ NL , , (Ω) for each ∈ A and that It now follows that which is a contradiction. Thus (55) holds. The fact that ML 0 , , (Ω) is dense in NL , , (Ω) now follows from the fact that NL(Ω) is order separable [8].
The extension property for uniformly continuous mappings follows by a straightforward argument.
is uniformly continuous. Since the mappings (61) and (62) coincide on a dense subset of NL , , (Ω), it follows that (61) is simply the inclusion mapping (62). This is related to the issue of consistency of generalized solutions of (29) to (30), which we construct in the sequel, with solutions in the space NL (Ω) ; see [10]. We will discuss this in some detail in what follows, after the uniform convergence structure on ML g (Ω) has been introduced. In this regard, the uniform convergence structure J g on ML g (Ω) is defined as the initial uniform convergence structure with respect to the mappings (35) to (37). That is, a filter U on ML g (Ω) × ML g (Ω) belongs to J g if and only if ∀ = 1, . . . , : and ∀ 0 ≤ < : ∀ ∈ N −1 , 0 < + ≤ : ∀ = 1, . . . , : Clearly the family consisting of the mappings (35) through (37) separates the points of ML g (Ω). As such, the uniform convergence structure J g is uniformly Hausdorff. In particular, see [14], the mapping The meaning of the diagram (68) is twofold. Firstly, it explains the regularity of generalized functions in NL g (Ω). In particular, each generalized partial derivative of a generalized function u ♯ ∈ NL g (Ω) is a nearly finite normal lower semicontinuous function. Therefore, each such generalized function may be represented as an element of the space (∏NL , (Ω))×NL(Ω) in a canonical way. Secondly, these diagrams state that each generalized function u ♯ ∈ NL g (Ω) satisfies the initial condition (30) in the sense that ∀ = 1, . . . , : ∀ 0 ≤ < :

Existence of Generalized Solutions
With the system of nonlinear PDEs (29) we may associate a mapping We arrive at the notion of generalized solution of the initial value problem (29) and (30) in the context of the space NL g (Ω) by suitably extending the mapping (70) to a mapping Such an extension is obtained through the uniform continuity of the mapping (70). In this regard, we have the following.

Theorem 8. The mapping (70) is uniformly continuous.
Proof. It follows from (60) through (61) that the inclusion mapping is uniformly continuous. The result now follows from the commutative diagram and the uniform continuity of T 0 , which is the mapping defined on ML (Ω) through the nonlinear partial differential operator; see [10].
In view of Theorem 8 the mapping (70) extends in a unique way to a uniformly continuous mapping (72). As such, the generalized initial value problem corresponding to (29) and (30) is given by the single equation where 0 denotes the element in NL(Ω) with all components identically 0. A solution of (75) is interpreted as a generalized solution of (29) through (30) based on the fact that each solution of (75) satisfies the initial condition in a suitable generalized sense, as mentioned in (69). Furthermore, in view of (60) to (62) and the diagram (74) we obtain the commutative diagram with ♯ injective and T ♯ 0 the uniformly continuous extension of the mapping associated with the system of nonlinear PDEs (29). In particular, the mapping ♯ is the inclusion mapping. As such, each solution u ♯ ∈ NL g (Ω) of (75) is a generalized solution of the system of nonlinear PDEs (29) in the sense of the spaces of generalized functions introduced in [10]. The main result of this section is the following.
We have shown that the initial value problem (29) through (30) admits a generalized solution in the space NL g (Ω). In particular, and in view of the commutative diagram (74), the generalized solution constructed in Theorem 9 is a generalized solution of the system of nonlinear PDEs in the sense of the spaces of generalized functions introduced in [10]. Furthermore, this solution satisfies the initial condition (30) in the sense that of the solution u ♯ is of first Baire category. In particular, there exists a residual subset of Ω such that each generalized partial derivative of u ♯ is continuous at every point of . As mentioned in Section 1, it is known [3] that if the mapping G : Ω×R → R in (29), as well as the initial data (30) is real analytic, then the initial value problem admits an analytic solution on a neighborhood of the noncharacteristic hypersurface S. Furthermore, Rosinger [4] showed that such an initial value problem admits a generalized solution u, in a suitable differential algebra of generalized functions, which is analytic everywhere except on a closed nowhere dense set. Since this solution is analytic in a neighborhood of S, it follows that u ∈ ML g (Ω) and satisfies As such, u is also a solution in the sense discussed in this paper.
It should be noted that the customary spaces of generalized functions that are typical in the study of nonlinear PDEs may fail to contain generalized solutions of the initial value problem (29) to (30). Indeed, it has been shown that some of these spaces, such as spaces of distributions, fail to contain generalized solutions even of C ∞ -smooth linear PDEs; see, for instance [17]. Theorem 9 is therefore a first in the literature. Namely, it is the first extension of the Cauchy-Kovalevskaia theorem, within a suitable space of generalized functions, on its own general and type independent grounds, to equations that are not analytic.
The extent to which the solution constructed here may be interpreted as a classical solution on some part of the domain of definition of the system of equations is unknown at present. Furthermore, it is unknown whether or not, in general, the solution may be interpreted in terms of any of the spaces of generalized functions that are typical in the study of linear and nonlinear PDEs.

Conclusion
In this paper we have shown how the methods developed in [10] may be modified in order to incorporate also initial and/or boundary value problems. In this regard, generalized solutions of a large class of nonlinear initial value problems are constructed. It should be noted that the techniques used to obtain the existence of solution are essentially the same as those used in [10] for the free problem. In this way, we come to appreciate another advantage of solving nonlinear PDEs in the spaces of generalized functions used here and in [10]. Namely, and in contradistinction with the usual functional analytic methods, initial value problems do not result in significant additional complications in the solution method.