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

Authors: Jan Harm van der Walt
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 1. Introduction

It is a virtual consensus among mathematicians specializing in nonlinear partial differential equations (PDEs) that ageneral 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 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 [8–10], 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 Section2 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.


## 2. 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 [8–10].In this regard, letΩ be an open subset of ℝk, and denote by 𝒜(Ω) the set of extended real valued functions on Ω. That is, 𝒜(Ω)={u:Ω→ℝ¯}, where ℝ¯=ℝ∪{±∞} is the extended real line. The lower and upper Baire operators I:𝒜(Ω)→𝒜(Ω) and S:𝒜(Ω)→𝒜(Ω) are defined through
(1)I(u):Ω∋x↦sup{inf{u(y):y∈V}:V∈𝒱x}∈ℝ¯,(2)S(u):Ω∋x↦inf{sup{u(y):y∈V}:V∈𝒱x}∈ℝ¯,
respectively, where 𝒱x denotes the neighborhood filter at x∈Ω, see [11] or [12] for a recent presentation. The mappings (1) and (2) satisfy
(3)∀u∈𝒜(Ω):I(u)≤u≤S(u).Furthermore, the operatorsI, S and their compositions are idempotent and monotone with respect to the pointwise order on 𝒜(Ω). That is,
(4)∀u∈𝒜(Ω):(1)I(I(u))=I(u),(2)S(S(u))=S(u),(3)(I∘S)((I∘S)(u))=(I∘S)(u),and(5)∀u,v∈𝒜(Ω):u≤v⇒((1)I(u)≤I(v)(2)S(u)≤S(v)(3)(I∘S)(u)≤(I∘S)(v)).A functionu∈𝒜(Ω) is normal lower semicontinuous at x∈Ω whenever
(6)(I∘S)(u)(x)=u(x),
while u is normal lower semicontinuous on Ω provided it is normal lower semicontinuous at every point x∈Ω; see [12, 13]. A normal lower semicontinuous function is called nearly finite whenever
(7){x∈Ω:u(x)∈ℝ}isopenanddenseinΩ.The set of nearly finite normal lower semicontinuous functions onΩ 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
(8)𝒞0(Ω)⊆𝒩ℒ(Ω).Conversely, each functionu∈𝒩ℒ(Ω) is continuous on a residual set. That is,
(9)∀u∈𝒩ℒ(Ω):∃B⊂ΩoffirstBairecategory:x∈Ω∖B⇒uiscontinuousatx.The following useful property of continuous functions extends to𝒩ℒ(Ω):(10)∀u,v∈𝒩ℒ(Ω):∀D⊆Ωdense:(∀x∈D:u(x)≤v(x))⇒u≤v.With respect to the pointwise order(11)u≤v⟺(∀x∈Ω:u(x)≤v(x))
the set 𝒩ℒ(Ω) is a Dedekind complete lattice. In particular, the supremum and infimum of a set 𝒜⊂𝒩ℒ(Ω) is given by
(12)inf𝒜=(I∘S)(φ),(13)sup𝒜=(I∘S)(ψ),
respectively, where φ:Ω∋x↦inf{u(x):u∈𝒜} and ψ:Ω∋x↦sup{u(x):u∈𝒜}. Furthermore, the lattice 𝒩ℒ(Ω) is fully distributive. That is,
(14)∀v∈𝒩ℒ(Ω):∀𝒜⊂𝒩ℒ(Ω):u0=sup𝒜⇒sup{inf{u,v}:u∈𝒜}=inf{u0,v}.A useful characterization of order bounded sets in terms of pointwise bounded sets is given as follows. If a set𝒜⊂𝒩ℒ(Ω) satisfies (15)∃B⊂ΩoffirstBairecategory:∀x∈Ω∖B:sup{u(x):u∈𝒜}<∞,
then
(16)∃u0∈𝒩ℒ(Ω):∀u∈𝒜:u≤u0.The dual statement for sets bounded from below also holds.Form∈ℕ∪{0}, we consider the set
(17)ℳℒm(Ω)={u∈𝒩ℒ(Ω)∣∃Γ⊂Ωclosednowheredense:u∈𝒞m(Ω∖Γ)}.Each of the spacesℳℒm(Ω) is a sublattice of 𝒩ℒ(Ω); see [14]. In particular, ℳℒ0(Ω) is σ-order dense in 𝒩ℒ(Ω). That is, for each u∈𝒩ℒ(Ω) we have
(18)∃(λn),(μn)⊂ℳℒ0(Ω):(1)λn≤λn+1≤u≤μn+1≤μn,n∈ℕ,(2)sup{λn:n∈ℕ}=u=inf{μn:n∈ℕ}.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 ℳℒ0(Ω) in the following way.Definition 1.
LetΣ consist of all nonempty order intervals in ℳℒ0(Ω). Let 𝒥o denote the family of filters on ℳℒ0(Ω)×ℳℒ0(Ω) that satisfy the following. There exists k∈ℕ such that
(19)∀j=1,…,k:∃Σi=(Inj)⊆Σ:∃uj∈𝒩ℒ(Ω):(1)In+1j⊆Inj,n∈ℕ,(2)sup{infInj:n∈ℕ}=uj=inf{supInj:n∈ℕ},(3)([Σ1]×[Σ1])∩⋯∩([Σk]×[Σk])⊆𝒰.The uniform convergence structure𝒥o is first countable and uniformly Hausdorff. Furthermore, a filter ℱ on ℳℒ0(Ω) converges to u∈ℳℒ0(Ω) with respect to 𝒥o if and only if
(20)∃(λn),(μn)⊂ℳℒ0(Ω):(1)n∈ℕ⇒λn≤λn+1≤μn+1≤μn,(2)sup{λn:n∈ℕ}=u=inf{μn:n∈ℕ},(3)[{[λn,μn]:n∈ℕ}]⊆ℱ.The completion of the spaceℳℒ0(Ω) with respect to the uniform convergence structure 𝒥o 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.Definition 2.
A filter𝒰 on 𝒩ℒ(Ω)×𝒩ℒ(Ω) belongs to the family 𝒥o♯ whenever, for some positive integer k, we have the following:
(21)∀i=1,…,k:∃(λni),(μni)⊂ℳℒ0(Ω):∃ui∈𝒩ℒ(Ω):(1)λni≤λn+1i≤μn+1i≤μni,n∈ℕ,(2)sup{λni:n∈ℕ}=ui=inf{μni:n∈ℕ},(3)⋂i=1k‍(([Σi]×[Σi])∩([ui]×[ui]))⊆𝒰.
Here Σi={Ini:n∈ℕ} with Ini={u∈ℳℒ0:λni≤u≤μni}.Form≥1 the usual linear partial differential operators Dα:𝒞m(Ω)→𝒞0(Ω),|α|≤m extend uniquely to mappings
(22)𝒟α:ℳℒm(Ω)→ℳℒ0(Ω),|α|≤m,
which may be defined as
(23)𝒟α:ℳℒm(Ω)∋u↦(I∘S)(Dαu)∈ℳℒ0(Ω),|α|≤m.The spaceℳℒm(Ω) is equipped with the initial uniform convergence structure 𝒥m with respect to the family of mappings (22). That is,
(24)𝒰∈𝒥m⟺(∀|α|≤m:(𝒟α×𝒟α)(𝒰)∈𝒥o).Clearly𝒥m 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 𝒥o that the uniform convergence structure 𝒥m is also first countable. Furthermore, the family of mappings (22) separates the points of ℳℒm(Ω), that is:
(25)∀u,v∈ℳℒm(Ω):∃|α|≤m:𝒟αu≠𝒟αv,
so that the uniform convergence structure 𝒥m is uniformly Hausdorff. As such, we may construct its completion, which we denote by 𝒩ℒm(Ω). This notation is due to the fact that, as we will shortly see, we may identify the completion of ℳℒm(Ω) in a canonical way with a subspace of 𝒩ℒ(Ω)M, for a suitable integer M. Indeed, see [14], the mapping
(26)D:ℳℒm(Ω)∋u↦(𝒟αu)|α|≤m∈ℳℒ0(Ω)M,
with ℳℒ0(Ω)M 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
(27)D♯:𝒩ℒm(Ω)∋u♯↦(𝒟α♯u♯)|α|≤m∈𝒩ℒ(Ω)M.Here the mappings(28)𝒟α♯:𝒩ℒm(Ω)→𝒩ℒ(Ω),|α|≤m
are the unique uniformly continuous extensions of the mappings (22).


## 3. Spaces of Generalized Functions

The space of generalized functions𝒩ℒm(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 𝒩ℒm(Ω) slightly.In this regard, consider a system ofK nonlinear PDEs:
(29)Dtmu(t,y)=G(t,y,…,DyqDtpui(t,y),…)
with t∈ℝ,  y∈ℝk-1, m≥1, 0≤p<m, q∈ℕk-1, |q|+p≤m and with the Cauchy data
(30)Dtpu(t0,y)=gp(y),0≤p<m,(t0,y)∈S
on the hyperplane
(31)S={(t0,y):y∈ℝk-1}.We assume that the initial data (30) satisfies
(32)∀0≤p<m:gp∈𝒞m-p(ℝk-1)K.It follows immediately from the results presented in [10] that the system of nonlinear PDEs (29) admits a generalized solution in 𝒩ℒm(ℝk)K. 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 ℳℒgm(Ω) the set
(33)ℳℒgm(Ω)={u∈ℳℒm(Ω)K|∀i=1,…,K:∀0≤p<uum:∀0≤psasa∀0≤p<m:∀0≤p<uum:jwi≤psdsasa∀q∈ℕk-1,0≤|q|+p≤m:iu:∀≤pasa(1)𝒟ytqpui(y,t0)=Dqgp,i(y),y∈ℝk-1(2)𝒟ytqpuiiscontinuousat(y,t0)≤asa},
where Ω=ℝk-1×ℝ. For each i=1,…,K, every 0≤p<m and each q∈ℕk-1 such that 0≤|q|+p≤m, we consider the space ℳℒi,q,p0(Ω), which is defined through
(34)ℳℒi,q,p0(Ω)={u∈ℳℒ0(Ω)|∀y∈ℝk-1:∀0≤p:∀0≤iass(1)u(y,t0)=Dqgp,i(y)∀<(2)uiscontinuousat(y,t0)}.Clearly, for every0≤p<m, and p∈ℕk-1 such that 0≤|q|+p≤m, and each i=1,…,K we may define the partial differential operators
(35)𝒟i,ytqp:ℳℒgm(Ω)→ℳℒi,q,p0(Ω),
as in Section 2 through
(36)𝒟i,ytqpu=(I∘S)(Dytqpui).The partial differential operator𝒟i,tm is defined in a similar way, namely, as
(37)𝒟i,tm:ℳℒgm(Ω)∋u↦(I∘S)(Dtmui)∈ℳℒ0(Ω).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 ℳℒgm(Ω), equipped with a suitable uniform convergence structure. In this regard, we introduce the following uniform convergence structure on ℳℒi,q,p0(Ω).Definition 3.
LetΣ consist of all nonempty order intervals in ℳℒi,q,p0(Ω). Let 𝒥i,q,p denote the family of filters on ℳℒi,q,p0(Ω)×ℳℒi,q,p0(Ω) that satisfy the following. There exists k∈ℕ such that
(38)∀j=1,…,k:∃Σj=(Inj)⊆Σ:∃uj∈𝒩ℒ(Ω):(1)In+1j⊆Inj,n∈ℕ,(2)sup{infInj:n∈ℕ}=uj=inf{supInj:n∈ℕ},(3)([Σ1]×[Σ1])∩⋯∩([Σk]×[Σk])⊆𝒰.Proposition 4.
The family of filters𝒥i,q,p on ℳℒi,q,p0(Ω)×ℳℒi,q,p0(Ω) is a Hausdorff uniform convergence structure.
Furthermore, a filterℱ on ℳℒi,q,p0(Ω) converges to u∈ℳℒi,q,p0(Ω) if and only if there exists a family Σℱ=(In) of nonempty order intervals on ℳℒi,q,p0(Ω) such that
(39)(1)In+1⊆In,n∈ℕ,(2)∀V⊆Ωnonemptyandopen:⋂n∈ℕIn∣V={u}∣V
and [Σℱ]⊆ℱ.Proof.
The first four axioms of the definition of a uniform convergence structure [15] are clearly fulfilled, so it remains to verify
(40)∀𝒰,𝒱∈𝒥o:𝒰∘𝒱exists⇒𝒰∘𝒱∈𝒥o.
In this regard, take any𝒰,𝒱∈𝒥o such that 𝒰∘𝒱 exists, and let Σ1,…,Σk and Σ1′,…,Σl′ be the collections of order intervals associated with 𝒰 and 𝒱, respectively, through Definition 3. Set
(41)Φ={(l,j):[Σl]∘[Σj′]exists}.
Then, by [15, Lemma 2.1.1]
(42)𝒰∘𝒱⊇⋂‍{([Σl]×[Σl])∘([Σj]×[Σj]):(l,j)∈Φ}.
Now(l,j)∈Φ if and only if
(43)∀m,n∈ℕ:Iml∩Inj≠∅.
For any (l,j)∈Φ, set Σl,j=(Inl,j) where, for each n∈ℕ(44)Inl,j=[inf(Inl)∧inf(Inj),sup(Inl)∨sup(Inj)].
Now, using (42), we find
(45)𝒰∘𝒱⊇⋂‍{[Σl]×[Σj]:(l,j)∈Φ}⊇⋂‍{[Σl,j]×[Σl,j]:(l,j)∈Φ}.
Sinceℳℒ0(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 proof of Proposition4 relies on the following.Lemma 5.
The setℳℒi,q,p0(Ω) is a lattice with respect to the pointwise order.Proof.
Consider functionsu,v∈ℳℒi,q,p0(Ω), and set w=sup{u,v}∈ℳℒ0(Ω). In view of (13) it follows that w(x)=(I∘S)(φ)(x), x∈Ω where φ(x)=sup{u(x),v(x)}, x∈Ω. Assume that
(46)∃y0∈ℝk,a∈ℝ:w(y0,t0)>a>Dqgp,i(y0).It then follows that S(φ)(y0,t0)>a>Dqgp,i(y0). Therefore
(47)∀δ>0:∃(yδ,tδ)∈Bδ(y0,t0):φ(yδ,tδ)>a>Dqgp,i(y0)
so that we obtain a sequence (yn,tn) in Ω which converges to (y0,t0) and satisfies
(48)∀n∈ℕ:u(yn,tn)>a>Dqgi,p(y0)=u(y0,t0),
or
(49)∀n∈ℕ:v(yn,tn)>a>Dqgi,p(y0)=v(y0,t0).But both u and v are continuous at (y,t0) for each y∈ℝk, which contradicts (48) and (49). Hence (46) cannot hold, so that w(y,t0)=Dqgi,p(y) for each y∈ℝk-1. Furthermore, since both u and v are continuous at (y,t0) for each y∈ℝk-1, it follows that φ is continuous at each of these points. As such, φ is normal lower semicontinuous at each point (y,t0) so that we have
(50)w(y,t0)=(I∘S)(φ)(y,t0)=φ(y,t0)=Dqgi,p(y,t0),y∈ℝk-1.
In the same way, we see thatw is upper semicontinuous at every point (y,t0). Therefore w is continuous at (y,t0) for every y∈ℝk-1 so that w∈ℳℒi,q,p0(Ω).
The existence of the infimum ofu and v follows in the same way.The completion ofℳℒi,q,p0(Ω) may be represented as a suitable space of nearly finite normal lower semicontinuous functions. In particular, consider the space
(51)𝒩ℒi,q,p(Ω)={u∈𝒩ℒ(Ω)∣∃λ,μ∈ℳℒi,q,p0(Ω):λ≤u≤μ}.
Note that ℳℒi,q,p0(Ω)⊂𝒩ℒi,q,p(Ω). As such, in order to show that 𝒩ℒi,q,p(Ω) is the completion of ℳℒi,q,p0(Ω), we must introduce a Hausdorff uniform convergence structure 𝒥i,q,p♯ on 𝒩ℒi,q,p(Ω) in such a way that the following conditions are satisfied.(1)
𝒩ℒi,q,p(Ω) is complete with respect to 𝒥i,q,p♯.(2)
𝒩ℒi,q,p(Ω) contains ℳℒi,q,p0(Ω) as a dense subspace.(3)
IfY is a complete, Hausdorff uniform convergence space, then any uniformly continuous mapping φ:ℳℒi,q,p0(Ω)→Y extends in a unique way to a uniformly continuous mapping φ♯:𝒩ℒi,q,p(Ω)→Y.The appropriate definition of the uniform convergence structure on𝒩ℒi,q,p(Ω) is similar to Definition 2.Definition 6.
Let𝒥i,q,p♯ denote the family of filters on 𝒩ℒi,q,p(Ω)×𝒩ℒi,q,p(Ω) that satisfy the following. There exists k∈ℕ such that
(52)∀j=1,…,k:∃(λnj),(μnj)⊆ℳℒi,q,p0(Ω):∃uj∈𝒩ℒi,q,p(Ω):(1)λnj≤λn+1j≤μn+1j≤μnj,n∈ℕ(2)⋂j=1k‍(([Σj]×[Σj])∩([uj]×[uj]))⊆𝒰,
where each uj∈𝒩ℒi,q,p(Ω) satisfies uj=sup{λnj:n∈ℕ}=inf{μnj:n∈ℕ}, and Σj={Inj:n∈ℕ} with
(53)Inj={u∈ℳℒi,q,p(Ω):λnj≤u≤μnj}.That the family of filters𝒥i,q,p♯ does indeed constitute a Hausdorff uniform convergence structure on 𝒩ℒi,q,p(Ω) can easily be seen. Indeed, 𝒥i,q,p♯ is nothing but the uniform convergence structure associated with the following Hausdorff convergence structure through [15, Proposition 2.1.7]. A filter ℱ on 𝒩ℒi,q,p(Ω) converges to u∈𝒩ℒi,q,p(Ω) if and only if
(54)∃(λn),(μn)⊂ℳℒi,q,p0(Ω):(1)λn≤λn+1≤μn+1≤μn,n∈ℕ,(2)⋂n∈ℕ[λn,μn]|V={u}|V,V⊆Ωopen,(3)[{[λn,μn]:n∈ℕ}]⊆ℱ.Theorem 7.
The space𝒩ℒi,q,p(Ω) equipped with the uniform convergence structure 𝒥i,q,p♯ is the uniform convergence space completion of ℳℒi,q,p0(Ω).Proof.
That𝒩ℒi,q,p(Ω) is complete follows immediately by the above remarks. Furthermore, it is clear that the subspace uniform convergence structure on ℳℒi,q,p0(Ω) is equal to 𝒥i,q,p. To see that ℳℒi,q,p0(Ω) is dense in 𝒩ℒi,q,p(Ω), consider any u∈𝒩ℒi,q,p(Ω). We claim
(55)u=sup𝒜,
where
(56)𝒜={v∈ℳℒi,q,p0(Ω):u≤v}.
Suppose that this were not the case. Then, since𝒩ℒ(Ω) is Dedekind order complete, it follows that there is some u0∈𝒩ℒ(Ω) so that u0=sup𝒜. Since (55) does not hold, it follows that u0 must satisfy u>u0. Furthermore, it is clear from (51) that u0∈𝒩ℒi,q,p(Ω). In view of (10) and the semicontinuity of u and u0 we have
(57)∃W⊆Ωnonemptyandopen:∃ϵ>0:∀(y,t)∈W:u0(y,t)+ϵ<u(y,t).
From (51) it follows that W∩ℝk-1×{t0}=∅. As such, it follows that v+(ϵ/2)χW∈𝒩ℒi,q,p(Ω) for each v∈𝒜 and that
(58)v+ϵ2χW≤u0<u,v∈𝒜.
It now follows that(59)v≤(I∘S)(u0-ϵ2χW)<u0,v∈𝒜,
which is a contradiction. Thus (55) holds. The fact that ℳℒi,q,p0(Ω) is dense in 𝒩ℒi,q,p(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.An important property of the uniform convergence spaceℳℒi,q,p0(Ω) and its completion 𝒩ℒi,q,p(Ω) relates to the inclusion mapping
(60)i:ℳℒi,q,p0(Ω)→ℳℒ0(Ω)
and its extension through uniform continuity
(61)i♯:𝒩ℒi,q,p(Ω)→𝒩ℒ(Ω).Indeed, it is clear form Definitions1 and 3 that the mapping (60) is in fact uniformly continuous. Similarly, the inclusion mapping
(62)i0:𝒩ℒi,q,p(Ω)→𝒩ℒ(Ω)
is uniformly continuous. Since the mappings (61) and (62) coincide on a dense subset of 𝒩ℒi,q,p(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 𝒩ℒm(Ω)K; see [10]. We will discuss this in some detail in what follows, after the uniform convergence structure on ℳℒgm(Ω) has been introduced.In this regard, the uniform convergence structure𝒥g on ℳℒgm(Ω) is defined as the initial uniform convergence structure with respect to the mappings (35) to (37). That is, a filter 𝒰 on ℳℒgm(Ω)×ℳℒgm(Ω) belongs to 𝒥g if and only if
(63)∀i=1,…,K:(𝒟i,tm×𝒟i,tm)(𝒰)∈𝒥o,and(64)∀0≤p<m:∀q∈ℕk-1,0<|q|+p≤m:∀i=1,…,K:(𝒟i,ytqp×𝒟i,ytqp)(𝒰×𝒰)∈𝒥i,q,p.Clearly the family consisting of the mappings (35) through (37) separates the points of ℳℒgm(Ω). As such, the uniform convergence structure 𝒥g is uniformly Hausdorff. In particular, see [14], the mapping
(65)D:ℳℒgm(Ω)→(∏‍ℳℒi,q,p0(Ω))×ℳℒ0(Ω)K
which is defined through
(66)D(u)=(…,𝒟i,ytqpu,…,𝒟i,tmu,…)
is a uniformly continuous embedding. As such, it follows that the mapping (66) extends to an injective, uniformly continuous mapping
(67)D♯:𝒩ℒgm(Ω)→(∏‍𝒩ℒi,q,p(Ω))×𝒩ℒ(Ω)K,
where 𝒩ℒgm(Ω) denotes the uniform convergence space completion of ℳℒgm(Ω). In particular, for each i=1,…,K, every 0≤p<m, and each q∈ℕk-1 such that 0≤p+|q|≤m the diagrams
(68)
commute, with πi,q,p and πi the appropriate projections and 𝒟i,ytqp♯ and 𝒟i,tm♯ the extensions through uniform continuity of the mappings (35) and (37), respectively.The meaning of the diagram (68) is twofold. Firstly, it explains the regularity of generalized functions in 𝒩ℒgm(Ω). In particular, each generalized partial derivative of a generalized function u♯∈𝒩ℒgm(Ω) is a nearly finite normal lower semicontinuous function. Therefore, each such generalized function may be represented as an element of the space (∏‍𝒩ℒi,p(Ω))×𝒩ℒ(Ω)K in a canonical way. Secondly, these diagrams state that each generalized function u♯∈𝒩ℒgm(Ω) satisfies the initial condition (30) in the sense that
(69)∀i=1,…,K:∀0≤p<m:∀q∈ℕk-1,0≤p+|q|≤m:𝒟i,ytqp♯u♯(t0,y)=Dqgp,i(t0,y),y∈ℝk-1.


## 4.  Existence of Generalized Solutions

With the system of nonlinear PDEs (29) we may associate a mapping
(70)T:ℳℒgm(Ω)→ℳℒ0(Ω)K,
the components of which are defined through
(71)Tj:ℳℒgm(Ω)∋u↦(I∘S)(𝒟j,tmu+Gj(·,·,…,𝒟i,ytpqu,…))∈ℳℒ0(Ω).
We arrive at the notion of generalized solution of the initial value problem (29) and (30) in the context of the space  𝒩ℒgm(Ω) by suitably extending the mapping (70) to a mapping
(72)T♯:𝒩ℒgm(Ω)→𝒩ℒ(Ω)K.
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
(73)i:ℳℒgm(Ω)→ℳℒm(Ω)K
is uniformly continuous. The result now follows from the commutative diagram
(74)
and the uniform continuity of T0, which is the mapping defined on ℳℒm(Ω)K through the nonlinear partial differential operator; see [10].In view of Theorem8 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
(75)T♯u♯=0,
where 0 denotes the element in 𝒩ℒ(Ω)K 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
(76)
with i♯ injective and T0♯ the uniformly continuous extension of the mapping
(77)T0:ℳℒm(Ω)K→ℳℒ0(Ω)K
associated with the system of nonlinear PDEs (29). In particular, the mapping i♯ is the inclusion mapping. As such, each solution u♯∈𝒩ℒgm(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.Theorem 9.
For each0≤p<m, let gp∈𝒞m-p(ℝk-1)K. Then there is some u♯∈𝒩ℒgm(Ω) so that
(78)T♯u♯=0.Proof.
Let us expressΩ=ℝk-1×ℝ as
(79)Ω=⋃ν∈ℕ‍Cν,
where, for ν∈ℕ, the compact set Cν is an n-dimensional interval
(80)Cν=[aν,bν]
with aν=(aν,1,…,aν,n), bν=(bν,1,…,bν,n)∈ℝk, and aν,j≤bν,j for every j=1,…,n. We assume that {Cν:ν∈ℕ} is locally finite, that is:
(81)∀x∈Ω:∃V⊆Ωaneighborhoodofx:{ν∈ℕ:Cν∩V≠∅}isfinite.
Such a partition ofΩ exists; see, for instance [16]. We also assume that, for each ν∈ℕ,
(82)𝒮∩Cν=∅,
or
(83)𝒮∩IntCν≠∅,
where 𝒮 is the noncharacteristic hypersurface 𝒮={(y,t0):y∈ℝk-1}. For the sake of convenience, let us write x=(y,t) for each (y,t)∈ℝk-1×ℝ. Let F:Ω×ℝM→ℝK be the mapping that defines the nonlinear operator T through T(x,D)u(x)=F(x,…,Dαui(x),…). Fix ν∈ℕ such that (83) is satisfied. In view of the fact that the mapping F is both open and surjective, we have
(84)∀x1=(y1,t1)∈Cν:∃ξ(x1)∈ℝM,F(x1,ξ(x1))=0:∃δ,ϵ>0:(1){(x,0):∥x-x1∥<δ}(1)⊂int{(x,F(x,ξ))|∥x-x1∥<δ∥ξ-ξ(x1)∥<ϵ},(2)F:Bδ(x1)×B2ϵ(ξ(x1))→ℝKopen.
In particular, if t1=t0, we may take ξ(x1)=(ξiq,p,ξim) such that
(85)∀i=1,…,K:∀0≤p<m:∀q∈ℕk-1,0<|q|+p≤m:ξiq,p=Dqgp,i(y1).
For eachx1∈Cν, fix ξ(x1)∈ℝM in (84) so that (85) is satisfied in case t1=t0. Since Cν is compact, it follows from (84) that
(86)∃δ>0:∀x1∈Cν:∃ϵx1>0:(1){(x,0):∥x-x1∥<δ}(1)⊂int{(x,F(x,ξ))|∥x-x1∥<δ∥ξ-ξ(x1)∥<ϵx1},(2)F:Bδ(x1)×B2ϵx0(ξ(x1))→ℝKopen.
Subdivide Cν into n-dimensional intervals Iν,1,…,Iν,μν with diameter not exceeding δ such that their interiors are pairwise disjoint and, for each j=1,…,μν,
(87)Iν,j∩𝒮=∅,
or
(88)intIν,j∩𝒮≠∅.
Ifaν,j with j=1,…,μν is the center of the interval Iν,j that satisfies (87), then by (86) we have
(89)∃ϵν,j>0:(1){(x,0):x∈Iν,j}(1)⊂int{(x,F(x,ξ)){x∈Iν,j∥ξ-ξ(aν,j)∥<ϵν,j}|x∈Iν,j∥ξ-ξ(aν,j)∥<ϵν,j},(2)F:Iν,j×B2ϵν,j(ξ(aν,j))→ℝKopen.
On the other hand, if Iν,j satisfies (88), set aν,j equal to the midpoint of 𝒮∩Iν,j. Then we obtain (89) by (86) such that (85) also holds. Take 0<γ<1 arbitrary but fixed. In view of [9, Lemma 5] and (89), we have
(90)∀x1∈Iν,j:∃Ux1=U∈𝒞m(ℝk)K:∃δ=δx1>0:x∈Bδ(x1)∩Iν,j⇒((1)(DαUi(x))i≤K|α|≤m∈Bϵν,j(ξ(aν,j))(2)i≤K⇒γ<Ti(x,D)U(x)<0),
with α=(q,p). Furthermore, if Iν,j satisfies (88), then we also have
(91)∀i=1,…,K:∀0≤p<m:∀q∈ℕk-1,0<|q|+p≤m:∀y∈ℝk-1:DytpqUi(y,t0)=Dqgp,i(y).
Indeed, in this case we may simply set(92)Ui(y,t)=∑p=0m-1‍(t-t0)pgp,i(y)+wi(t)
for a suitable function wi∈𝒞m(ℝ) that satisfies
(93)∀0≤p<m:wi(p)(t0)=0.
As above, we may subdivideIν,j into pairwise disjoint,  n-dimensional intervals Jν,j,1,…,Jν,j,μν,j so that for k=1,…,μν,j we have
(94)∃Uν,j,k=U∈𝒞m(ℝk)K:∀x∈Jν,j,k:(1)(DαUi(x)i≤K|α|≤m)∈Bϵν,j(ξ(aν,j)),|α|≤m,(2)i≤K⇒fi(x)-γ<Ti(x,D)U(x)<fi(x),
and
(95)Jν,j,k∩𝒮=∅,
or
(96)intIν,j,k∩𝒮≠∅.
Furthermore, whenever Jν,j,k satisfies (96), we have
(97)∀i=1,…,K:∀0≤p<m:∀q∈ℕk-1,0<|q|+p≤m:∀y∈ℝk-1:DytqpUi(y,t0)=Dqgp,i(y).
Set
(98)Γ1=Ω∖(⋃ν∈ℕ‍(⋃j=1μν‍(⋃k=1μν,j‍intJν,j,k))),V1=∑ν∈ℕ‍(∑j=1μν‍(∑k=1μν,j‍χJν,j,kUν,j,k)),
where χJν,j,k is the characteristic function of Jν,j,k. Then Γ1 is closed nowhere dense, and V1∈𝒞m(Ω∖Γ1)K. Furthermore, 𝒮∩Γ1 is closed nowhere dense in 𝒮 and
(99)∀i=1,…,K:∀0≤p<m:∀q∈ℕk-1,0<|q|+p≤m:∀(y,t0)∈𝒮∖(𝒮∩Γ1):DytqpV1,i(y,t0)=Dqgp,i(y).
In view of (94) we have, for each i=1,…,K(100)-γ<Ti(x,D)V1(x)<0,x∈Ω∖Γ1.
Furthermore, for eachν∈ℕ, for each j=1,…,μν, each k=1,…,μν,j, each |α|≤m, and every i=1,…,K we have
(101)x∈intJν,j,k⇒ξiα(aν,j)-ϵ<DαV1,i(x)<ξiα(aν,j)+ϵ.
For 0≤p<m, define the functions λ1,iα,μ1,iα∈𝒞0(Ω∖Γ1), where α=(p,q) with |q|=0, as
(102)λ1,iα(x)={ξiα(aν,j)-2ϵν,jifx∈intIν,j,k,Iν,j,k∩𝒮=∅DtpV1,i(y,t)-vν,j(t)ifx∈intIν,j,k,Iν,j,k∩𝒮≠∅,μ1,iα(x)={ξiα(aν,j)+2ϵν,jifx∈intIν,j,k,Iν,j,k∩𝒮=∅DtpV1,i(y,t)+vν,j(t)ifx∈intIν,j,k,Iν,j,k∩𝒮≠∅.
Herevν,j is a continuous, real valued function on ℝ such that
(103)vν,j(t0)=0,0<vν,j(t)<2ϵν,j,t≠0.
For all other α, consider the functions
(104)λ1,iα(x)=ξiα(aν,j)-2ϵν,jifx∈intIν,j,μ1,iα(x)=ξiα(aν,j)+2ϵν,jifx∈intIν,j.
Then it follows by (101) that
(105)λ1,iα(x)<DαV1,i(x)<μ1,iα(x),x∈Ω∖Γ1,μ1,iα(x)-λ1,iα(x)<4ϵν,j,x∈intIν,j.
Applying (89) restricted to Ω∖Γ1, and proceeding in a fashion similar as above, we may construct, for each n∈ℕ such that n>1, a closed nowhere dense set Γn⊂Ω such that
(106)Γn∩𝒮closednowheredensein𝒮,
a function Vn∈𝒞m(Ω∖Γn)K and functions λn,iα,μn,iα∈𝒞0(Ω∖Γn) so that, for each i=1,…,K,
(107)-γn<Ti(x,D)Vn(x)<0,x∈Ω∖Γn
and for every |α|≤m(108)λn-1,iα(x)<λn,iα(x)<DαVn,i(x)<μn,iα(x)<μn-1,iα(x),x∈Ω∖Γn,(109)μn,iα(x)-λn,iα(x)<4ϵν,jn,x∈(intIν,j)∩(Ω∖Γn).
Furthermore, for each0≤p<m and q∈ℕk-1 so that 0≤|q|+p≤m we have
(110)DytqpVn,i(y,t0)=λn,iα(y,t0)=μn,iα(y,t0)=Dqgp,i(y),(y,t0)∉𝒮∩Γn,
where α=(p,q).
Notice that the functionsun, the components of which are defined through
(111)un,i=(I∘S)(Vn,i),
belong to ℳℒgm(Ω). In view of (108) it follows that the functions λ¯n,iα,μ¯n,iα∈ℳℒ0(Ω), which are defined as
(112)λ¯n,iα=(I∘S)(λn,iα),μ¯n,iα=(I∘S)(μn,iα),
satisfy
(113)λ¯n-1,iα≤λ¯n,iα≤𝒟αun,i≤μ¯n,iα≤μ¯n-1,iα.
Furthermore, in caseα=(p,q) with 0≤p<m and q∈ℕk-1 such that 0≤p+|q|≤m, then λ¯n,iα,μ¯n,iα∈ℳℒi,q,p0(Ω). It now follows from (109) that the sequence (un) is a Cauchy sequence in ℳℒgm(Ω). Moreover, (107) implies that the sequence (Tun) converges to 0 in ℳℒ0(Ω)K. The result now follows from Theorem 8.We have shown that the initial value problem (29) through (30) admits a generalized solution in the space 𝒩ℒgm(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
(114)∀0≤p<m:∀q∈ℕk-1,0≤|q|+p≤m:∀y∈ℝk-1:𝒟yt,iqp♯u♯(y,t0)=Dqgp,i(y).
Furthermore, it follows by (9) that the singularity set
(115){(y,t)∈Ω{∃∃𝒟iα#}|∃|α|≤m:∃i=1,…,K:𝒟iα♯unotcontinuousat(y,t)}
of the solution u♯ is of first Baire category. In particular, there exists a residual subset R of Ω such that each generalized partial derivative of u♯ is continuous at every point of R.As mentioned in Section1, it is known [3] that if the mapping G:Ω×ℝM→ℝK 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∈ℳℒgm(Ω) and satisfies
(116)Tu=0.
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.


## 5. 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.



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