Solutions of Smooth Nonlinear Partial Differential Equations

Abstract

The method of order completion provides a general and type-independent theory for the existence and basic regularity of the solutions of large classes of systems of nonlinear partial differential equations (PDEs). Recently, the application of convergence spaces to this theory resulted in a significant improvement upon the regularity of the solutions and provided new insight into the structure of solutions. In this paper, we show how this method may be adapted so as to allow for the infinite differentiability of generalized functions. Moreover, it is shown that a large class of smooth nonlinear PDEs admit generalized solutions in the space constructed here. As an indication of how the general theory can be applied to particular nonlinear equations, we construct generalized solutions of the parametrically driven, damped nonlinear Schrödinger equation in one spatial dimension.

1. Introduction

In the 1994 monograph [1] Oberguggenberger and Rosinger presented a general and type-independent theory for the existence and basic regularity of the solutions of a large class of systems of nonlinear PDEs, based on the Dedekind order completion of spaces of piecewise smooth functions. In the mentioned monograph, it is shown that the solutions satisfy a blanket regularity property. Namely, the solutions may be assimilated with usual real measurable functions, or even nearly finite Hausdorff continuous functions [2], defined on the Euclidean domain of definition the respective system of equations. The latter result is based on the highly nontrivial characterization of the Dedekind order completion of sets of continuous functions in terms of spaces of Hausdorff continuous interval valued functions [3]. Recently, the regularity of the solutions constructed through the order completion method has been significantly improved upon by introducing suitable uniform convergence structures on appropriate spaces of piecewise smooth functions; see [4–7]. This new approach also gives new insight into the structure of the solutions obtained through the original order completion method [1].

The generality and type independence of the solution method introduced in [1, 4–6] has to date not been obtained in any of the usual theories of generalized solutions of linear and nonlinear PDEs. Indeed, and perhaps as a result of the insufficiency of the spaces of generalized functions that are typical in the study of generalized solutions of PDEs, at least from the point of view of the existence of solutions of PDEs, it is often believed that such a general theory is not possible, see for instance [8, 9]. Within the setting of the linear topological spaces of generalized functions that form the basis for most studies of PDEs, this may perhaps turn out to be the case. As a clarification and motivation of the above remarks, the following general comments may be of interest.

For over 135 years by now, there has been a general and type-independent existence and regularity result for the solutions of systems of analytic nonlinear PDEs. Indeed, in 1875 Kovalevskaia [10], upon the suggestion of Weierstrass, gave a rigorous proof of an earlier result of Cauchy, published in 1821 in his Course d'Analyse. This result, although restricted to the realm of analytic PDEs, is completely general as far as the type of nonlinearities involved are concerned. The analytic solutions of such a systems of PDEs can, however, be guaranteed to exists only on a neighborhood of the noncharacteristic analytic hypersurface on which the analytic initial data is specified. The nonexistence of solutions of a system of analytic PDEs on the whole domain of definition of the respective system of equations is not due to the particular techniques used in the proof of the result, but may rather be attributed to the very nature of nonlinear PDEs. Indeed, rather simple examples, such as the nonlinear conservation law𝑈_𝑡+𝑈_𝑥𝑈=0,𝑡>0,𝑥∈ℝ(1.1) with the initial condition𝑈(0,𝑥)=𝑢(𝑥),𝑥∈ℝ(1.2) show that, irrespective of the smoothness of the initial data (1.2), the solution of the initial value problem may fail to exist on the whole domain of definition [0,∞)×ℝ of the equation, see for instance [11, 12]. Furthermore, in the case of the nonlinear conservation law (1.1), it is exactly the points where the solution fails to exists that are of interest, since these may represent the formation and propagation of shock waves, as well as other chaotic phenomena such as turbulence.

In view of the above remarks, it is clear that any general and type-independent theory for the existence and regularity of the solutions of nonlinear PDEs must alow for sufficiently singular objects to act as generalized solutions of such equations. In particular, the solutions may fail to be continuous, let alone sufficiently smooth, on the whole domain of definition of respective system of equations. In many cases, it happens that the spaces of generalized functions that are used in the study of PDEs do not admit such sufficiently singular objects. Indeed, we may recall that, due to the well-known Sobolev Embedding theorem, see for instance [13], the Sobolev Space 𝐻^𝑘(ℝ^𝑛) will, for sufficiently large values of 𝑘, contain only continuous functions.

Moreover, even in case a given system of PDEs admits a solution which is classical, indeed even analytic, everywhere except at a single point of its Euclidean domain of definition, it may happen that such a solution does not belong to any of the customary spaces of generalized functions. For example, given a function𝑢∶ℂ⧵𝑧₀⟶ℂ(1.3) which is analytic everywhere except at the single point 𝑧₀∈ℂ, and with an essential singularity at 𝑧₀, Picard's Theorem states that 𝑢 attains every complex value, with possibly one exception, in every neighborhood of 𝑧₀. Clearly such a function does not satisfy any of the usual growth conditions that are, rather as a rule, imposed on generalized functions. Indeed, we may recall that the elements of a Sobolev space are locally integrable, while the elements of the Colombeau algebras [14], which contain the 𝒟^ distributions, must satisfy certain polynomial type growth conditions near singularities. Therefore these concepts of generalized functions cannot accommodate the mentioned singularity of the function in (1.3).

In this paper, we present further developments of the general and type-independent solution method presented in [1], and in particular the uniform convergence spaces of generalized functions introduced in [4–6]. Furthermore, and in contradistinction with the spaces of generalized functions introduced in [6], we construct here a space of generalized functions that admit generalized partial derivatives of arbitrary order. While, following the methods introduced in [6], one may easily construct such a space of generalized functions, the existence of generalized solutions of systems of nonlinear PDEs in this space is nontrivial. Here we present the mentioned construction of the space of generalized functions, and show how generalized solutions of a large class of 𝒞^∞-smooth nonlinear PDEs may be obtained in this space.

As an application of the general theory, we discuss also the existence and regularity of generalized solutions of the parametrically driven, damped nonlinear Schrödinger equation in one spatial dimension. In this regard, we show that for a large class of 𝒞^∞-smooth initial values, the mentioned Schrödinger equation admits a generalized solution that satisfies the initial condition in a suitable generalized sense. We also introduce the concept of a strongly generic weak solution of this equation, and show that the solution we construct is such a weak solution.

The paper is organized as follows. In Section 2 we recall some basic facts concerning normal lower semicontinuous functions from the literature. The construction of spaces of generalized functions is given in Section 3, while Section 4 is concerned with the existence of generalized solutions of 𝒞^∞-smooth nonlinear PDEs. Lastly, in Section 5, we apply the general method to the parametrically driven, damped nonlinear Schrödinger equation in one spatial dimension. For all details on convergence spaces we refer the reader to the excellent book [15] and the paper [16].

2. Normal Lower Semicontinuous Functions

The concept of a normal lower semicontinuous function was first introduced by Dilworth [17] in connection with his attempts at characterizing the Dedekind order completion of spaces of continuous functions, a problem that was solved only recently by Anguelov [3]. Here we recall some facts concerning normal semicontinuous functions. For more details, and the proofs of some of the results, we refer the reader to the more recent presentations in [4, 18].

Denote by ℝ=ℝ∪{±∞} the extended real line, ordered as usual. The set of all extended real-valued functions on a topological space 𝑋 is denoted 𝒜(𝑋). A function 𝑢∈𝒜(𝑋) is said to be nearly finite whenever{𝑥∈𝑋∶𝑢(𝑥)∈ℝ}isopenanddense.(2.1) Two fundamental operations on the space 𝒜(𝑋) are the Lower and Upper Baire Operators𝑆𝐼∶∶𝒜𝒜((𝑋𝑋))⟶⟶𝒜𝒜((𝑋𝑋)),,(2.2) introduced by Baire [19], see also [3], which are defined by𝑆𝐼((𝑢𝑢))∶∶𝑋𝑋∋∋𝑥𝑥⟼⟼sinufpsinufp{{𝑢𝑢((𝑦𝑦))∶∶𝑦𝑦∈∈𝑉𝑉}}∶∶𝑉𝑉∈∈𝒱𝒱_𝑥𝑥,(2.3),(2.4) respectively, with 𝒱_𝑥 denoting the neighborhood filter at 𝑥∈𝑋. Clearly, the Baire operators 𝐼 and 𝑆 satisfy𝐼(𝑢)≤𝑢≤𝑆(𝑢),𝑢∈𝒜(𝑋),(2.5) when 𝒜(𝑋) is equipped with the usual pointwise order𝑢≤𝑣⟺(∀𝑥∈𝑋∶𝑢(𝑥)≤𝑣(𝑥)).(2.6) Furthermore, the Baire operators, as well as their compositions, are idempotent and monotone with respect to the pointwise order. That is,∀∀𝑢𝑢𝑢(((,≤123∈𝑣)))𝑣∈𝒜𝐼𝑆(𝐼⟹𝒜(((Ω∘𝐼𝑆(𝑆)(Ω(𝑢)∶⎛⎜⎜⎜⎝𝑢))())(((()∶312𝐼==)))∘𝐼𝐼𝑆(𝑆𝑆𝐼(()𝑢((∘𝑢𝑢(𝑢))𝑢𝑆)),)≤,≤))(=𝐼𝑢𝑆)(((𝑣≤𝐼𝑣))∘(𝑆𝐼)∘(𝑆𝑢)),(𝑣)⎞⎟⎟⎟⎠.(2.7)

The operators 𝐼 and 𝑆, as well as their compositions 𝐼∘𝑆 and 𝑆∘𝐼, are useful tools for the study of (extended) real-valued functions. In this regard, we may mention that these mappings characterize certain continuity properties of functions in 𝒜(𝑋). In particular, we have𝑢𝑢∈∈𝒜𝒜((𝑋𝑋))iissulopwpeerrsseemmiiccoonnttiinnuuoouuss⟺⟺𝑆𝐼((𝑢𝑢))==𝑢𝑢,.(2.8) Furthermore, a function 𝑢∈𝒜(𝑋) is normal lower semicontinuous on 𝑋 whenever(𝐼∘𝑆)(𝑢)=𝑢.(2.9)

We denote the set of nearly finite normal lower semicontinuous functions on 𝑋 by 𝒩ℒ(𝑋). The concept of normal lower semicontinuity of extended real-valued functions extends that of continuity of usual real-valued functions. In particular, each continuous function is nearly finite and normal lower semicontinuous so that we have the inclusion𝒞(𝑋)⊆𝒩ℒ(𝑋).(2.10) Conversely, a normal lower semicontinuous function is generically continuous in the sense that∀∃𝑥𝑢𝐵∈∈⊆𝒩𝑋𝑋ℒ⧵o𝐵f(𝑋fi⟹r)st∶B𝑢aciroentciantueoguosryat∶𝑥.(2.11) In particular, in case 𝑋 is a Baire space, it follows that each 𝑢∈𝒩ℒ(𝑋) is continuous on some residual set, which is dense in 𝑋. Furthermore, the following well-known property of continuous functions holds also for normal lower semicontinuous functions. Namely, we have ∀𝑢(,∀𝑣𝑥∈∈𝒩𝐷ℒ∶(𝑢𝑋()𝑥,)≤𝐷𝑣⊆(𝑥)𝑋)⟹dens𝑢e≤∶𝑣.(2.12)

With respect to the usual pointwise order, the space 𝒩ℒ(𝑋) is a fully distributive lattice. That is, suprema and infima of finite sets always exists and∀𝑢𝐴⊂𝒩ℒ(𝑋),𝑣∈𝒩ℒ(𝑋)∶₀=sup𝐴⟹inf𝑣,𝑢₀=sup{inf{𝑢,𝑣}∶𝑢∈𝐴}.(2.13) Furthermore, 𝒩ℒ(𝑋) is Dedekind order complete. That is, every set which is order bounded from above, respectively, below, has a least upper bound, respectively, greatest lower bound. In particular, the supremum and infimum of a set 𝐴⊂𝒩ℒ(𝑋), is given bysinufp𝐴𝐴==((𝐼𝐼∘∘𝑆𝑆))((𝜓𝜑)),,(2.14) respectively, with 𝜑 and 𝜓 given by𝜑𝜓∶∶𝑋𝑋∋∋𝑥𝑥⟼⟼siunpf{{𝑢𝑢((𝑥𝑥))∶∶𝑢𝑢∈∈𝐴𝐴}}∈∈ℝ,(2.15)ℝ.(2.16) A useful characterization of order bounded sets in 𝒩ℒ(𝑋) is the following: If 𝑋 is a Baire space, then for any set 𝐴⊂𝒩ℒ(𝑋) we have ∃𝑢𝑢₀≤∈𝑢𝒩₀ℒ(𝑋)∶,𝑢∈𝐴(2.17) if and only if∃∀𝑅𝑥su∈⊆p𝑅𝑋{𝑢∶a(𝑥re)s∶id𝑢ua∈ls𝐴et}∶<∞.(2.18) Indeed, suppose that a set 𝐴⊂𝒩ℒ(𝑋) satisfies (2.18). Then the function 𝜑 associated with 𝐴 through (2.15) satisfies𝜑(𝑥)<∞,𝑥∈𝑅.(2.19) The function 𝑢₀=(𝐼∘𝑆)(𝜑) is normal lower semicontinuous and satisfies𝑢≤𝑢₀,𝑢∈𝐴.(2.20) It is sufficient to show that 𝑢₀ is finite on a dense subset of 𝑋. To see that 𝑢₀ satisfies this condition, assume the opposite. That is, we assume ∃∀𝑢𝑉𝑥∈⊆𝑉𝑋∶₀nonemptyandopen∶(𝑥)=∞.(2.21) It follows by (2.5) that𝑆(𝜑)(𝑥)=∞,𝑥∈𝑉.(2.22) Thus (2.4) implies that∀∃𝑥𝑥_𝑀∈𝑉,𝑊∈𝒱_𝑥,𝑀>0∶𝜑𝑥∈𝑉∩𝑊∶_𝑀>𝑀.(2.23) Note that, since each 𝑢∈𝐴 is lower semicontinuous, the function 𝜑 is also lower semicontinuous. Therefore (2.23) implies∀∃𝑀𝐷_𝑀>0∶𝜑(𝑥⊆)>𝑉𝑀op,en𝑥a∈nd𝐷d_𝑀ensein𝑉∶.(2.24) There is therefore a residual set 𝑅^⊆𝑉 such that𝜑(𝑥)=∞,𝑥∈𝑅^.(2.25) Since 𝑋 is a Baire space, so is the open set 𝑉 in the subspace topology. Furthermore, 𝑅∩𝑉 is a residual set in 𝑉. But 𝑅∩𝑉⊆𝑉⧵𝑅^ so that 𝑅∩𝑉 must be of first Baire category in 𝑉, which is a contradiction. Therefore 𝑢₀ is finite on a dense subset of 𝑋. The dual statement for sets bounded from below also holds.

A subspace of 𝒩ℒ(𝑋) which is of particular interest to us here is the space ℳℒ(𝑋) which consists of all functions in 𝒩ℒ(𝑋) which are real-valued and continuous on an open and dense subset of 𝑋. That is,ℳℒ(𝑋)={𝑢∈𝒩ℒ(𝑋)∣∃Γ⊂𝑋closednowheredense∶𝑢∈𝒞(𝑋⧵Γ)}.(2.26) The space ℳℒ(𝑋) is a sublattice of 𝒩ℒ(𝑋). As such, it is also fully distributive. Furthermore, whenever 𝑋 is a metric space, the following order denseness property is satisfied:∀𝑢∈𝒩ℒ(𝑋)∶sup{𝑣∈ℳℒ(𝑋)∶𝑣≤𝑢}=𝑢=inf{𝑤∈ℳℒ(𝑋)∶𝑢≤𝑤}.(2.27) Moreover, the following sequential version of (2.27) holds:∃∀𝑢𝜆∈𝒩ℒ(𝑋)∶_𝑛(1)𝜆,_𝑛𝜇_𝑛⊂ℳℒ(𝑋)∶(2)sup≤𝜆𝜆_𝑛+1𝑛≤𝑢≤𝜇_𝑛+1≤𝜇_𝑛,𝑛∈ℕ,∶𝑛∈ℕ=𝑢=inf𝜇_𝑛∶𝑛∈ℕ.(2.28) We may note that the space ℳℒ(𝑋), and suitable subspaces of it, also play an important role in the theory of rings of continuous functions [20], and the theory of differential algebras of generalized functions [21] where such spaces arise in connection with the so-called closed nowhere dense ideals. In the next section we construct a space of generalized functions as the completion of a suitable uniform convergence space, the elements of which are functions in ℳℒ(𝑋), when 𝑋 is an open subset of Euclidean 𝑛-space ℝ^𝑛.

3. Spaces of Generalized Functions

We now consider the construction of spaces of generalized functions based on the spaces of normal lower semicontinuous functions discussed in Section 2. This follows closely the method used in [6], with the exception that we consider here the case of infinitely differentiable functions, this being the main topic of the current investigation.

In this regard, let Ω be an open, nonempty and possibly unbounded subset of ℝ^𝑛. For 𝑚∈ℕ∪{∞} we denote by ℳℒ^𝑚(Ω) the set of those functions in 𝒩ℒ(Ω) that are 𝒞^𝑚-smooth everywhere except on some closed nowhere dense set Γ⊂Ω. That is,ℳℒ^𝑚(Ω)=𝑢∈𝒩ℒ(Ω)∣∃Γ⊂Ωclosednowheredense∶𝑢∈𝒞^𝑚(Ω⧵Γ).(3.1) One should note that, while the singularity set Γ associated with a function 𝑢∈ℳℒ^𝑚(Ω) through (3.1) is a topologically small set, it may be large in the sense of measure [22]. That is, the set Γ may have arbitrarily large positive Lebesgue measure. Furthermore, a function 𝑢∈ℳℒ^𝑚(Ω) typically does not satisfy any of the usual growth conditions that are imposed on generalized functions. In particular, 𝑢 is, in general, not locally integrable on any neighborhood of any point 𝑥∈Γ. Moreover, 𝑢 will typically not satisfy any of the polynomial type growth conditions that are imposed on elements of the Colombeau algebras of generalized functions [14].

For 𝑚=0, the space (3.1) reduces to ℳℒ⁰(Ω)=ℳℒ(Ω), as defined in (2.26). For 𝑚=∞, the usual partial differential operators𝐷^𝛼∶𝒞^∞(Ω)⟶𝒞^∞(Ω)⊂𝒞⁰(Ω),𝛼∈ℕ^𝑛(3.2) extend in a straight forward way to mappings𝒟^𝛼∶ℳℒ^∞(Ω)⟶ℳℒ^∞(Ω)⊆ℳℒ⁰(Ω),𝛼∈ℕ^𝑛(3.3) which are defined through𝒟^𝛼∶ℳℒ^∞(Ω)∋𝑢⟼(𝐼∘𝑆)(𝐷^𝛼𝑢)∈ℳℒ^∞(Ω),𝛼∈ℕ^𝑛.(3.4)

The space of generalized functions we consider here are constructed as the completion of the space ℳℒ^∞(Ω) equipped with a suitable uniform convergence structure. In this regard, see [4], we consider on ℳℒ⁰(Ω) the uniform order convergence structure.

Definition 3.1. Let Σ consist of all nonempty order intervals in ℳℒ⁰(Ω). The family 𝒥_𝑜 of filters on ℳℒ⁰(Ω)×ℳℒ⁰(Ω) consists of all filters that satisfy the following: there exists 𝑘∈ℕ such that ∀∃𝑖Σ=_𝑖1,…,𝑘∶(1=)𝐼𝐼^𝑖𝑛_𝑖𝑛+1⊆Σ,𝑢_𝑖∈𝒩ℒ(Ω)∶(2)sup⊆in𝐼f^𝑖𝑛𝐼,_𝑖𝑛𝑛∈ℕ,(3)Σ₁∶𝑛∈ℕ=𝑢_𝑖=infsup𝐼^𝑖𝑛∶𝑛∈ℕ,×Σ₁∩⋯∩Σ_𝑘×Σ_𝑘⊆𝒰.(3.5)

The family of filters 𝒥_𝑜 is a uniformly Hausdorff and first countable uniform convergence structure. Furthermore, the induced convergence structure is the order convergence structure [23]. That is, a filter ℱ on ℳℒ⁰(Ω) converges to 𝑢∈ℳℒ⁰(Ω) if and only if∃(𝜆1_𝑛)𝜆,_𝑛𝜇_𝑛⊂ℳℒ⁰(Ω)∶(2)sup≤𝜆𝜆_𝑛+1𝑛≤𝑢≤𝜇_𝑛+1≤𝜇_𝑛,𝑛∈ℕ,(3)𝜆_𝑛∶𝑛∈ℕ=𝑢=inf𝜇_𝑛∶𝑛∈ℕ,,𝜇_𝑛∈ℱ,𝑛∈ℕ.(3.6)

The completion of the uniform convergence space ℳℒ⁰(Ω) may be characterized in terms of the space 𝒩ℒ(Ω) equipped with a suitable uniform convergence structure 𝒥^♯𝑜, see [4]. In particular, this means that 𝒩ℒ(Ω) is complete and contains ℳℒ⁰(Ω) as a dense subspace. Furthermore, given any complete Hausdorff uniform convergence space 𝑌, and any uniformly continuous mappingΨ∶ℳℒ⁰(Ω)⟶𝑌,(3.7) there exists a unique uniformly continuous mappingΨ^♯∶𝒩ℒ(Ω)⟶𝑌(3.8) which extends Ψ.

The space ℳℒ^∞(Ω) is equipped with the initial uniform convergence structure with respect to the family of mappings (3.3). That is,𝒰∈𝒥_𝐷⟺∀𝛼∈ℕ^𝑛∶𝒟^𝛼×𝒟^𝛼(𝒰)∈𝒥_𝑜.(3.9) Each of the mappings (3.3) is uniformly continuous with respect to the uniform convergence structures 𝒥_𝐷 and 𝒥_𝑜 on ℳℒ^∞(Ω) and ℳℒ⁰(Ω), respectively. In fact, (3.9) is the finest uniform convergence structure on ℳℒ^∞(Ω) making the mappings (3.3) uniformly continuous.

Since the family of mappings (3.3) is countable and separates the points of ℳℒ^∞(Ω), that is,∃∀𝛼𝑢,∈𝑣∈ℕ_𝑛ℳℒ^∞(Ω)∶𝒟_𝛼∶𝑢≠𝒟^𝛼𝑣,(3.10) it follows from the corresponding properties of ℳℒ⁰(Ω) that the uniform convergence structure 𝒥_𝐷 is uniformly Hausdorff and first countable. Furthermore, a filter ℱ on ℳℒ^∞(Ω) converges to 𝑢∈ℳℒ^∞(Ω) with respect to the convergence structure 𝜆_𝐷 induced by 𝒥_𝐷 if and only if∀𝒟𝛼_𝛼∈ℕ^𝑛∶(ℱ)convergesto𝒟^𝛼𝑢inℳℒ⁰(Ω).(3.11)

The completion of ℳℒ^∞(Ω), which we denote by 𝒩ℒ^∞(Ω), is related to the completion 𝒩ℒ(Ω) of ℳℒ⁰(Ω) in the following way. Since ℳℒ^∞(Ω) carries the initial uniform convergence structure with respect to the family of mappings (3.3), it follows [16] that the mapping𝐃∶ℳℒ^∞(Ω)∋𝑢⟼𝒟^𝛼𝑢_{𝛼∈ℕ𝑛}∈ℳℒ⁰(Ω)^ℕ𝑛(3.12) is a uniformly continuous embedding, with ℳℒ⁰(Ω)^ℕ𝑛 equipped with the product uniform convergence structure with respect to 𝒥_𝑜. In particular, the diagram (3.13) commutes for each 𝛼∈ℕ^𝑛, with 𝜋_𝛼 the projection. In view of the uniform continuity of the mappings (3.3) and (3.12), there are unique extensions of these mappings to the completion 𝒩ℒ^∞(Ω) of ℳℒ^∞(Ω). That is, we have uniformly continuous mappings𝒟^𝛼♯𝐃∶𝒩ℒ^∞(Ω)⟶𝒩ℒ(Ω),𝛼∈ℕ^𝑛,(3.14)_♯∶𝒩ℒ^∞(Ω)⟶ℳℒ⁰(Ω)^ℕ𝑛^♯(3.15) which extend the mappings (3.3) and (3.12), respectively. Here (ℳℒ⁰(Ω)^ℕ𝑛)^♯ denotes the completion of ℳℒ⁰(Ω)^ℕ𝑛. In particular, the mapping (3.15) is injective. Note that there exists a canonical, bijective uniformly continuous mapping𝜄∶ℳℒ⁰(Ω)^ℕ𝑛^♯⟶𝒩ℒ(Ω)^ℕ𝑛,(3.16) see [16]. We may therefore consider (3.15) as an injective uniformly continuous mapping𝐃^♯∶𝒩ℒ^∞(Ω)⟶𝒩ℒ(Ω)^ℕ𝑛,(3.17) where 𝒩ℒ(Ω)^ℕ𝑛 carries the product uniform convergence structure with respect to 𝒥^♯𝑜. Furthermore, the commutative diagram (3.13) extends to the diagram(3.18)

The interpretation of the existence of the injective, uniformly continuous mapping (3.17) and the commutative diagram (3.18) is as follows. Each generalized function 𝑢^♯∈𝒩ℒ^∞(Ω) may be represented in a canonical way through its generalized partial derivatives 𝒟^𝛼♯𝑢^♯, which are usual nearly finite normal lower semicontinuous functions. This gives a first clarification of the structure of generalized functions. Furthermore, this also provides a basic blanket regularity for the generalized functions in 𝒩ℒ^∞(Ω). Namely, each such generalized function is identified with the vector of normal lower semicontinuous functions𝐃^♯𝑢^♯=𝒟^𝛼♯𝑢^♯.(3.19) Now, in view of (2.11), we have∀∃∀𝑢𝑅𝛼^♯∈⊆∈ℕΩ𝒩_𝑛ℒar^∞es(iΩdu)a∶lset∶𝑥∈𝑅⟹∶𝒟_𝛼♯𝑢^♯continuousat𝑥.(3.20) Thus the singularity set associated with each generalized function 𝑢^♯∈𝒩ℒ^∞(Ω), that is, the set where 𝑢^♯ or any of its generalized partial derivatives are discontinuous, is of first Baire category. This set, while small in a topological sense, may be dense in Ω. Furthermore, it may have arbitrarily large positive Lebesgue measure [22]. We note that such highly singular objects may be of interest in connection with turbulence in fluids and other types chaotical phenomena.

4. Existence of Generalized Solutions

In the previous section we discussed the structure of spaces of generalized functions which are obtained as the completion of suitable uniform convergence spaces, the elements of which are nearly finite normal lower semicontinuous functions. This construction is an extension of that given in [6] for spaces 𝒩ℒ^𝑚(Ω) of generalized functions which admit only generalized partial derivatives of an arbitrary but fixed finite order 𝑚, to the case of infinitely differentiable functions.

It is shown in [6] that a large class of systems of nonlinear PDEs admit solutions, in a suitable generalized sense, in the spaces 𝒩ℒ^𝑚(Ω). In this section we discuss the existence of such generalized solutions in the space 𝒩ℒ^∞(Ω). In this regard, consider nonlinear PDE of order 𝑚 of the form𝑇(𝑥,𝐷)𝑢(𝑥)=𝑓(𝑥),𝑥∈Ω.(4.1) Here the right hand term 𝑓 is supposed to be 𝒞^∞-smooth on Ω, while the nonlinear partial differential operator 𝑇(𝑥,𝐷) is defined by a 𝒞^∞-smooth mapping𝐹∶Ω×ℝ^𝐾⟶ℝ(4.2) through𝑇(𝑥,𝐷)𝑢(𝑥)=𝐹(𝑥,…,𝐷^𝛼𝑢(𝑥),…),|𝛼|≤𝑚(4.3) for any sufficiently smooth function 𝑢 defined on Ω. For each 𝛽∈ℕ^𝑛, we denote by 𝐹^𝛽 the mapping𝐹^𝛽∶Ω×ℝ^𝑀𝛽⟶ℝ(4.4) such that𝐷^𝛽(𝑇(𝑥,𝐷)𝑢(𝑥))=𝐹^𝛽(𝑥,…,𝐷^𝛼𝑢(𝑥),…),|𝛼|≤𝑚+||𝛽||(4.5) for all functions 𝑢∈𝒞^∞(Ω). Consider the mapping𝐹^∞∶Ω×ℝ^ℕ𝑛∋𝑥,𝜉_𝛼_𝛼∈ℕ^𝑛⟼𝐹^𝛽𝑥,…,𝜉_𝛼,…_{𝛽∈ℕ𝑛}∈ℝ^ℕ𝑛.(4.6) We will assume that the nonlinear PDE (4.1) satisfies the condition∀∃𝑥𝜉(∈𝑥)Ω∈∶ℝ_ℕ𝑛∃𝑉∈𝒱_𝑥,𝐹^∞(𝑥,𝜉(𝑥))=𝐷^𝛽𝑓(𝑥)_{𝛽∈ℕ𝑛}∶𝐹_∞,𝑊∈𝒱_𝜉(𝑥)∶∶𝑉×𝑊⟶ℝ^ℕ𝑛open,(4.7) where ℝ^ℕ𝑛 is equipped with the product topology.

With the nonlinear operator (4.3) we may associate a mapping𝑇∶𝒞^∞(Ω)⟶𝒞^∞(Ω).(4.8) This mapping may be extended so as to act on ℳℒ^∞(Ω). In this regard, we set𝑇∶ℳℒ^∞(Ω)∋𝑢⟼(𝐼∘𝑆)𝐹⋅,𝑢,…,𝒟^𝛼𝑢,…∈ℳℒ^∞(Ω).(4.9) Furthermore, the partial derivatives 𝒟^𝛽(𝑇𝑢) of 𝑇𝑢, for 𝑢∈ℳℒ^∞(Ω), may be represented through the mappings (4.4). In particular,∀∀𝛽𝑢∈∈ℳℕ^𝑛ℒ∶_∞𝒟(Ω)∶_𝛽(𝑇𝑢)=𝑇^𝛽𝑢,(4.10) where the 𝑇^𝛽, with 𝛽∈ℕ^𝑛, are the mappings defined in terms of (4.4) as𝑇^𝛽∶ℳℒ^∞(Ω)∋𝑢⟼(𝐼∘𝑆)𝐹^𝛽⋅,…,𝒟^𝛼𝑢,…∈ℳℒ^∞(Ω)⊂ℳℒ⁰(Ω),(4.11) where 𝛼≤𝑚+|𝛽|. We denote by 𝑇^∞ the mapping𝑇^∞∶ℳℒ^∞(Ω)∋𝑢⟼𝑇^𝛽𝑢_{𝛽∈ℕ𝑛}∈ℳℒ⁰(Ω)^ℕ𝑛.(4.12) From (4.10) to (4.12) if follows that the diagram(4.13) commutes, with 𝐃 the mapping (3.12).

Through the mapping (4.9) we obtain a first extension of the nonlinear PDE (4.1). Namely, the equation𝑇𝑢=𝑓,(4.14) where 𝑇 is the mapping (4.9) and the unknown 𝑢 is supposed to belong to ℳℒ^∞(Ω). Equation (4.14) generalizes (4.1) in the sense that any solution 𝑢∈𝒞^∞(Ω) of (4.1) is also a solution of (4.14). Conversely, any solution 𝑢∈ℳℒ^∞(Ω) of (4.14) satisfies (4.1) everywhere except on the closed nowhere dense set Γ associated with 𝑢 through (3.1). That is,𝑇(𝑥,𝐷)𝑢(𝑥)=𝑓(𝑥),𝑥∈Ω⧵Γ.(4.15) A further generalization of (4.1) is obtained by extending the mapping (4.9) to the completion 𝒩ℒ^∞(Ω) of ℳℒ^∞(Ω). In order for the concept of generalized solution of (4.1) obtained through such an extension to be a sensible one, the extension of (4.9) to 𝒩ℒ^∞(Ω) must be constructed in a canonical way. In this regard, the following is the fundamental result.

Theorem 4.1. The mapping 𝑇∶ℳℒ^∞(Ω)→ℳℒ^∞(Ω) associated with the nonlinear partial differential operator (4.3) through (4.9) is uniformly continuous.

Proof. In view of the commutative diagram (4.13) it is sufficient to show that the mapping (4.12) is uniformly continuous. In this regard, we claim that each of the mappings 𝑇^𝛽∶ℳℒ^∞(Ω)⟶ℳℒ⁰(Ω)(4.16) is uniformly continuous. To see that this is so, we represent each mapping 𝑇^𝛽 through the diagram (4.17) where 𝐃^|𝛽|∶ℳℒ^∞(Ω)∋𝑢⟼𝒟^𝛼𝑢_{|𝛼|≤𝑚+|𝛽|}∈ℳℒ⁰(Ω)^𝑀|𝛽|,(4.18)𝐹^𝛽∶ℳℒ⁰(Ω)^𝑀𝛽∋𝐮=𝑢_𝛼_{|𝛼|≤𝑚+|𝛽|}⟼(𝐼∘𝑆)𝐹⋅,…,𝑢_𝛼,…∈ℳℒ⁰(Ω).(4.19) Clearly the mapping (4.18) is uniformly continuous. As such, it suffices to show that (4.19) is uniformly continuous. In this regard, for each |𝛼|≤𝑚+|𝛽| consider a sequence (𝐼^𝛼𝑛) of order intervals that satisfies (1) and (2) of (3.5). For each 𝑛∈ℕ there is an order interval 𝐼_𝑛 in ℳℒ⁰(Ω) such that 𝐹^𝛽_{|𝛼|≤𝑚+|𝛽|}𝐼_𝑛⊆𝐼^𝑛.(4.20) Indeed, there exists a closed nowhere dense set Γ⊂Ω such that ∀∃𝐾𝑀_𝐾⊂Ω⧵Γcompact∶∀𝑛∈ℕ>,0𝑥∶∈𝐾,𝐮_𝑛−𝑀_𝐾=𝑢_𝑛,𝛼∈_{|𝛼|≤𝑚+|𝛽|}𝐼^𝛼𝑛∶<𝑢_𝛼(𝑥)<𝑀_𝐾,|𝛼|≤𝑚+||𝛽||.(4.21) The inclusion (4.20) now follows by the continuity of the mapping (4.4) and the definitions of the operators 𝐼 and 𝑆, respectively. For each 𝑛∈ℕ, set 𝜆_𝑛=inf𝐹^𝛽_{|𝛼|≤𝑚+|𝛽|}𝐼^𝛼𝑛𝜇∈𝒩ℒ(Ω),_𝑛=sup𝐹^𝛽_{|𝛼|≤𝑚+|𝛽|}𝐼^𝛼𝑛∈𝒩ℒ(Ω).(4.22) Clearly the sequence (𝜆_𝑛) is increasing, while (𝜇_𝑛) is decreasing and 𝜆_𝑛≤𝜇_𝑛,𝑛∈ℕ.(4.23) We show that 𝑢=sup𝜆_𝑛∶𝑛∈ℕ=inf𝜇_𝑛∶𝑛∈ℕ=𝑢.(4.24) From condition (2) of (3.5) it follows that ∀∃𝜖𝑁>_𝜖,𝑉0,𝑉⊆Ωnonemptyandopen∶∀𝑛≥𝑁∈_𝜖,𝑉ℕ,𝑥∈𝑉∶𝑢_𝛼∶(𝑥)−𝜖<𝑢^𝑛,𝛼(𝑥)<𝑢^𝛼(𝑥)+𝜖,𝐮_𝑛=𝑢_𝑛,𝛼∈_{|𝛼|≤𝑚+|𝛽|}𝐼^𝛼𝑛,(4.25) where 𝑢^𝛼∈𝒩ℒ(Ω) is the function associated with (𝐼^𝛼𝑛) through (2) of (3.5). In fact, due to (2.12), the inequality in (4.25) holds on a nonempty, open subset of 𝑉. That is, ∀∃𝑉𝑥^∈⊆𝑉𝑉_nonemptyandopen∶𝑢_𝛼∶(𝑥)−𝜖<𝑢^𝑛,𝛼(𝑥)<𝑢^𝛼(𝑥)+𝜖,𝐮_𝑛=𝑢_𝑛,𝛼∈_{|𝛼|≤𝑚+|𝛽|}𝐼^𝛼𝑛.(4.26) Therefore the continuity of the mapping (4.4) implies ∀∃𝜖𝑁>_𝜖,𝑉0,𝑉⊆Ωnonemptyandopen∶∀𝑥∈𝑉∈_ℕ,𝑉^⊆𝑉nonemptyandopen∶|||,𝑛≥𝑁_𝜖,𝑉∶𝐹^𝛽𝐮_𝑛(𝑥)−𝐹^𝛽𝐯_𝑛(𝑥)|||<𝜖,𝐮_𝑛,𝐯_𝑛∈_{|𝛼|≤𝑚+|𝛽|}𝐼^𝛼𝑛(4.27) so that ∀∃𝜖𝑁>_𝜖,𝑉0,𝑉⊆Ωnonemptyandopen∶∀𝑥∈𝑉∈_ℕ,𝑉^⊆𝑉nonemptyandopen∶0<𝜆_𝑛,𝑛≥𝑁_𝜖,𝑉∶(𝑥)−𝜇_𝑛(𝑥)<𝜖(4.28) which verifies (4.24). From the sequential order denseness (2.28) of ℳℒ⁰(Ω) in 𝒩ℒ(Ω) we obtain ∃∀𝑛𝜆∈ℕ∶_𝑛,𝑚(1)𝜆,_𝑛,𝑚𝜇_𝑛,𝑚⊂ℳℒ⁰(Ω)∶(2)sup≤𝜆𝜆_{𝑛,𝑚+1𝑛,𝑚}≤𝜇_𝑛,𝑚+1≤𝜇_𝑛,𝑚,𝑚∈ℕ,∶𝑚∈ℕ=𝜆_𝑛,inf𝜇_𝑛,𝑚∶𝑚∈ℕ=𝜇_𝑛.(4.29) By [23, Lemma 36] it follows that ∃(𝜆1^𝑛)𝜆,_𝑛𝜇^𝑛⊂ℳℒ⁰(Ω)∶(2)𝜆_𝑛≤𝜆^𝑛+1≤𝜇^𝑛+1≤𝜇^𝑛,𝑛∈ℕ,(3)sup≤𝜆𝜆_𝑛𝑛≤𝜇_𝑛≤𝜇^𝑛,𝑛∈ℕ,∶𝑛∈ℕ=𝑢=𝑢=inf𝜇^𝑛∶𝑛∈ℕ.(4.30) Therefore the sequence of order intervals (𝐼_𝑛)=([𝜆^𝑛,𝜇^𝑛]) satisfies (1) and (2) of (3.5) and ∀𝑛∈ℕ∶𝐹^𝛽_{|𝛼|≤𝑚+|𝛽|}𝐼^𝛽𝑛⊆𝐼^𝑛(4.31) which shows that 𝐹^𝛽 is uniformly continuous. Therefore, according to the diagram (4.17) each of the mappings 𝑇^𝛽 is also uniformly continuous.
The uniform continuity of the mapping (4.12) now follows by the commutative diagram (4.32) This completes the proof.

As a consequence of Theorem 4.1 we obtain a canonical extension of the mapping (4.9) to a mapping𝑇^♯∶ℳℒ^∞(Ω)⟶ℳℒ^∞(Ω).(4.33) Indeed, since 𝑇 is uniformly continuous, there exists a unique uniformly continuous extension of (4.9). This extension of the nonlinear partial differential operator to the space of generalized functions gives rise to a concept of generalized solution of (4.1). Namely, any solution 𝑢^♯∈𝒩ℒ^∞(Ω) of the extended equation𝑇^♯𝑢^♯=𝑓(4.34) corresponding to the nonlinear PDE (4.1) is interpreted as a generalized solution of (4.1).

In proving the uniform continuity of the mapping (4.9) in Theorem 4.1, we also showed that the mappings (4.11) and (4.12) are uniformly continuous. Since the mapping (3.12) is a uniformly continuous embedding, the diagram (4.13) may be extended to(4.35) where 𝑇^∞♯ is the uniformly continuous extension of (4.12). Furthermore, each of the mappings (4.11) extend uniquely to uniformly continuous mappings𝑇^𝛽♯∶𝒩ℒ^∞(Ω)⟶𝒩ℒ(Ω),𝛽∈ℕ^𝑛.(4.36) Since (4.10) coincides with 𝒟^𝛽♯∘𝑇^♯ on the dense subspace ℳℒ^∞(Ω) of 𝒩ℒ^∞(Ω), it follows [24] that∀𝑇𝑢^♯∈𝒩ℒ^∞(Ω)∶_𝛽♯𝑢^♯=𝒟^𝛽♯𝑇^♯𝑢^♯.(4.37) From the commutative diagram (4.35), and the identity (3.15) it follows that the mapping 𝑇^∞♯ may be represented as𝑇^∞♯∶𝒩ℒ^∞(Ω)∋𝑢⟼𝑇^𝛽♯𝑢^♯_{𝛽∈ℕ𝑛}∈𝒩ℒ(Ω)^ℕ𝑛.(4.38) The meaning of this is that the usual situation encountered when dealing with classical, 𝒞^∞-smooth solution of (4.1), namely,∀𝐷𝑢_𝛽∈𝒞^∞(Ω)asolutionof(4.1),𝛽∈ℕ^𝑛∶(𝑇(𝑥,𝐷)𝑢)(𝑥)=𝐷^𝛽𝑓(𝑥),𝑥∈Ω,(4.39) remains valid, in a generalized sense, for any generalized solution 𝑢^♯∈𝒩ℒ^∞(Ω) of (4.34). That is,∀𝑇𝛽∈_𝛽♯ℕ^𝑛∶𝑢^♯=𝒟^𝛽♯𝑇^♯𝑢^♯=𝐷^𝛽𝑓.(4.40) The main result of this paper, concerning the existence of solutions of (4.34), is the following.

Theorem 4.2. Consider a nonlinear PDE of the form (4.1). If the nonlinear operator (4.9) satisfies (4.7), then there exists some 𝑢^♯∈𝒩ℒ^∞(Ω) that satisfies (4.34).

Proof. Let us express Ω as Ω=_𝜈∈ℕ𝐶_𝜈,(4.41) where, for 𝜈∈ℕ, the compact sets 𝐶_𝜈 are 𝑛-dimensional intervals 𝐶_𝜈=𝑎_𝜈,𝑏_𝜈(4.42) with 𝑎_𝜈=(𝑎_𝜈,1,…,𝑎_𝜈,𝑛), 𝑏_𝜈=(𝑏_𝜈,1,…,𝑏_𝜈,𝑛)∈ℝ^𝑛 and 𝑎_𝜈,𝑗≤𝑏_𝜈,𝑗 for every 𝑗=1,…,𝑛. We assume that {𝐶_𝜈∶𝜈∈ℕ} is locally finite, that is, ∀∃𝑥𝑉∈⊆𝜈∈ΩΩℕ∶an∶e𝐶ig_𝜈hborhoodof𝑥∶∩𝑉≠∅isfinite.(4.43) Such a partition of Ω exists, see for instance [25].
Fix 𝜈∈ℕ. To each 𝑥₀∈𝐶_𝜈 we apply (4.7) so that we obtain∃∀𝑥𝜉₀𝑥∈₀𝐶_𝜈∶∃𝛿_𝑥0=𝜉^𝛼𝑥₀∈ℝ^ℕ𝑛,𝐹^∞𝑥₀,𝜉𝑥₀=𝐷^𝛽𝑓𝑥₀_𝛽∈ℕ^𝑛∶(1,)𝜖𝐹_𝑥0∞>0∶(2)𝐷_𝛽∶𝐵_𝛿𝑥0𝑥₀×𝑊^1𝑥₀⟶ℝ^ℕopen𝑓(𝑥)_{𝛽∈ℕ𝑛}∈𝐹^∞{𝑥}×𝑊^1𝑥₀,𝑥∈𝐵_𝛿𝑥0𝑥₀,(4.44) where 𝑊^1𝜖_𝑥0 is the neighborhood of 𝜉(𝑥₀) defined as 𝑊^1𝑥₀=_{|𝛼|≤𝑀𝑥0,1}𝐵_𝜖𝑥0𝜉^𝛼𝑥₀×ℝ^ℕ𝑛(4.45) for 𝑀₁>𝑚 a sufficiently large integer. Note that our choice of integer 𝑀₁ does not depend on the set 𝐶_𝜈 or the point 𝑥₀∈𝐶_𝜈. Consider now a function 𝑈_𝑥0∈𝒞^∞(Ω) such that ∀𝐷𝛼_𝛼∈ℕ^𝑛∶𝑈_𝑥0𝑥₀=𝜉^𝛼𝑥₀.(4.46) From the continuity of 𝑈_𝑥0 and (4.44) it follows that ∃∀𝑥𝛿_0𝑥0∈𝐶_𝜈∶∀𝑥∈>𝐵0_𝛿𝑥0∶(1)𝐹_∞𝑥₀,||𝛽||≤𝑀₁−𝑚∶(2)𝐷_𝛽∶𝐵_𝛿𝑥0𝑥₀×𝑊^1𝑥₀⟶ℝ^ℕopen,(3)𝐷𝑓_𝛼(𝑥)−𝜖_𝑥0<𝑇^𝛽𝑈_𝑥0(𝑥)<𝐷^𝛽𝑓(𝑥)+𝜖_𝑥0,(4)𝐷_𝛽𝑈_𝑥0(𝑥)_{𝛼∈ℕ𝑛}∈𝑊^1𝑥₀,𝑓(𝑥)_{𝛽∈ℕ𝑛}∈𝐹^∞{𝑥}×𝑊^1𝑥₀.(4.47) Since 𝐶_𝜈 is compact (4.47) may be strengthened to ∃∀𝛿𝑥_𝜈0>0∶(1∈)𝐹𝐶_𝜈∞,𝑥∈𝐵_𝛿𝜈𝑥₀,||𝛽||≤𝑀₁−𝑚∶(2)𝐷_𝛽∶𝐵_𝛿𝜈𝑥₀×𝑊^1𝑥₀⟶ℝ^ℕopen,(3)𝐷𝑓_𝛼(𝑥)−𝜖_𝑥0<𝑇^𝛽𝑈_𝑥0(𝑥)<𝐷^𝛽𝑓(𝑥)+𝜖_𝑥0,(4)𝐷_𝛽𝑈_𝑥0(𝑥)_{𝛼∈ℕ𝑛}∈𝑊^1𝑥₀,𝑓(𝑥)_{𝛽∈ℕ𝑛}∈𝐹^∞{𝑥}×𝑊^1𝑥₀.(4.48) Now subdivide 𝐶_𝜈 into locally finite, compact 𝑛-dimensional intervals 𝐶_𝜈,1,…,𝐶_{𝜈,𝐾𝜈} with pairwise disjoint interiors such that each 𝐶_𝜈,𝑖 has diameter not exceeding 𝛿_𝜈. Let 𝑎_𝜈,𝑖 denote the midpoint of 𝐶_𝜈,𝑖. Then, from (4.48), it follows that ∃∀𝑈𝑥∈_𝜈,𝑖∈𝐶𝒞^∞_𝜈,𝑖(Ω),𝜖_𝜈,𝑖>0∶(1)𝐹_∞,||𝛽||≤𝑀₁−𝑚∶(2)𝐷_𝛽∶int𝐶_𝜈,𝑖×𝑊^1𝜈,𝑖⟶ℝ^ℕopen,(3)𝐷𝑓_𝛼(𝑥)−𝜖_𝜈,𝑖<𝑇^𝛽𝑈_𝜈(𝑥)<𝐷^𝛽𝑓(𝑥)+𝜖_𝜈,𝑖,(4)𝐷_𝛽𝑈_𝜈(𝑥)_{𝛼∈ℕ𝑛}∈𝑊^1𝜈,𝑖,𝑓(𝑥)_{𝛽∈ℕ𝑛}∈𝐹^∞{𝑥}×𝑊^1𝜈,𝑖,(4.49) where 𝑊^1𝜈,𝑖=_{|𝛼|≤𝑀𝜈,1}𝐵_𝜖𝜈𝜉^𝛼𝑎_𝜈,𝑖×ℝ^ℕ𝑛.(4.50) Now consider the function 𝑉₁=_𝜈∈ℕ⎛⎜⎜⎝^𝐾𝜈_𝑖=1𝜒_𝜈,𝑖𝑈_𝜈,𝑖⎞⎟⎟⎠,(4.51) with 𝜒_𝜈,𝑖 the characteristic function of int𝐶_𝜈,𝑖, the interior of 𝐶_𝜈,𝑖. Clearly we have 𝑉₁∈𝒞^∞(Ω⧵Γ₁) where Γ₁⊂Ω is the closed nowhere dense set Γ₁=Ω⧵⎛⎜⎜⎝_𝜈∈ℕ⎛⎜⎜⎝^𝐾𝜈_𝑖=1int𝐶_𝜈,𝑖⎞⎟⎟⎠⎞⎟⎟⎠.(4.52) Upon application of (4.49) we find ∀∀𝜈𝑥∈∈ℕin,t𝐶𝑖=_𝜈,𝑖1,…,𝐾_𝜈∶𝐷,||𝛽||≤𝑀₁−𝑚∶_𝛽𝑓(𝑥)−𝜖_𝜈,𝑖<𝐷^𝛽𝑇(𝑥,𝐷)𝑉₁(𝑥)<𝐷^𝛽𝑓(𝑥)+𝜖_𝜈,𝑖.(4.53) Furthermore, ∀∀𝜈𝑥∈∈ℕin,t𝐶𝑖=_𝜈,𝑖1,…,𝐾_𝜈∶𝜆_𝛼1,|𝛼|≤𝑀₁∶(𝑥)<𝐷^𝛼𝑉₁(𝑥)<𝜇^𝛼1(𝑥),(4.54) where 𝜆^𝛼1,𝜇^𝛼1∈𝒞⁰(Ω) are defined as 𝜇𝜆^𝛼1_𝛼1(𝑥)=𝜉^𝛼𝑎_𝜈,𝑖−𝜖_𝜈,𝑖if𝑥∈𝐶_𝜈,𝑖,(𝑥)=𝜉^𝛼𝑎_𝜈,𝑖+𝜖_𝜈,𝑖if𝑥∈𝐶_𝜈,𝑖.(4.55) Clearly the functions 𝜆^𝛼1 and 𝜇^𝛼1 satisfy 𝜇^𝛼1(𝑥)−𝜆^𝛼1(𝑥)=2𝜖_𝜈,𝑖if𝑥∈𝐶_𝜈,𝑖.(4.56) Continuing in this way, we may construct a sequence (Γ_𝑛) of closed nowhere dense subsets of Ω such that Γ_𝑛⊆Γ_𝑛+1 for each 𝑛∈ℕ, a strictly increasing sequence of integers (𝑀_𝑛) and functions 𝑉_𝑛∈𝒞^∞(Ω⧵Γ_𝑛) such that ∀∀𝜈𝑥∈∈ℕin,t𝐶𝑖=_𝜈,𝑖1,…,𝐾_𝜈∶𝐷⧵Γ_𝑛,||𝛽||≤𝑀_𝑛−𝑚∶_𝛽𝑓(𝑥)−𝜖^𝜈,𝑖𝑛<𝑇^𝛽𝑉_𝑛(𝑥)<𝐷^𝛽𝑓(𝑥)+𝜖^𝜈,𝑖𝑛.(4.57) Furthermore, for each 𝑛∈ℕ we have ∃∀𝜆|𝛼_𝛼𝑛|≤𝑀_𝑛∶∀𝑥∈,𝜇Ω^𝛼𝑛∈⧵Γ𝒞⁰_𝑛Ω⧵Γ_𝑛∶(1)𝜆_𝛼𝑛,𝜈∈ℕ,𝑖=1,…,𝐾_𝜈∶(2)𝜇_𝛼𝑛(𝑥)<𝐷^𝛼𝑉_𝑛(𝑥)<𝜇^𝛼𝑛(𝑥),(𝑥)−𝜆^𝛼𝑛(𝑥)<𝜖^𝜈,𝑖𝑛,𝑥∈int𝐶^𝜈,𝑖(3)𝜇_𝛼𝑛⧵Γ_𝑛,≤𝜇^𝛼𝑛+1≤𝜆^𝛼𝑛+1≤𝜆^𝛼𝑛.(4.58) Consider now the functions 𝑢_𝑛=(𝐼∘𝑆)𝑉_𝑛∈ℳℒ^∞(Ω).(4.59) From (4.57) as well as the monotonicity and idempotency of the operator 𝐼∘𝑆, it follows that the sequence (𝑇𝑢_𝑛) converges to 𝑓 in ℳℒ^∞(Ω). Furthermore, (4.58) implies that the sequence (𝑢_𝑛) is a Cauchy sequence in ℳℒ^∞(Ω). As such, it follows by Theorem 4.1 that there is some 𝑢^♯∈𝒩ℒ^∞(Ω) that satisfies (4.34).

It should be noted that the concept of generalized solution of nonlinear PDEs introduced here is similar to many of those that are typical in the literature, at least as far as the way in which the concept of a generalized solution is arrived at. In this regard, we may recall a construction of generalized solutions of PDEs that is representative of many of those methods that are customary in the study of PDEs. In order to construct a generalized solution of a nonlinear PDE𝑇(𝑥,𝐷)𝑢(𝑥)=𝑓(𝑥),𝑥∈Ω⊆ℝ^𝑛(4.60) one considers some relatively small space 𝑋 of usual, sufficiently smooth functions on Ω, and a space 𝑌 of functions on Ω such that 𝑓∈𝑌. With the partial differential operator 𝑇(𝑥,𝐷) one associates a mapping𝑇∶𝑋⟶𝑌(4.61) in the usual way, namely, for every 𝑢∈𝑋 one has𝑇𝑢(𝑥)=𝑇(𝑥,𝐷)𝑢(𝑥),𝑥∈Ω.(4.62) The spaces 𝑋 and 𝑌 are equipped with uniform topologies, in fact, usually metrizable locally convex linear space topologies. The mapping (4.61) is assumed to be suitably compatible with the topologies on 𝑋 and 𝑌. In particular, 𝑇 is supposed to be uniformly continuous, which enables one to extend the mapping (4.61) in canonical way to the completions 𝑋^♯ and 𝑌^♯ of the spaces 𝑋 and 𝑌 with respect to their respective uniform topologies. In this case, one ends up with a mapping𝑇^♯∶𝑋^♯⟶𝑌^♯.(4.63) A solution 𝑢^♯∈𝑋^♯ of the generalized equation𝑇^♯𝑢^♯=𝑓(4.64) is now interpreted as a generalized solution of (4.60). Showing that such a generalized solution exists is often a rather difficult task, and may involve highly nontrivial ideas from function analysis and topology. Furthermore, a method that applies to a particular equation, may fail completely if the equation is changed slightly. This is in contradistinction with the generality and type independence of the solution method presented here. Moreover, one may note that the sequence of approximating solutions obtained in the proof of Theorem 4.2 is constructed using only basic properties of continuous, real-valued functions and elementary topology of Euclidean space.

5. An Application

In this section we show how the general theory developed in Sections 3 and 4 may be applied to particular equations, in fact, systems of equations. Furthermore, it is also demonstrated how the techniques of the preceding sections may be adapted so as to also incorporate initial and/or boundary conditions that may be associated with a given system of PDEs. In this way, we come to appreciate yet another advantage of solving linear and nonlinear PDEs by the methods introduced in this paper, as well as in [5–7]. Namely, and in contradistinction with the customary linear functional analytic methods, initial and/or boundary value problems are solved by essentially the same techniques that apply to the free problem. Indeed, the basic theory need only be adjusted in a minimal way in order to incorporate such additional conditions.

In this regard, we consider the one-dimensional parametrically driven, damped nonlinear Schrödinger equation𝑖𝜓_𝑡+𝜓_𝑥𝑥+2||𝜓||²𝜙=ℎ𝜓^∗𝑒^2𝑖𝑡−𝑖𝛾𝜓,(5.1) with 𝜓^∗ denoting complex conjugation, which, upon setting 𝜓=𝑢+𝑖𝑣, may be written as a system of equations−𝑢𝑣_𝑡+𝑢_𝑥𝑥+2𝑢³+2𝑣²𝑢=ℎ𝑢cos(2𝑡)+ℎ𝑣sin(2𝑡)+𝛾𝑣,_𝑡+𝑣_𝑥𝑥+2𝑣³+2𝑢²𝑣=−ℎ𝑣cos(2𝑡)+ℎ𝑢sin(2𝑡)−𝛾𝑢,(5.2) subject to the initial condition𝑣𝑢((𝑥𝑥,,00))==𝑢𝑣₀₀(𝑥),(𝑥),𝑥∈ℝ,(5.3) where 𝑢₀,𝑣₀∈𝒞^∞(ℝ). We will show that the initial value problem (5.2) and (5.3) admits a generalized solution (𝑢^♯,𝑣^♯)∈𝒩ℒ^∞(Ω)², where Ω=ℝ×[0,∞). Furthermore, this solution is shown to be a strongly generic weak solution of (5.2) in the following sense.

Definition 5.1. A pair of functions (𝑢,𝑣)∈𝐿^∞loc(Ω)² is a strongly generic weak solution of (5.2) if there exists a closed nowhere dense set Γ⊂ℝ×(0,∞) so that (𝑢,𝑣) satisfies (5.2) weakly on (ℝ×(0,∞))⧵Γ.

The motivation for this definition comes from systems theory. In this regard, recall [26] that a property of a system defined on an open subset Ω of ℝ^𝑛 is strongly generic if it holds on an open and dense subset of Ω.

We may note that a large variety of nonlinear resonant phenomena in various physical media is described by the system of equations (5.1). Among these, we may count the Faraday resonance in fluid dynamics [27], instabilities in plasma [28], oscillons in granular materials [29] and anisotropic XY model of ferromagnetism [30–33]. In these applications, it is often the so-called soliton solutions that are of interest, and a lot of work has been carried out on the analysis of such solutions, see for instance [34–36]. Here we are concerned with just the basic existence and regularity results for solutions of the initial value problem (5.2) to (5.3), for a large class of initial conditions. Of course, in case the initial data in (5.3) is analytic, the Cauchy-Kovalevskaia Theorem guarantees that a solution exists, at least locally, while the global version of that theorem [21] gives existence of a generalized solution, in a suitable algebra of generalized functions, which is analytic everywhere except possibly on a closed nowhere dense subset of Ω. However, in the case of arbitrary 𝒞^∞-smooth initial values, we are not aware of any general existence or regularity results.

Before we consider the problem of existence of generalized solutions of (5.2) and (5.3), let us express the problem in the notation of Section 4. In particular, we write the system of equations (5.2) in the form𝐓(𝑥,𝑡,𝐷)(𝑢,𝑣)(𝑥,𝑡)=𝟎(𝑥,𝑡),(𝑥,𝑡)∈Ω,(5.4) where 𝟎 denotes the two-dimensional vector valued function which is identically 𝟎 on Ω. As in Section 4, the operator 𝐓(𝑥,𝑡,𝐷) is defined through a jointly continuous, 𝒞^∞-smooth mapping𝐅∶Ω×ℝ¹⁰⟶ℝ²(5.5) as𝐓(𝑥,𝑡,𝐷)(𝑢,𝑣)(𝑥,𝑡)=𝐅(𝑥,𝑡,…,𝐷^𝛼𝑢(𝑥,𝑡),…,𝐷^𝛼𝑣(𝑥,𝑡),…),|𝛼|≤2.(5.6) In particular, the mapping 𝐅 takes the form𝐅(𝑥,𝑡,𝜉)=−𝜉⁴+𝜉₉+2𝜉³¹+2𝜉₁𝜉²²−ℎ𝜉₁cos(2𝑡)−ℎ𝜉₂sin(2𝑡)−𝛾𝜉₂−𝜉₃+𝜉₁₀+2𝜉³²+2𝜉²¹𝜉₂+ℎ𝜉₂cos(2𝑡)−ℎ𝜉₁sin(2𝑡)+𝛾𝜉₁.(5.7) With the nonlinear operator 𝐓(𝑥,𝑡,𝐷) we may associate a mapping𝐓∶ℳℒ^∞(Ω)²⟶ℳℒ^∞(Ω)²,(5.8) the components of which are defined as𝑇_𝑗∶ℳℒ^∞(Ω)²∋(𝑢,𝑣)⟼(𝐼∘𝑆)𝐹_𝑗⋅,⋅,…,𝒟^𝛼𝑢,…,𝒟^𝛼𝑣,…∈ℳℒ^∞(Ω)(5.9) for |𝛼|≤2 and 𝑗=1,2, where 𝐹₁ and 𝐹₂ are the components of the mapping (5.5).

Furthermore, we express the partial derivatives of 𝐓(𝑢,𝑣) through ∀∀𝛽(𝑢∈,𝑣ℕ)²∈∶ℳℒ_∞𝒟_𝛽(Ω)²∶𝑇_𝑗(𝑢,𝑣)=𝑇^𝛽𝑗(𝑢,𝑣),𝑗=1,2.(5.10) Here, for 𝛽∈ℕ² and 𝑗=1,2,𝑇^𝛽𝑗∶ℳℒ^∞(Ω)²⟶ℳℒ^∞(Ω)(5.11) is defined as𝑇^𝛽𝑗∶ℳℒ^∞(Ω)²∋(𝑢,𝑣)⟼(𝐼∘𝑆)𝐹^𝛽𝑗⋅,⋅,…,𝒟^𝛼𝑢,…,𝒟^𝛼𝑣,…∈ℳℒ^∞(Ω),(5.12) where 𝐹^𝛽𝑗∶Ω×ℝ^2𝑀𝛽→ℝ is the 𝒞^∞-smooth mapping such that𝐷^𝛽𝐹_𝑗(⋅,⋅,….,𝐷^𝛼𝑢,…,𝐷^𝛼𝑣,…)(𝑥,𝑡)=𝐹^𝛽𝑗(𝑥,𝑡,….,𝐷^𝛼𝑢(𝑥,𝑡),…,𝐷^𝛼𝑣(𝑥,𝑡),…),(𝑥,𝑡)∈Ω(5.13) for every 𝑢∈𝒞^∞(Ω). Just as is done in Section 4, we may use the mappings (5.11) to obtain a representation of the the operator 𝐓 through the mappings𝐓^∞𝐃∶₂ℳℒ^∞(Ω)²∋(𝑢,𝑣)⟼𝑇^𝛽1(𝑢,𝑣),𝑇^𝛽2(𝑢,𝑣)_𝛽∈ℕ2∈ℳℒ⁰(Ω)^2ℕ2,∶ℳℒ^∞(Ω)²∋(𝑢,𝑣)⟼𝒟^𝛽𝑢,𝒟^𝜌𝑣_{𝛽,𝜌∈ℕ2}∈ℳℒ^∞(Ω)^2ℕ2.(5.14) In particular, the diagram(5.15) commutes.

We may associate with the mapping (5.5) the continuous mapping𝐅^∞∶Ω×ℝ^2ℕ2⟼ℝ^2ℕ2(5.16) defined as𝐅^∞𝑥,𝑡,𝜉^𝛼1,𝜉_𝜌,2_𝛽∈ℕ²=⎛⎜⎜⎝𝐹^𝛽1𝐹_𝛽2𝑥,𝑡,𝜉^𝛼,1,𝜉_𝜌,2_{𝛼,𝜌∈ℕ}²𝑥,𝑡,𝜉^𝛼,1,𝜉_𝜌,2_{𝛼,𝜌∈ℕ}²⎞⎟⎟⎠.(5.17) Note that the mapping (5.8) is linear in the components corresponding to 𝑢_𝑡, 𝑣_𝑡, 𝑢_𝑥𝑥 and 𝑣_𝑥𝑥. Furthermore, 𝐹₁ does not depend on the components corresponding to 𝑢_𝑡 and 𝑣_𝑥𝑥, while 𝐹₂ does not depend on the components corresponding to 𝑣_𝑡 and 𝑢_𝑥𝑥. From this it follows quite easily that 𝐅^∞ satisfies the following two-dimensional version of (4.7):∀∃𝑉(𝑥∈,𝑡)𝒱∈Ω_(𝑥,𝑡)∶∃𝑊∈𝒱,_{𝜉(𝑥,𝑡)}𝜉(𝑥,𝑡)∈ℝ^2ℕ2,𝐅^∞(𝑥,𝑡,𝜉(𝑥,𝑡))=𝟎∶𝐅_∞∶∶𝑉×𝑊→ℝ^ℕ𝑛open.(5.18)

By the same arguments used in the proof of Theorem 4.1, which applies so single equations, we obtain the following existence result for generalized solution of the system of equations (5.2).

Theorem 5.2. If ℳℒ^∞(Ω)² is equipped with the product uniform convergence structure with respect to the uniform convergence structure (3.9) on ℳℒ^∞(Ω), then the mapping (5.8) is uniformly continuous.

In view of Theorem 5.2, it follows that the mapping (5.8) extends uniquely to a uniformly continuous mapping𝐓^♯∶ℳℒ^∞(Ω)²^♯⟶ℳℒ^∞(Ω)²^♯,(5.19) where (ℳℒ^∞(Ω)²)^♯ denotes the completion of ℳℒ^∞(Ω)². We may identify (ℳℒ^∞(Ω)²)^♯ in a canonical way with the product of the completion 𝒩ℒ^∞(Ω) of ℳℒ^∞(Ω). That is, there exists a unique bijective uniformly continuous mapping𝜄^♯∶ℳℒ^∞(Ω)²^♯⟶𝒩ℒ^∞(Ω)²(5.20) which extends the identity on ℳℒ^∞(Ω)². We will throughout use this identification, and hence write𝐓^♯∶𝒩ℒ^∞(Ω)²⟶𝒩ℒ^∞(Ω)²(5.21) instead of (5.19). As in the general case considered in Section 4, we call any solution (𝑢^♯,𝑣^♯)∈𝒩ℒ^∞(Ω)² of the extended equation𝐓^♯𝑢^♯,𝑣^♯=𝟎(5.22) a generalized solution of (5.1). Since the systems of PDEs (5.2) satisfies the two-dimensional version (5.18) of (4.7) we obtain, by the methods of Theorem 4.2, the following basic existence result.

Theorem 5.3. There exists some (𝑢^♯,𝑣^♯)∈𝒩ℒ^∞(Ω)² that satisfies (5.22).

In order to incorporate the initial condition (5.3) into the solution method, the way in which approximations are constructed must be altered near 𝑡=0. This can be done in a rather straightforward way, as is seen next.

Theorem 5.4. For any 𝑢₀,𝑣₀∈𝒞^∞(ℝ), there exists a solution (𝑢^♯,𝑣^♯)∈𝒩ℒ^∞(Ω)² of (5.22) that satisfies ∀∀𝛼𝑥(1∈=)ℝ𝒟𝛼∶₁,_𝛼♯0∈ℕ²∶(2)𝒟_𝛼♯𝑢^♯(𝑥,0)=𝐷^𝛼𝑥𝑢₀(𝑥),(3)𝒟_𝛼♯𝑣^♯(𝑥,0)=𝐷^𝛼𝑥𝑣₀(𝑥),𝑢^♯,𝒟^𝛼♯𝑣^♯arecontinuousat(𝑥,0).(5.23)

Proof. We express Ω as Ω=_{𝜈∈ℤ×ℕ}𝐼_𝜈,(5.24) where, for 𝜈=(𝜈₁,𝜈₂)∈ℤ×ℕ𝐼_𝜈=𝜈₁,𝜈₁+1×𝜈₂,𝜈₂+1.(5.25) Consider an arbitrary but fixed set 𝐼_𝜈. Since the mapping 𝐅^∞ satisfies (5.18), it follows that ∀𝑥₀,𝑡₀∈𝐼_𝜈∃∃𝜉𝛿=𝑥₀𝛿,𝑡₀_𝑥0,𝑡0=𝜉^𝛼𝑖𝑥₀,𝑡₀∈ℝ^2ℕ2,𝐅^∞𝑥₀,𝑡₀,𝜉𝑥₀,𝑡₀=𝟎∈ℝ^2ℕ2∶(1)𝐅_∞,𝜖=𝜖_𝑥0,𝑡0>0∶(2)𝟎∈∶𝐅_∞𝐵_𝛿𝑥₀,𝑡₀×𝑊^1𝜖𝜉𝑥₀,𝑡₀⟶ℝ^2ℕ2open,(𝑥,𝑡)×𝑊^1𝜖𝜉𝑥₀,𝑡₀,(𝑥,𝑡)∈𝐵_𝛿𝑥₀,𝑡₀,(5.26) where 𝑊^1𝜖(𝜉(𝑥₀,𝑡₀)) is the neighborhood of 𝜉(𝑥₀,𝑡₀) defined as 𝑊^1𝜖𝜉𝑥₀,𝑡₀=^𝑖=1,2_|𝛼_|≤𝑀1𝐵_𝜖𝜉^𝛼𝑖𝑥₀,𝑡₀×ℝ^2ℕ2(5.27) for 𝑀₁>2 a sufficiently large integer. Note that our choice of the constant 𝑀₁ does not depend on the point (𝑥₀,𝑡₀), or the set 𝐼_𝜈. If 𝑡₀=0, we may chose 𝜉(𝑥₀,𝑡₀) in such a way that ∀𝛼(1=)𝜉𝛼_1𝛼1,0∈ℕ²∶(2)𝜉_𝛼2𝑥₀,𝑡₀=𝐷^𝛼1_𝑥𝑢₀𝑥₀,𝑥₀,𝑡₀=𝐷^𝛼1_𝑥𝑣₀𝑥₀.(5.28) For each (𝑥₀,𝑡₀), let us fix 𝜉(𝑥₀,𝑡₀)∈ℝ^2ℕ2 in (5.26) so that (5.28) holds if 𝑡₀=0. For (𝑥₀,𝑡₀)∈𝐼_𝜈 consider 𝒞^∞-smooth functions 𝑈=𝑈_𝑥0,𝑡0 and 𝑉=𝑉_𝑥0,𝑡0 on Ω such that ∀𝛼(1∈)ℕ𝐷²_𝛼∶(2)𝐷_𝛼𝑈𝑥₀,𝑡₀=𝜉^𝛼1𝑥₀,𝑡₀,𝑉𝑥₀,𝑡₀=𝜉^𝛼2𝑥₀,𝑡₀.(5.29) In particular, if 𝑡₀=0 we may set 𝑈(𝑥,𝑡)=𝑢₀(𝑥)+𝜑^1𝑥_0,𝑡0(𝑡)+^{𝛼1,𝛼2≠0}_𝛼𝛼=_{1,𝛼2∈ℕ2}1𝛼₁!𝛼₂!𝑡^𝛼2𝑥−𝑥₀^𝛼1𝑉(𝑥,𝑡)=𝑣₀𝜉^𝛼1𝑥₀,𝑡₀,(𝑥)+𝜑^2𝑥_0,𝑡0(𝑡)+^{𝛼1,𝛼2≠0}_𝛼𝛼=_{1,𝛼2∈ℕ2}1𝛼₁!𝛼₂!𝑡^𝛼2𝑥−𝑥₀^𝛼1𝜉^𝛼2𝑥₀,𝑡₀,(5.30) where 𝜑^1𝑥_0,𝑡0,𝜑^2𝑥_0,𝑡0∈𝒞^∞(ℝ) satisfy ∀𝛼(1=)𝐷0,_𝛼2𝑡𝛼₂∈ℕ²∶(2)𝐷_𝛼2𝑡𝜑^1𝑥_0,𝑡0(0)=𝜉^𝛼1𝑥₀,𝑡₀,𝜑^2𝑥_0,𝑡0(0)=𝜉^𝛼2𝑥₀,𝑡₀.(5.31) From the continuity of 𝑈,𝑉 and their derivatives, as well as the components 𝐹^𝛽𝑗 of 𝐅^∞, and (5.26) it follows that ∀∃𝛿𝑥=₀,𝛿𝑡₀_𝑥0,𝑡0∈𝐼_𝜈1,𝜈2∶∀(𝑥,𝑡)∈𝐵>_𝛿0,𝜖=𝜖_𝑥0,𝑡0>0∶(1)𝐅_∞𝑥₀,𝑡₀,||𝛽||≤𝑀₁−2,𝑗=1,2∶(2)−1∶<𝐵𝐷_𝛿𝛽𝑥₀,𝑡₀×𝑊^1𝜖𝑥₀,𝑡₀⟶ℝ^2ℕ2open,(3)(𝐷_𝛼𝑇_𝑗(𝑥,𝑡,𝐷)(𝑈,𝑉)(𝑥,𝑡)<1,(4)𝟎∈𝑈𝐅_∞(𝑥,𝑡),𝐷^𝜌𝑉(𝑥,𝑡))_{𝛼,𝜌∈ℕ}²∈𝑊^1𝜖𝑥₀,𝑡₀,(𝑥,𝑡)×𝑊^1𝜖𝑥₀,𝑡₀(5.32) Since 𝐼_𝜈 is compact, (5.32) may be strengthened to ∃∀𝛿_𝜈𝑥>0∶₀∃𝜖=,𝜖𝑡₀_𝑥0,𝑡0∈𝐼_𝜈∶∀(𝑥,𝑡)∈𝐵>_𝛿𝜈0∶(1)𝐅_∞𝑥₀,𝑡₀,||𝛽||≤𝑀₁−2,𝑗=1,2∶(2)−1∶<𝐵𝐷_𝛿𝜈𝛽𝑥₀,𝑡₀×𝑊^1𝜖𝑥₀,𝑡₀⟶ℝ^2ℕ2open,(3)(𝐷_𝛼𝑇_𝑗(𝑥,𝑡,𝐷)(𝑈,𝑉)(𝑥,𝑡)<1,(4)𝟎∈𝑈𝐅_∞(𝑥,𝑡),𝐷^𝜌𝑉(𝑥,𝑡))_{𝛼,rho∈ℕ}²∈𝑊^1𝜖𝑥₀,𝑡₀,(𝑥,𝑡)×𝑊^1𝜖𝑥₀,𝑡₀.(5.33) Now write 𝐼_𝜈 as 𝐼_𝜈=^𝑁𝜈_𝑘=1⎛⎜⎜⎝^𝑁𝜈_𝑙=1𝐼_{𝜈,𝑘,𝑙}⎞⎟⎟⎠,(5.34) where 𝑁_𝜈>√2/𝛿_𝜈 and 𝐼_{𝜈,𝑘,𝑙}=𝜈₁+(𝑘−1)𝑁_𝜈,𝜈₁+𝑘𝑁_𝜈×𝜈²+(𝑙−1)𝑁_𝜈,𝜈₂+𝑙𝑁_𝜈.(5.35) Denote by 𝑎_{𝜈,𝑘,𝑙} the midpoint of the set 𝐼_{𝜈,𝑘,𝑙} if 𝑙≠1. Otherwise, 𝑎_{𝜈,𝑘,𝑙}=𝜈₁+2𝑘−12𝑁_𝜈1,𝜈2,0.(5.36) Then the functions 𝑈_{𝜈,𝑘,𝑙}=𝑈_{𝑎𝜈12,𝜈,𝑘,𝑙} and 𝑉_{𝜈,𝑘,𝑙}=𝑉_{𝑎𝜈12,𝜈,𝑘,𝑙} satisfy ∃∀𝜖(𝑥=,𝑡𝜖)∈_{𝜈,𝑘,𝑙}𝐼>0∶_{𝜈1,𝜈2,𝑘,𝑙}(1)𝐅_∞,||𝛽||≤𝑀₁−2,𝑗=1,2∶(2)−1∶<𝐼𝐷_{𝜈,𝑘,𝑙𝛽}×𝑊^1𝜖𝑎_{𝜈,𝑘,𝑙}⟶ℝ^2ℕ2open,(3)𝐷_𝛼𝑇_𝑗(𝑥,𝑡,𝐷)𝑈_{𝜈,𝑘,𝑙},𝑉_{𝜈,𝑘,𝑙}(𝑥,𝑡)<1,(4)𝟎∈𝑈𝐅_{𝜈,𝑘,𝑙∞}(𝑥,𝑡),𝐷^𝜌𝑉_{𝜈,𝑘,𝑙}(𝑥,𝑡)_{𝛼,rho∈ℕ2}∈𝑊^1𝜖𝑎_{𝜈,𝑘,𝑙},(𝑥,𝑡)×𝑊^1𝜖𝑎_{𝜈,𝑘,𝑙}.(5.37) Set Γ₁=Ω⧵⎛⎜⎜⎝_{𝜈∈ℤ×ℕ}⎛⎜⎜⎝^𝑁𝜈_𝑘=1⎛⎜⎜⎝^𝑁𝜈_𝑙=1int𝐼_{𝜈,𝑘,𝑙}𝑈₁⎞⎟⎟⎠⎞⎟⎟⎠⎞⎟⎟⎠,=⎛⎜⎜⎝_{𝜈∈ℤ×ℕ}⎛⎜⎜⎝^𝑁𝜈_𝑘=1⎛⎜⎜⎝^𝑁𝜈_𝑙=1𝜒_{𝜈,𝑘,𝑙}𝑈_{𝜈,𝑘,𝑙}𝑉₁⎞⎟⎟⎠⎞⎟⎟⎠⎞⎟⎟⎠,=⎛⎜⎜⎝_{𝜈∈ℤ×ℕ}⎛⎜⎜⎝^𝑁𝜈_𝑘=1⎛⎜⎜⎝^𝑁𝜈_𝑙=1𝜒_{𝜈,𝑘,𝑙}𝑉_{𝜈,𝑘,𝑙}⎞⎟⎟⎠⎞⎟⎟⎠⎞⎟⎟⎠,(5.38) where 𝜒_{𝜈,𝑘,𝑙} is the characteristic function of int𝐼_{𝜈,𝑘,𝑙}. Then, for 𝑗=1,2 and |𝛽|≤𝑀₁−2 we have −1≤𝐷^𝛽𝑇_𝑗(𝑥,𝑡,𝐷)𝑈₁,𝑉₁(𝑥,𝑡)≤1,(𝑥,𝑡)∈Ω⧵Γ₁.(5.39) Furthermore, for |𝛼|≤𝑀₁ we have 𝜆𝜆^1𝛼,1(𝑥,𝑡)≤𝐷^𝛼𝑈₁≤𝜇^1𝛼,1(𝑥,𝑡),(𝑥,𝑡)∈Ω⧵Γ₁,(5.40)_1𝛼,2(𝑥,𝑡)≤𝐷^𝛼𝑉₁≤𝜇^1𝛼,2(𝑥,𝑡),(𝑥,𝑡)∈Ω⧵Γ₁,(5.41) where 𝜆^1𝛼,𝑖,𝜇^1𝛼,𝑖∈𝒞⁰(Ω⧵Γ₁), for 𝑖=1,2, are defined as 𝜆𝜇^1𝛼,𝑖_1𝛼,𝑖(𝑥,𝑡)=𝜉^𝛼,𝑖𝑎_{𝜈,𝑘,𝑙}−𝜖_{𝜈,𝑘,𝑙},(𝑥,𝑡)∈int𝐼_{𝜈,𝑘,𝑙},(𝑥,𝑡)=𝜉^𝛼,𝑖𝑎_{𝜈,𝑘,𝑙}+𝜖_{𝜈,𝑘,𝑙},(𝑥,𝑡)∈int𝐼_{𝜈,𝑘,𝑙}(5.42) for 𝛼=(𝛼₁,𝛼₂) with 𝛼₂≠0, otherwise 𝜆^1𝛼,1(𝑥,𝑡)=𝜉^𝛼,1𝐷𝑎_{𝜈,𝑘,𝑙}−𝜖_{𝜈,𝑘,𝑙}if(𝑥,𝑡)∈int𝐼_{𝜈,𝑘,𝑙},𝑙≠1,