Research Article | Open Access
Solutions of Smooth Nonlinear Partial Differential Equations
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.
In the 1994 monograph  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 , 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 . 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 .
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 , 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 law with the initial condition 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 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 , the Sobolev Space will, for sufficiently large values of , 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 function which is analytic everywhere except at the single point , and with an essential singularity at , Picard's Theorem states that attains every complex value, with possibly one exception, in every neighborhood of . 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 , which contain the 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 , 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 , we construct here a space of generalized functions that admit generalized partial derivatives of arbitrary order. While, following the methods introduced in , 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 -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 -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 -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  and the paper .
2. Normal Lower Semicontinuous Functions
The concept of a normal lower semicontinuous function was first introduced by Dilworth  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 . 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 the extended real line, ordered as usual. The set of all extended real-valued functions on a topological space is denoted . A function is said to be nearly finite whenever Two fundamental operations on the space are the Lower and Upper Baire Operators introduced by Baire , see also , which are defined by respectively, with denoting the neighborhood filter at . Clearly, the Baire operators and satisfy when is equipped with the usual pointwise order Furthermore, the Baire operators, as well as their compositions, are idempotent and monotone with respect to the pointwise order. That is,
The operators and , as well as their compositions and , 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 . In particular, we have Furthermore, a function is normal lower semicontinuous on whenever
We denote the set of nearly finite normal lower semicontinuous functions on by . 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 inclusion Conversely, a normal lower semicontinuous function is generically continuous in the sense that In particular, in case is a Baire space, it follows that each is continuous on some residual set, which is dense in . Furthermore, the following well-known property of continuous functions holds also for normal lower semicontinuous functions. Namely, we have
With respect to the usual pointwise order, the space is a fully distributive lattice. That is, suprema and infima of finite sets always exists and Furthermore, 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 , is given by respectively, with and given by A useful characterization of order bounded sets in is the following: If is a Baire space, then for any set we have if and only if Indeed, suppose that a set satisfies (2.18). Then the function associated with through (2.15) satisfies The function is normal lower semicontinuous and satisfies It is sufficient to show that is finite on a dense subset of . To see that satisfies this condition, assume the opposite. That is, we assume It follows by (2.5) that Thus (2.4) implies that Note that, since each is lower semicontinuous, the function is also lower semicontinuous. Therefore (2.23) implies There is therefore a residual set such that Since is a Baire space, so is the open set in the subspace topology. Furthermore, is a residual set in . But so that must be of first Baire category in , which is a contradiction. Therefore is finite on a dense subset of . The dual statement for sets bounded from below also holds.
A subspace of which is of particular interest to us here is the space which consists of all functions in which are real-valued and continuous on an open and dense subset of . That is, The space is a sublattice of . As such, it is also fully distributive. Furthermore, whenever is a metric space, the following order denseness property is satisfied: Moreover, the following sequential version of (2.27) holds: We may note that the space , and suitable subspaces of it, also play an important role in the theory of rings of continuous functions , and the theory of differential algebras of generalized functions  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 , when is an open subset of Euclidean -space .
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 , 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 be an open, nonempty and possibly unbounded subset of . For we denote by the set of those functions in that are -smooth everywhere except on some closed nowhere dense set . That is, One should note that, while the singularity set associated with a function through (3.1) is a topologically small set, it may be large in the sense of measure . That is, the set may have arbitrarily large positive Lebesgue measure. Furthermore, a function typically does not satisfy any of the usual growth conditions that are imposed on generalized functions. In particular, is, in general, not locally integrable on any neighborhood of any point . Moreover, will typically not satisfy any of the polynomial type growth conditions that are imposed on elements of the Colombeau algebras of generalized functions .
The space of generalized functions we consider here are constructed as the completion of the space equipped with a suitable uniform convergence structure. In this regard, see , we consider on the uniform order convergence structure.
Definition 3.1. Let consist of all nonempty order intervals in . The family of filters on consists of all filters that satisfy the following: there exists such that
The family of filters is a uniformly Hausdorff and first countable uniform convergence structure. Furthermore, the induced convergence structure is the order convergence structure . That is, a filter on converges to if and only if
The completion of the uniform convergence space may be characterized in terms of the space equipped with a suitable uniform convergence structure , see . In particular, this means that is complete and contains as a dense subspace. Furthermore, given any complete Hausdorff uniform convergence space , and any uniformly continuous mapping there exists a unique uniformly continuous mapping which extends .
The space is equipped with the initial uniform convergence structure with respect to the family of mappings (3.3). That is, Each of the mappings (3.3) is uniformly continuous with respect to the uniform convergence structures and on and , respectively. In fact, (3.9) is the finest uniform convergence structure on making the mappings (3.3) uniformly continuous.
Since the family of mappings (3.3) is countable and separates the points of , that is, it follows from the corresponding properties of that the uniform convergence structure is uniformly Hausdorff and first countable. Furthermore, a filter on converges to with respect to the convergence structure induced by if and only if
The completion of , which we denote by , is related to the completion of in the following way. Since carries the initial uniform convergence structure with respect to the family of mappings (3.3), it follows  that the mapping is a uniformly continuous embedding, with equipped with the product uniform convergence structure with respect to . In particular, the diagram (3.13) commutes for each , with 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 of . That is, we have uniformly continuous mappings which extend the mappings (3.3) and (3.12), respectively. Here denotes the completion of . In particular, the mapping (3.15) is injective. Note that there exists a canonical, bijective uniformly continuous mapping see . We may therefore consider (3.15) as an injective uniformly continuous mapping where carries the product uniform convergence structure with respect to . 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 may be represented in a canonical way through its generalized partial derivatives , 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 . Namely, each such generalized function is identified with the vector of normal lower semicontinuous functions Now, in view of (2.11), we have Thus the singularity set associated with each generalized function , that is, the set where 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 . Furthermore, it may have arbitrarily large positive Lebesgue measure . 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  for spaces of generalized functions which admit only generalized partial derivatives of an arbitrary but fixed finite order , to the case of infinitely differentiable functions.
It is shown in  that a large class of systems of nonlinear PDEs admit solutions, in a suitable generalized sense, in the spaces . In this section we discuss the existence of such generalized solutions in the space . In this regard, consider nonlinear PDE of order of the form Here the right hand term is supposed to be -smooth on , while the nonlinear partial differential operator is defined by a -smooth mapping through for any sufficiently smooth function defined on . For each , we denote by the mapping such that for all functions . Consider the mapping We will assume that the nonlinear PDE (4.1) satisfies the condition where is equipped with the product topology.
With the nonlinear operator (4.3) we may associate a mapping This mapping may be extended so as to act on . In this regard, we set Furthermore, the partial derivatives of , for , may be represented through the mappings (4.4). In particular, where the , with , are the mappings defined in terms of (4.4) as where . We denote by the mapping From (4.10) to (4.12) if follows that the diagram(4.13) commutes, with the mapping (3.12).
Through the mapping (4.9) we obtain a first extension of the nonlinear PDE (4.1). Namely, the equation where is the mapping (4.9) and the unknown is supposed to belong to . Equation (4.14) generalizes (4.1) in the sense that any solution of (4.1) is also a solution of (4.14). Conversely, any solution of (4.14) satisfies (4.1) everywhere except on the closed nowhere dense set associated with through (3.1). That is, A further generalization of (4.1) is obtained by extending the mapping (4.9) to the completion of . 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 must be constructed in a canonical way. In this regard, the following is the fundamental result.
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
is uniformly continuous. To see that this is so, we represent each mapping through the diagram
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 consider a sequence of order intervals that satisfies (1) and (2) of (3.5). For each there is an order interval in such that
Indeed, there exists a closed nowhere dense set such that
The inclusion (4.20) now follows by the continuity of the mapping (4.4) and the definitions of the operators and , respectively. For each , set
Clearly the sequence is increasing, while is decreasing and
We show that
From condition (2) of (3.5) it follows that
where is the function associated with through (2) of (3.5). In fact, due to (2.12), the inequality in (4.25) holds on a nonempty, open subset of . That is,
Therefore the continuity of the mapping (4.4) implies
which verifies (4.24). From the sequential order denseness (2.28) of in we obtain
By [23, Lemma 36] it follows that
Therefore the sequence of order intervals satisfies (1) and (2) of (3.5) and
which shows that is uniformly continuous. Therefore, according to the diagram (4.17) each of the mappings 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 mapping Indeed, since 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 of the extended equation 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 is the uniformly continuous extension of (4.12). Furthermore, each of the mappings (4.11) extend uniquely to uniformly continuous mappings Since (4.10) coincides with on the dense subspace of , it follows  that From the commutative diagram (4.35), and the identity (3.15) it follows that the mapping may be represented as The meaning of this is that the usual situation encountered when dealing with classical, -smooth solution of (4.1), namely, remains valid, in a generalized sense, for any generalized solution of (4.34). That is, The main result of this paper, concerning the existence of solutions of (4.34), is the following.
Proof. Let us express as
where, for , the compact sets are -dimensional intervals
with , and for every . We assume that is locally finite, that is,
Such a partition of exists, see for instance .
Fix . To each we apply (4.7) so that we obtain where is the neighborhood of defined as for a sufficiently large integer. Note that our choice of integer does not depend on the set or the point . Consider now a function such that From the continuity of and (4.44) it follows that Since is compact (4.47) may be strengthened to Now subdivide into locally finite, compact -dimensional intervals with pairwise disjoint interiors such that each has diameter not exceeding . Let denote the midpoint of . Then, from (4.48), it follows that where Now consider the function with the characteristic function of , the interior of . Clearly we have where is the closed nowhere dense set Upon application of (4.49) we find Furthermore, where are defined as Clearly the functions and satisfy Continuing in this way, we may construct a sequence of closed nowhere dense subsets of such that for each , a strictly increasing sequence of integers and functions such that Furthermore, for each we have Consider now the functions From (4.57) as well as the monotonicity and idempotency of the operator , it follows that the sequence converges to in . Furthermore, (4.58) implies that the sequence is a Cauchy sequence in . As such, it follows by Theorem 4.1 that there is some 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 PDE one considers some relatively small space of usual, sufficiently smooth functions on , and a space of functions on such that . With the partial differential operator one associates a mapping in the usual way, namely, for every one has The spaces and 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 and . In particular, is supposed to be uniformly continuous, which enables one to extend the mapping (4.61) in canonical way to the completions and of the spaces and with respect to their respective uniform topologies. In this case, one ends up with a mapping A solution of the generalized equation 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 equation with denoting complex conjugation, which, upon setting , may be written as a system of equations subject to the initial condition where . We will show that the initial value problem (5.2) and (5.3) admits a generalized solution , where . Furthermore, this solution is shown to be a strongly generic weak solution of (5.2) in the following sense.
The motivation for this definition comes from systems theory. In this regard, recall  that a property of a system defined on an open subset of is strongly generic if it holds on an open and dense subset of .
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 , instabilities in plasma , oscillons in granular materials  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  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 . However, in the case of arbitrary -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 form where denotes the two-dimensional vector valued function which is identically on . As in Section 4, the operator is defined through a jointly continuous, -smooth mapping as In particular, the mapping takes the form With the nonlinear operator we may associate a mapping the components of which are defined as for and , where and are the components of the mapping (5.5).
Furthermore, we express the partial derivatives of through Here, for and , is defined as where is the -smooth mapping such that for every . Just as is done in Section 4, we may use the mappings (5.11) to obtain a representation of the the operator through the mappings In particular, the diagram(5.15) commutes.
We may associate with the mapping (5.5) the continuous mapping defined as Note that the mapping (5.8) is linear in the components corresponding to , , and . Furthermore, does not depend on the components corresponding to and , while does not depend on the components corresponding to and . From this it follows quite easily that satisfies the following two-dimensional version of (4.7):
In view of Theorem 5.2, it follows that the mapping (5.8) extends uniquely to a uniformly continuous mapping where denotes the completion of . We may identify in a canonical way with the product of the completion of . That is, there exists a unique bijective uniformly continuous mapping which extends the identity on . We will throughout use this identification, and hence write instead of (5.19). As in the general case considered in Section 4, we call any solution of the extended equation 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 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 . This can be done in a rather straightforward way, as is seen next.
Theorem 5.4. For any , there exists a solution of (5.22) that satisfies
Proof. We express as where, for Consider an arbitrary but fixed set . Since the mapping satisfies (5.18), it follows that where is the neighborhood of defined as for a sufficiently large integer. Note that our choice of the constant does not depend on the point , or the set . If , we may chose in such a way that For each , let us fix in (5.26) so that (5.28) holds if . For consider -smooth functions and on such that In particular, if we may set where satisfy From the continuity of and their derivatives, as well as the components of , and (5.26) it follows that Since is compact, (5.32) may be strengthened to