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Summary
In this thesis, the Order Completion Method for nonlinear partial differential equation, in the setting of convergence spaces, is interpreted in terms of the algebraic theory of gen- eralised functions. In particular, certain spaces of generalised functions that are involved in the construction of generalised solutions for nonlinear partial differential equations through the Order Completion Method are identied with a differential chain of algebras of generalise functions. By so doing, the generalised solutions for smooth nonlinear partial differential equation obtained through Order Completion Method are interpreted as chain generalised solutions. Moreover, the mentioned differential chain is shown to be related to the Rosinger’s chain of nowhere dense algebras of generalised functions. This leads to an interpretation of the existence result for the solution of smooth nonlinear partial differ- ential equations obtained through the order completion method in the chain of nowhere dense algebras.
Using techniques introduced by Verneave, we construct a chain of almost everywhere algebras of generalised functions and show how the chain of algebras of generalised func- tions associated with the order completion method is related to this chain of almost everywhere algebras of generalised functions. We also discuss the embedding of the dis- tributions into the chain of almost everywhere algebras of generalised functions. We further show that the generalised solutions of nonlinear partial differential equations ob- tained through the order completion method corresponds to a chain generalised solution in the chain of nowhere dense algebras of generalized functions.
Finally, using the theory of chains of algebras of generalized functions, we construct algebras of generalised functions that can handle certain types of singularities occurring on sets of rst Baire category, so called, space-time foam algebras.
I Introduction
1 Algebraic nonlinear theories of generalised functions
1.1 Deciencies of D′(Ω)
1.2 Vector spaces of generalised functions
1.3 Differential algebras of generalised functions
1.4 Embedding D′(Ω) into differential algebras
1.5 Chains of algebras of generalised functions
1.5.1 First method for constructing chains of algebras of generalised functions
1.5.2 An alternative way to construct chains of algebra containing the distributions
1.5.3 Limitations of Embedding D′(Ω) into chains of algebra of generalized functions
1.6 Nonlinear Partial Differential Operators
2 The order completion method
2.1 Solutions of continuous nonlinear PDEs through order completion
2.2 Structure and regularity of generalized solutions
2.2.1 Convergence spaces
2.2.2 Order convergence
2.2.3 Normal lower semi-continuous functions
2.2.4 Structure and regularity results
II Differential algebraic interpretation of the order completion method
3 The spaces NLl(Ω) as a chain of algebras
3.1 NLl(Ω) as an algebra of generalised functions
3.2 Chain Structure of fNLl(Ω) : l 2 Ng
4 Nowhere Dense Algebras
4.1 Two Constructions of Nowhere Dense Algebras
4.1.1 Rosinger’s Nowhere Dense Algebra
4.1.2 Verneave’s Almost Everywhere Algebra
4.2 The chain of almost everywhere algebras
4.3 Functions with Nowhere Dense Singularities
4.3.1 Embedding MLl(Ω) into And
4.3.2 Embedding MLl(Ω) into Aae
4.4 NLl(Ω) and Nowhere Dense Algebras
4.5 Space-time Foam Algebras
5 Concluding Remarks
5.1 Main results
