The Order Completion Method for Systems of Nonlinear PDEs Revisited |
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Authors: | Jan Harm van der Walt |
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Institution: | (1) Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa |
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Abstract: | In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion as presented
in Oberguggenberger and Rosinger (Solution of continuous nonlinear PDEs through order completion, North-Holland, Amsterdam,
1994). In particular, spaces of generalized functions are constructed that contain generalized solutions to a large class of systems
of continuous, nonlinear PDEs. In terms of the existence and uniqueness results previously obtained for such systems of equations
(van der Walt, Acta Appl. Math. 103:1–17, 2008), one may interpret the existence of generalized solutions presented here as a regularity result. Furthermore, it is indicated
how the methods developed in this paper may be adapted to solve initial and/or boundary value problems. In particular, we
consider the Navier-Stokes equations in three spacial dimensions, subject to an initial condition on the velocity. In this
regard, we obtain the existence of a generalized solution to a large class of such initial value problems.
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Keywords: | Nonlinear PDEs Order completion Uniform convergence space |
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