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- Title
The Order Completion Method for Systems of Nonlinear PDEs Revisited.
- Authors
van der Walt, Jan Harm
- 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, ). 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, ), 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.
- Subjects
NONLINEAR partial differential operators; PARTIAL differential equations; NONLINEAR evolution equations; COMPLEX variables; BOUNDARY value problems; INITIAL value problems
- Publication
Acta Applicandae Mathematicae, 2009, Vol 106, Issue 2, p149
- ISSN
0167-8019
- Publication type
Article
- DOI
10.1007/s10440-008-9287-2