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Zbl 1177.35055
van der Walt, Jan Harm
The order completion method for systems of nonlinear PDEs revisited. (English)
[J] Acta Appl. Math. 106, No. 2, 149-176 (2009). ISSN 0167-8019; ISSN 1572-9036/e

Summary: We present further developments regarding the enrichment of the basic theory of order completion as presented in {\it M. B. Oberguggenberger} and {\it E. E. Rosinger} [Solution of continuous nonlinear PDEs through order completion, Amsterdam: Elsevier Science (1994; Zbl 0821.35001)]. 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, 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.

MSC 2000:: *35G20 General theory of nonlinear higher-order PDE
54A20 Convergence in general topology
06B30 Topological lattices
46E05 Lattices of functions defined by smoothness properties
46F30 Generalized functions for nonlinear analysis

Keywords: initial and/or boundary value problems; Navier-Stokes equations in three spacial dimensions

Citations: Zbl 0821.35001

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Zentralblatt MATH Berlin [Germany]