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The order completion method for systems of nonlinear PDEs revisited. (English) Zbl 1177.35055
Summary: We present further developments regarding the enrichment of the basic theory of order completion as presented in M. B. Oberguggenberger and 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:
35G20General theory of nonlinear higher-order PDE
54A20Convergence in general topology
06B30Topological lattices, order topologies
46E05Lattices of continuous, differentiable or analytic functions
46F30Generalized functions for nonlinear analysis
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