[Paper Review] Solving large classes of nonlinear systems of PDEs

Roumen Anguelov, Elemér E Rosinger|ArXiv.org|May 31, 2005

Mathematical and Theoretical Analysis16 references4 citations

TL;DR

This paper presents a novel method based on order completion to solve large classes of nonlinear systems of PDEs, including the Navier-Stokes equations and constitutive relations in continuum mechanics, without relying on functional analysis or generalized function spaces. The solutions are shown to be Hausdorff continuous functions, ensuring universal regularity and existence across arbitrary domains and nonlinearities with minimal continuity assumptions on coefficients.

ABSTRACT

It is shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff continuous functions. The usual Navier-Stokes equations, as well as their various modifications aiming at a realistic modelling are included as particular cases. The same holds for the critically important constitutive relations in various branches of Continuum Mechanics. The solution method does not involve functional analysis, nor various Sobolev or other spaces of distributions or generalized functions. The general and type independent existence and regularity results regarding solutions presented here are a first in the literature.

Motivation & Objective

• To establish a general existence and regularity theory for nonlinear systems of PDEs without relying on Sobolev or distributional frameworks.
• To extend the solution concept beyond classical solutions to include generalized solutions within the space of Hausdorff continuous functions.
• To demonstrate that the method applies universally to systems like Navier-Stokes and constitutive relations in continuum mechanics.
• To provide a constructive framework for solving PDEs using order completion of function spaces, particularly C^m(Ω) and C^0(Ω).
• To clarify the structure of the Dedekind order completion of continuous function spaces via Hausdorff-continuous functions.

Proposed method

• The method employs the order completion technique, originally developed by Oberguggenberger and Rosinger, to extend classical PDE operators to generalized solution spaces.
• It constructs commutative diagrams that embed classical PDE operators T(x,D): C^m(Ω) → C^0(Ω) into extended operators T: M^m_T(Ω) → M^0(Ω), ensuring surjectivity.
• The solution space M^m_T(Ω) is defined as the Dedekind order completion of C^m_{nd}(Ω), the space of piecewise C^m functions with closed, nowhere dense discontinuity sets.
• Solutions are represented as interval-valued functions in the space H(Ω) of Hausdorff-continuous functions, which are order complete and generalize continuous functions.
• The method avoids functional analytic tools such as Sobolev spaces or distributions, relying instead on order-theoretic constructions and interval-valued representations.
• The approach leverages the fact that Hausdorff-continuous functions are uniquely determined by their values on dense subsets, ensuring solution uniqueness and stability.

Experimental results

Research questions

• RQ1 Can large classes of nonlinear PDEs, including those with discontinuous coefficients, be solved without using Sobolev or distributional frameworks?
• RQ2 What is the nature of the generalized solutions obtained via order completion, and how do they relate to classical solutions?
• RQ3 Can the Dedekind order completion of C^m(Ω) be effectively characterized using Hausdorff-continuous functions?
• RQ4 Do the solutions obtained via this method possess universal regularity properties independent of the PDE type?
• RQ5 How does the order completion method ensure existence and surjectivity for nonlinear PDEs where classical operators fail to be surjective?

Key findings

• The method guarantees the existence of solutions for all nonlinear PDEs of the form F(x, U, D^pU) = f(x) with F and f jointly continuous, even when classical solutions do not exist.
• Solutions are shown to be Hausdorff continuous functions, which are a natural generalization of continuous functions and possess a universal regularity property.
• The space of Hausdorff-continuous functions H(Ω) is Dedekind order complete and order complete, enabling the construction of solutions via order-theoretic completion.
• The discontinuity sets of Hausdorff-continuous functions are closed and nowhere dense for each ε > 0, ensuring that solutions are well-behaved despite possible discontinuities.
• The method avoids functional analytic tools such as Sobolev spaces, distributions, or hyperfunctions, providing a purely order-theoretic solution framework.
• The approach applies universally to systems including Navier-Stokes equations and constitutive relations in continuum mechanics, regardless of nonlinearity or coefficient discontinuities.