The Euler-Bernoulli equation with distributional coefficients and forces

TL;DR

This paper develops a framework for solving the Euler-Bernoulli beam equation with irregular coefficients and forces, establishing well-posedness of very weak solutions and demonstrating their consistency with classical solutions.

Contribution

It introduces a notion of very weak solutions for the Euler-Bernoulli equation with distributional data, proving existence, uniqueness, and consistency with classical solutions.

Findings

Very weak solutions exist and are unique for irregular data.

Numerical analysis confirms the consistency with classical solutions.

Very weak solutions provide insights when classical solutions do not exist.

Abstract

In this work we investigate a very weak solution to the initial-boundary value problem of an Euler-Bernoulli beam model. We allow for bending stiffness, axial- and transversal forces as well as for initial conditions to be irregular functions or distributions. We prove the well-posedness of this problem in the very weak sense. More precisely, we define the very weak solution to the problem and show its existence and uniqueness. For regular enough coefficients we show consistency with the weak solution. Numerical analysis shows that the very weak solution coincides with the weak solution, when the latter exists, but also offers more insights into the behaviour of the very weak solution, when the weak solution doesn't exist.