Why more physics can help achieving better mathematics

TL;DR

This paper explores how adding physical complexity to models can improve their mathematical properties, challenging the idea that simplification always aids analysis.

Contribution

It provides examples demonstrating that more physically detailed models can have better mathematical and numerical characteristics than simplified ones.

Findings

Higher-order terms improve solution properties

Stochastic noise can enhance mathematical well-posedness

Nonlocal models can be more physically accurate and mathematically advantageous

Abstract

In this paper, we discuss the question whether a physical "simplification" of a model makes it always easier to study, at least from a mathematical and numerical point of view. To this end, we give different examples showing that these simplifications often lead to worse mathematical properties of the solution to the model. This may affect the existence and uniqueness of solutions as well as their numerical approximability and other qualitative properties. In the first part, we consider examples where the addition of a higher-order term or stochastic noise leads to better mathematical results, whereas in the second part, we focus on examples showing that also nonlocal models can often be seen as physically more exact models as they have a close connection to higher-order models.