Mathematicians finally understand the behavior of an important class of differential equations that describe everything from water pressure to oxygen levels in human tissues.

The trajectory of a storm, the evolution of stock prices, the spread of disease — mathematicians can describe any phenomenon that changes in time or space using what are known as partial differential equations. But there’s a problem: These “PDEs” are often so complicated that it’s impossible to solve them directly.

Mathematicians instead rely on a clever workaround. They might not know how to compute the exact solution to a given equation, but they can try to show that this solution must be “regular,” or well-behaved in a certain sense — that its values won’t suddenly jump in a physically impossible way, for instance. If a solution is regular, mathematicians can use a variety of tools to approximate it, gaining a better understanding of the phenomenon they want to study.

But many of the PDEs that describe realistic situations have remained out of reach. Mathematicians haven’t been able to show that their solutions are regular. In particular, some of these out-of-reach equations belong to a special class of PDEs that researchers spent a century developing a theory of — a theory that no one could get to work for this one subclass. They’d hit a wall.

Now, two Italian mathematicians have finally broken through, extending the theory to cover those messier PDEs. Their paper, published last summer, marks the culmination of an ambitious project that, for the first time, will allow scientists to describe real-life phenomena that have long defied mathematical analysis.

...read more at quantamagazine.org