# Mathematicians Finally Confirm That Melting Ice Remains Smooth

Drop ice cubes in a glass of water. You can imagine how it starts to melt. You also know that no matter what its shape, you can’t see it melt into something like a snowflake, which is composed anywhere with sharp edges and fine cusps.

Mathematicians model this melting process using equations. The equations worked well, but it took 130 years to prove that they were in line with obvious facts about reality. In a paper posted in March, Alessio Figalli and Joaquim Serra at the Swiss Federal Institute of Technology Zurich and Xavier Ros-Oton at the University of Barcelona established that the equations actually fit intuition. Snowflakes on the model may not be impossible, but they are rare and completely fleeting.

“These results open up a new perspective in the field,” he said Maria Colombo at the Swiss Federal Institute of Technology Lausanne. “There has never been such a deep and precise understanding of this phenomenon before.”

The question of how ice melts in water is called Stefan’s problem, named after physicist Josef Stefan, who pose it was in 1889. This is the most important example of a “free boundary” problem, in which mathematicians consider how a process such as heat dissipation makes a boundary move. In this case, the boundary is between ice and water.

For many years, mathematicians have tried to understand the complex models of these evolving boundaries. To advance, the new work draws inspiration from previous studies of a different kind of physical system: soap films. It builds them up to prove that along the evolving boundary between ice and water, sharp spots such as cusps or edges rarely form, and even if they do occur they disappear immediately.

These sharp spots are called singularities, and, it turns out, they are as ephemeral in the free boundaries of mathematics as they are in the physical world.

Melted Hourglasses

Consider, again, an ice cube in a glass of water. Both substances are made up of the same water molecules, but water is in two different phases: solid and liquid. There is a boundary where the two phases meet. But as the heat from the water transfers to the ice, the ice will melt and the boundary will move. Eventually, the ice — and the boundary with it — will disappear.

We can be told by intuition that this melted boundary has always remained smooth. After all, you can’t cut yourself on the sharp edges if you pull a piece of ice out of a glass of water. But with a little imagination, it is easy to imagine the scenarios where the sharp spots appeared.

Take a piece of ice in the shape of an hourglass and dip it. As the ice melts, the waist of the hourglass becomes thinner and thinner until the liquid eats away all the way. Once this happens, what was once a smooth waist becomes a two -point cusps, or singularity.

“It’s one of those problems that naturally presents singularities,” he said Giuseppe Mingione at the University of Parma. “It’s the physical reality that tells you that.”

Yet reality also tells us that singularities are controlled. We know that the cusps should not last long, because the hot water should quickly dissolve them. Maybe if you start with a large block of ice made around the clock, a snowflake can form. But it still won’t last more than a sudden.

In 1889 Stefan was subjected to the problem of mathematical analysis, spelling out two equations describing melted ice. One describes the dissipation of heat from hot water to cold ice, which reduces ice while expanding the water region. The second equation traces the changing interface between ice and water as the melting process progresses. (Actually, the equations could also describe the situation where ice is so cold that it causes water to freeze around — but in the current work, the researchers have ignored the possibility.)