Mathematicians finally confirm that the glacier is smooth

Throw ice cube in a glass of water. You can draw how it starts to melt. You also know that no matter what it is, you will never see it melt into something as snow-covered, made with sharp edges and fine cusps.

Mathematicians base this method on melting and equations. The equations work well, but it has taken 130 years to prove that they are consistent with the realities of reality. Mu a paper published in March, Alessio Figalli and Joaquim Serra of the Swiss Federal Institute of Technology Zurich and Xavier Ros-Oton The University of Barcelona has confirmed that the equations are similar in intuition. Snowflakes in the model may not be impossible, but they are very rare and durable.

“This result opens up new ideas in the field,” he said Maria Colombo and the Swiss Federal Institute of Technology Lausanne. “There was no deep and accurate understanding of the incident in the past.”

The question of how water dissolves in water is called the Stefan problem, by physicist Josef Stefan, who. he wrote it in 1889. It is the most important example of the problem of “free limits”, where mathematicians consider how methods such as the spread of heat cause the boundaries to shift. In this case, the boundary is between ice and water.

Over the years, mathematicians have tried to understand complex forms of changing boundaries. To move forward, this new work draws inspiration from previous studies on another type of body: soap films. It attaches to them to ensure that in the changing boundaries between the ice and the water, sharp spots like cusps or edges are not visible, and even then they disappear immediately.

These sharp dots are called singularities, and, they, seem like ephemeral within the free limits of mathematics as they are in the world.

Melt Hourglass

Consider ice in a glass of water. Both substances are made up of the same water molecules, but the water has two components: solid and liquid. The boundary stays where the two parts meet. But as the water temperature rises in the ice, the glaciers melt and the boundaries are moved. Eventually, the ice sheet — along with its boundaries — disappeared.

Intuition can tell us that the melting point remains smooth all the time. After all, you will not cut yourself on the sharp edges if you draw ice in a cup of water. But with a little forethought, it is easy to imagine a situation where sharp-edged spots appear.

Take a piece of ice in the form of an hourglass and immerse it. As the ice melts, the hourglass waist becomes smaller and thinner until the water is eaten all the way through. At this point this happens, what was once a smooth waist becomes two-dimensional, or different.

“It’s one of those problems that naturally manifests itself,” he said Giuseppe Mingone at Parma University. “It’s the real reality that tells you that.”

Yet reality also tells us that inconsistencies are driven. We know that cups should not last long, because warm water should melt quickly. Perhaps if you start with a large pillar made of temples, a snowflake may form. But it could not last for a moment.

In 1889 Stefan solved the problem by mathematical research, and wrote two equations describing the melting of ice. One describes the spread of heat from warm water to cold ice, which reduces the ice and causes the water area to expand. The second equation monitors the change in form between ice and water as the melting process continues. (Instead, equations can also explain how ice is so cold that the surrounding water freezes – but in the present case, researchers ignore this.)