Long-Sought Proof Tames Some of Math's Unruliest Equations(quantamagazine.org)
quantamagazine.org
Long-Sought Proof Tames Some of Math's Unruliest Equations
https://www.quantamagazine.org/long-sought-proof-tames-some-of-maths-unruliest-equations-20260206/
18 comments
https://arxiv.org/pdf/2401.07160
As I read this, I get stuck on the form of the solutions they wish to solve. For example, in the lava examples, presumably at a static time snapshot, the mathematicians wish to generate a function of space point in; temperature out. Then, maybe, do this at multiple time points, or evolve the system over time. Or maybe generalize classes of how a lava system could evolve.
This is a very complicated model of the real world, and I think this sort of problem comes up whenever we move from "spherical cow" physics and math to modeling or simulating something? There's chaos in the system, and sensitivity to initial conditions which aren't known. It's like reading about the "3 body problem is unsolvable".
Maybe you look at this without the framework of PDEs, and simulate it. But the article implies that lava is heterogeneous, so you don't know how to model how each part of it interacts with the rest. I struggle understanding, for example, how the author uses the word "equation" to describe something this complicated.
So, maybe the ideal solution is a set of coarse descriptions of the lava flow's temperature distribution, likelyhoods for each, predictions of which you get depending on how much you know about the initial conditions. Probably fractal?
This is a very complicated model of the real world, and I think this sort of problem comes up whenever we move from "spherical cow" physics and math to modeling or simulating something? There's chaos in the system, and sensitivity to initial conditions which aren't known. It's like reading about the "3 body problem is unsolvable".
Maybe you look at this without the framework of PDEs, and simulate it. But the article implies that lava is heterogeneous, so you don't know how to model how each part of it interacts with the rest. I struggle understanding, for example, how the author uses the word "equation" to describe something this complicated.
So, maybe the ideal solution is a set of coarse descriptions of the lava flow's temperature distribution, likelyhoods for each, predictions of which you get depending on how much you know about the initial conditions. Probably fractal?
It's often the case that describing how a complex system changes with its input variables is much easier than writing the function from the variable to the state.
A PDE is a precise description of some unknown function in terms of how it changes, so it's really the ideal framework for doing the kind of simulation you're talking about.
A PDE is a precise description of some unknown function in terms of how it changes, so it's really the ideal framework for doing the kind of simulation you're talking about.
I should have added this in my original note, but as a programmer, if you’re interested in learning about this I highly recommend Steve Brunton’s course in differential equations and dynamical systems on you tube https://youtu.be/9fQkLQZe3u8?si=Cu7F5QQyljjiG4K8
He’s a really amazing teacher and he goes through all the maths on a lightboard and then writes simulations of all the systems he talks about in python and matlab. All the course materials are available free online. It’s quite extraordinary.
He’s a really amazing teacher and he goes through all the maths on a lightboard and then writes simulations of all the systems he talks about in python and matlab. All the course materials are available free online. It’s quite extraordinary.
PDE can simulate Kelvin helmhotlz instability and if you want to go even smaller, you can go to particle in cell methods. And the distribution thing you are talking about is similar to lattice boltzmann methods.
Just so people know, the reason these are called elliptic is you can write the general form of a conic as
Ax^2 + Bxy + Cy^2 + Dx + Ey + F=0,
...for some constants A, B, C, D, E, and F, then an ellipse is where
B^2 - 4AC < 0.
Well, you can write the general form of a second order linear pde in two variables x and y as
Au_xx + Bu_xy + Cu_yy + Du_x + Eu_y+Fu = G.[1]
Where A, B, C, D, E, F, G are constants or functions of x and y. An elliptic PDE is where
B^2 - 4 AC < 0.
eg Laplace's equation (u_xx+u_yy=0) or the Schrodinger equation.
[1] In this notation, u(x,y) is the unknown function of x and y and u_xx denotes the second partial derivative of u with respect to x and you can extrapolate for the others.
Ax^2 + Bxy + Cy^2 + Dx + Ey + F=0,
...for some constants A, B, C, D, E, and F, then an ellipse is where
B^2 - 4AC < 0.
Well, you can write the general form of a second order linear pde in two variables x and y as
Au_xx + Bu_xy + Cu_yy + Du_x + Eu_y+Fu = G.[1]
Where A, B, C, D, E, F, G are constants or functions of x and y. An elliptic PDE is where
B^2 - 4 AC < 0.
eg Laplace's equation (u_xx+u_yy=0) or the Schrodinger equation.
[1] In this notation, u(x,y) is the unknown function of x and y and u_xx denotes the second partial derivative of u with respect to x and you can extrapolate for the others.
This obviously has implications for modeling physical systems. That's mentioned at the end of the article - though I'm proud of my very, very non-mathematician self for thinking of it earlier, lol - but not expanded. For those of you who do that sort of thing, how helpful will it be? What sort of improvement (in resolution? Fidelity? Efficiency? Anything else?) might your particular field expect?
Such equations are solved by searching an approximate solution which is a function that belongs to a restricted class of functions, where each function can be described by a finite number of parameters, for instance functions that are piecewise polynomials (splines), truncated polynomial series, truncated Fourier series etc. All the various methods for computing approximate solutions, e.g. finite differences, finite elements, boundary elements, spectral methods and so on, are equivalent with this.
When the equations are well-behaved, you can be certain that it is possible in principle to obtain an approximate solution that can be as close as you want to the true solution. Otherwise, it may happen that no function belonging to the restricted set of functions where you search solutions can approximate well enough the true solution, e.g. because the true solution can grow faster than any function in that set.
This research establishes conditions that can be verified for PDEs to ensure that the methods that you intend to use for solving them will work correctly, instead of providing misleading results.
When the equations are well-behaved, you can be certain that it is possible in principle to obtain an approximate solution that can be as close as you want to the true solution. Otherwise, it may happen that no function belonging to the restricted set of functions where you search solutions can approximate well enough the true solution, e.g. because the true solution can grow faster than any function in that set.
This research establishes conditions that can be verified for PDEs to ensure that the methods that you intend to use for solving them will work correctly, instead of providing misleading results.
Since it's a proof of an existing conjecture, it won't have practical implications. Anyone who needed the result has been relying on the conjecture.
Citations? Seems like this is a general assertion so it’d be nice to see if it’s true in any particular case.
This is just a general pattern: applied mathematicians are often using things pure mathematicians haven't proved to be true yet. The examples are widespread for the generalized Riemann hypothesis. There are statements we aren't sure about, but there's also a lot that we are sure about but not sure about the proof of.
How can they be sure they got it right when the previous canonical knowledge (Schauder) turned out to be false?
It wasn't wrong. It was proved for uniform materials. This paper extends it to non-uniform materials, with an additional condition.
In what way did Schauder's work turn out to be false? It simply doesn't apply to the situations discussed here.
There was this sentence in the article: "...he realized that nonuniformly elliptic PDEs that seem well behaved can have irregular solutions even when they satisfy the condition Schauder had identified"
Yes, nonuniformly elliptic PDEs. Schauder's theorem applies to uniformly elliptic PDEs.
Im amused by the article’s attempts to describe the proof in layman’s terms. It makes it sound easy!
Quanta-to-English translation:
The work concerns elliptic partial differential equations (PDEs), which describe systems that vary in space but are in equilibrium over time (e.g., stress distribution on a bridge, temperature in a static lava flow).
Mathematicians seek to prove that solutions to these equations are "regular."
Regularity means the solution is well-behaved, smooth, and lacks sudden, physically impossible jumps or singularities.
Establishing regularity is essential because it allows researchers to use approximation methods to solve complex equations that cannot be calculated directly.
The Standard Theory (Schauder Theory): In the 1930s, Juliusz Schauder proved that for uniformly elliptic PDEs (modeling "nice," homogeneous materials where properties like conductivity stay within fixed limits), regularity is guaranteed if the equation's coefficients change gradually.
This theory failed for nonuniformly elliptic PDEs.
These equations model heterogeneous materials (e.g., a mix of rock and gas) where physical properties can vary drastically and are unbounded.
For decades, mathematicians could not determine the conditions required to guarantee regular solutions for these messier equations.
Initial Discovery (2000): Giuseppe Mingione and colleagues discovered that Schauder’s condition (gradual change) was insufficient for nonuniform cases; equations satisfying Schauder's rules could still yield irregular solutions.
They proposed that regularity in nonuniform systems depends on a specific inequality.
This inequality acts as a precise threshold: it dictates that the more nonuniform the material is, the more tightly controlled the changes in the equation's coefficients must be.
Mathematicians Cristiana De Filippis and Giuseppe Mingione provided the proof using the following techniques:
The "Ghost Equation": Because the gradient (the function describing how fast the solution changes) of the original PDE could not be calculated directly, they derived a "ghost equation"—an approximation or "shadow" of the original PDE.
Gradient Recovery: They developed a multistep procedure to extract information from this ghost equation to recover the gradient of the actual solution.
Bounding the Gradient: To prove regularity, they had to show the gradient does not become infinitely large. They achieved this by splitting the gradient into smaller pieces and proving that each piece remains within a specific size limit.
The Result: De Filippis and Mingione proved that the inequality proposed 20 years prior is the exact, sharp boundary for regularity.
If a nonuniformly elliptic PDE satisfies this inequality, its solutions are guaranteed to be regular.
If it does not, regularity cannot be guaranteed.
This extends Schauder’s century-old theory to nonuniformly elliptic equations, allowing for the rigorous mathematical analysis of complex, real-world physical systems with extreme variations.
The work concerns elliptic partial differential equations (PDEs), which describe systems that vary in space but are in equilibrium over time (e.g., stress distribution on a bridge, temperature in a static lava flow).
Mathematicians seek to prove that solutions to these equations are "regular."
Regularity means the solution is well-behaved, smooth, and lacks sudden, physically impossible jumps or singularities.
Establishing regularity is essential because it allows researchers to use approximation methods to solve complex equations that cannot be calculated directly.
The Standard Theory (Schauder Theory): In the 1930s, Juliusz Schauder proved that for uniformly elliptic PDEs (modeling "nice," homogeneous materials where properties like conductivity stay within fixed limits), regularity is guaranteed if the equation's coefficients change gradually.
This theory failed for nonuniformly elliptic PDEs.
These equations model heterogeneous materials (e.g., a mix of rock and gas) where physical properties can vary drastically and are unbounded.
For decades, mathematicians could not determine the conditions required to guarantee regular solutions for these messier equations.
Initial Discovery (2000): Giuseppe Mingione and colleagues discovered that Schauder’s condition (gradual change) was insufficient for nonuniform cases; equations satisfying Schauder's rules could still yield irregular solutions.
They proposed that regularity in nonuniform systems depends on a specific inequality.
This inequality acts as a precise threshold: it dictates that the more nonuniform the material is, the more tightly controlled the changes in the equation's coefficients must be.
Mathematicians Cristiana De Filippis and Giuseppe Mingione provided the proof using the following techniques:
The "Ghost Equation": Because the gradient (the function describing how fast the solution changes) of the original PDE could not be calculated directly, they derived a "ghost equation"—an approximation or "shadow" of the original PDE.
Gradient Recovery: They developed a multistep procedure to extract information from this ghost equation to recover the gradient of the actual solution.
Bounding the Gradient: To prove regularity, they had to show the gradient does not become infinitely large. They achieved this by splitting the gradient into smaller pieces and proving that each piece remains within a specific size limit.
The Result: De Filippis and Mingione proved that the inequality proposed 20 years prior is the exact, sharp boundary for regularity.
If a nonuniformly elliptic PDE satisfies this inequality, its solutions are guaranteed to be regular.
If it does not, regularity cannot be guaranteed.
This extends Schauder’s century-old theory to nonuniformly elliptic equations, allowing for the rigorous mathematical analysis of complex, real-world physical systems with extreme variations.