Hunter Liu's Website

≪ Seminars and Talks

3D Isentropic Compressible Euler

Speaker: Eric Kim
Date of Talk: May 12, 2025

To be honest I’ve no clue what two of the four words in the title mean. What is isentropy?

At any rate, the question is, can one find smooth solutions to the PDE \[\begin{cases} \partial_t(\rho u) + \nabla\cdot \left( \rho u \otimes u \right) + \nabla p(\rho ) = 0, \\ \partial_t \rho + \nabla\cdot (\rho u) = 0,\end{cases}\] where \(p(\rho ) = \frac{\rho ^ \gamma }{\gamma }\) for some \(\gamma > 1\). By guessing that there should be a self-similar radially symmetric solution, one can convert this system of PDEs into a system of autonomous ODEs by using some super janky changes of variables.

Theorem 1. (Buckmaster, Lao-Cabora, Gómez-Serrano, 2025)

For any \(\gamma > 1\), there exists a smooth self-similar radially symmetric solution. For \(\gamma = \frac{7}{5}\), there are countably many solutions with distinct characteristic scales.

In 2022, Merle et al. actually showed the second claim for “almost every” \(\gamma \in (1 ,\infty)\), where “almost every” is actually when a certain magic function doesn’t vanish. However, This leaves out certain physically relevant values such as \(\frac{5}{3}\) (and presumably \(\frac{7}{5}\)).

The theme of the papers presented in the seminar looms in the background: the statement of the theorem is “there exists a solution” to an exact equation, something seemingly impossible for computers to do. We anticipate some heavy analysis of the system of autonomous ODEs to obtain an easily verifiable statement. Indeed, this is what happens:

  1. Do some trickery to convert the PDEs into a system of autonomous ODEs.
  2. Do more changes of variables to find the fixed points of the system of ODEs; one of these fixed points is “desirable” for smoothness and other good properties of solutions, and one wants to show that some trajectories in the phase portrait eventually end up there.
  3. Find a bounding region containing the desirable fixed point. This requires computer verification of a few algebraic equations to ensure that the sides of the bounding region really connect, etc.
  4. Golf this down to showing that some vector field (related to some Taylor series) is always pointing inward on one barrier boundary. This only involves checking positivity of several polynomials!
  5. Establish a recurrence on the Taylor coefficients, get good control over the largest coefficients, then just compute the finite “head” directly.

Somewhere along this chain of reductions, the Cauchy–Kovalevskaya theorem becomes important, but I’m not sure exactly where. (I have also never heard of this theorem before…)