Two Papers Discussing the Positive Case of Onsager’s Conjecture in Bounded Domains

Hey! My school just wrapped up it’s Spring Break. I hope you had an excellent Spring Break! (Mine always feels like it’s too short T_T haha)

Over the break I noticed two papers posted to ArXiv discussing the Onsager Conjecture in bounded domains. Here are links to each of the papers:

  1. Onsager’s Conjecture with Physical Boundaries and an Application to the Vanishing Viscosity Limit
  2. Onsager’s conjecture and anomalous dissipation on domains with boundary

It’s interesting to note that the second paper was posted a day after the first paper and they prove (roughly) the same result, so it appears to me that we are witnessing the aftermath of a race between two talented teams of mathematicians to be the first to publish.

Since I’m currently active in all things Onsager Conjecture related I decided to check out both of their results. Here’s my quick take on the two papers. Cheers!

1. First they are both trying to prove energy conservation results in bounded domains. As we know, when viscosity is present we can expect it to cause energy dissipation, but Onsager’s conjecture also shows that even without viscosity, one can still have energy dissipation due to insufficient regularity. This type of energy loss is called anomalous dissipation

2. The previous result by Titi et al. assumed Holder regularity > 1/3 in the entire bounded domain to show energy conservation. Both new papers prove the same result (in fact in a remark of the 2nd paper they explain how one can convert one of the paper’s results into the other paper’s result – so it really is the same result)

3. The new result is that you don’t need to assume regularity in the entire domain. You just need the regularity assumption in bulk of the domain and near the boundary you can instead assume an L^p boundedness on the velocity and the pressure, and that the normal derivative of the velocity at the boundary is well behaved (continuous).

4. By weakening the assumptions near the boundary, they can now take vanishing viscosity limits of their weakened solutions and get sensible results. Before if one used the stronger assumption of holder regularity in the entire domain, the an inviscid limit of these solutions would give a limiting velocity whose Holder norm blows up near the boundary. The new weakened solutions don’t have a holder norm near the boundary so they don’t have this problem and so both papers show that they can construct a solution to Euler by an inviscid limit and this solution does not have anomalous dissipation

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