real analysis – Band limited initial data : Regularity for Navier Stokes Equation defined on a Torus $mathbb{T}^m$

Consider the Navier stokes equation and the Euler equation defined on a Torus(periodic solutions).
Let the dimensionality of the space $mathbb{T}^m$ be $mge 3$.

Link to the Problem.

Has it been investigated partially or conclusively, the regularity of the solutions when the initial data $u_0(x) = u(x,0)$ is a trigonometric polynomial of a certain degree?

References to any closely related research is also appreciated.

web application – Path traversal limited

I trying to pentest website. I know that there are files named “file1” and “file2”. The server is nginx.

So I did this test:

  1. => 404

  2. => 200 and it showed me file2, so it worked!

After this test I assume that the website is vulnerable.

So I did this test:

  1. => 400

  2. => 400

  3. I tried windows files and linux files…so I did a lot of tests.

As it shows above, I am not able to get files from outside of the web root.

Is this considered vulnerable to path traversal even if I am not able to get any files from outside of the web root?

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How to do incremental backup with limited privilege in mysql?

  • Env:MySql Community v5.7 @AWS RDS
  • User Privs: Select_Priv, Create_View, Show_View, Show_Db, Execute, Create_tmp_table
  • binary_log: ON

Currently I can do full backup using mysqldump --single-transaction without locking tables. Is that possible to do incremental backups in this case? And how?

I’ve tried FLUSH LOGS but it requires RELOAD privilege which i don’t have. Also, some of the 3rd-party backup tools requires higher access levels.

Thanks in advance!