If we are given that a function $u$ in time and space is in $H^1(\Omega)$ where $\Omega$ is the space domain. What can we deduce about the regularity go $u_t$ (the derivative if $u$ in time) ? Can we say that $u_t \in H^1(\Omega)$?
2026-03-16 21:28:07.1773696487
Regularity of derived function
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Nothing can be informed about $u_t$ We can only have information about the first order derivative with respect to $x$, which is in $L^2$