partial differential equations – Propagation of Sobolev Regularity to Endpoints in Local Well-Posedness Theory


Let $F:H^srightarrow H^s$ and suppose I have proved LWP for some PDE

begin{align*}
partial_t u &= Fu qquad text{on }, mathbf{R}times (0,infty) \
u(x,0)&=u_0(x) qquad text{on }, mathbf{R}
end{align*}

and in particular, given $u_0in H^s$, there is a time $T$ such that $uin C((0,T);H^s(mathbf{R}))$. Furthermore, I have proven that

begin{align*}
sup_{0leqslant t < T}lvertlvert u(cdot,t)rvertrvert_{H^s} < infty
end{align*}

Is it the case that $u(x,T)in H^s(mathbf{R)}$?

I have argued yes, since letting $u(x,T)=:lim_{trightarrow T}u(x,t)$ means that by Fatou’s

begin{align*}
lvertlvert u(cdot,T)rvertrvert_{H^s} leqslant liminf_{trightarrow T},
lvertlvert u(cdot,t)rvertrvert_{H^s} leqslant sup_{0leqslant t < T}lvertlvert u(cdot,t)rvertrvert_{H^s} < infty
end{align*}

However, this doesn’t make much sense to me, because in this case couldn’t we easily get LWP for $uin C_t((0,T);H^s(mathbf{R}))$ and by a similar argument keep extending the regularity to the rest of the space? I always thought we lost LWP regularity since we could no longer control the $H^s$ norm at $T$ (via Grönwall or some similar argument).

If it helps I have also proved that there are solutions with $u_0in H^s$ which lose their $H^s$ regularity in finite time.

Where have I made a mistake? And is my intuition about losing control on a Sobolev norm giving us the endpoint for LWP correct?