# 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?