Suppose $$ Lu = (\partial_t^2 - g^{ij}\partial_i \partial_j + b ^i \partial_i + c)u$$ has an elliptic spatial part (i.e. $g^{ij}$ with $i, j = 1,...,n$ is uniformly bounded as a linear operator from above and below, and we reserve index $0$ for time). This is used in many lecture notes to prove that there exists a weak solution $u \in L^\infty([0, T], H^{s+ 1})$ for some $T>0$ with 0 initial data to $L u = F$ where $F$ is smooth and compactly supported. Then Sogge, Hormander, and Luk say that we can "use the equation" to show that in fact $u \in C^1([0, T], H^{s-1})$. The argument they give is to "let $v = \partial_t u$" and separate the temporal parts from the spatial parts of $L$ to get, in the weak sense: $$ \partial_t v - g^{0i}\partial_i v+ b^0v = g^{ij}\partial_i \partial_j u - b^i \partial_i u - c u + F \in L^\infty([0, T], H^{s - 1}). $$ Then Hormander says to solve this transport equation for $v$ (Sogge invokes elliptic regularity) to show that $\partial_t u = v \in L^\infty([0, T], H^{s-1})$. Then we can apparently apply the same trick for $\partial_t^2 u \in L^\infty([0, T], H^{s-1})$.
My questions are:
What exactly does $\partial_t u = v$ mean? Are we talking about the weak derivative? How do we know that $u$ even has weak derivatives? Does this follow from $\partial_t v - g^{0i}\partial_i v + b^0 v \in L^\infty$?
How does one use the equation to show that $v \in L^\infty$? For Hormander, the transport equation may not have a global solution if the characteristics are not complete. For the Sogge approach, how are we applying elliptic regularity exactly?
