# Does a weak solution have weak derivatives?

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:

1. 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 $$\partial_t v - g^{0i}\partial_i v + b^0 v \in L^\infty$$?

2. 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?