Regularity of solution to first order time dependent variational problem

Question

Consider the following first order evolution problem over some regular bounded domain $\Omega\subset\Bbb R^d$ $$\frac{\partial\phi}{\partial t}(\mathbf{x},t) +\vec V(\mathbf{x},t)\cdot \nabla\phi(\mathbf{x},t)=f(\mathbf{x},t),\;\;\;\;(\mathbf{x},t)\in \Omega\times [0,T),\;\;\;T\in\mathbb{R},\tag{1}\label{1}$$ where we are given $$\phi(\mathbf{x},0)=\phi_0\in H^1(\Omega),\;\; f\in L^2(0,T;H^1(\Omega)),\;\; \text{ and }\;\;\vec{V}\in L^2(0,T;H^1(\Omega))^2.$$ If we fix $s\in [0,T)$, the variational form of the above equation at time $s$ reads as $$ \int\limits_\Omega \frac{\partial \phi(s)}{\partial t}v+\int\limits_\Omega (\vec V(s)\cdot \nabla\phi(s))v = \int\limits_\Omega f(s)v,\;\;\;\forall v\in H^1(\Omega).$$ For any $t<T$, we can integrate the above equation from 0 to $t$ to get $$\int\limits_\Omega(\phi(t)-\phi_0)v=-\int\limits_0^t\int\limits_\Omega (\vec V(s)\cdot \nabla\phi(s))v +\int\limits_0^t\int\limits_\Omega f(s)v,\;\;\;\forall v\in H^1(\Omega).\tag{2}\label{2}$$ Suppose that we are given $\phi\in L^2(0,T;H^1(\Omega))$ which satisfies \eqref{2} for almost all $t\in[0,T)$. The question is: under the above assumptions, can we hope to show that $\phi\in H^1(0,T;L^2(\Omega))$ also, and try to recover a classical solution of \eqref{1}? If not, what conditions on $f$, $\vec{V}$ and $\phi_0$ are needed to recover a classical solution?