I have never seen the following PDE before $b \in L^\infty([0, T] \times \mathbb{R}^d)$: $$ \partial_t u + \mathrm{div}(bu) = \Delta u\text{ on }(0, T) \times \mathbb{R}^d, ~ u(0, x) = u_0(x) $$ Does anyone know its name and/or has references in which case a solution (probably in $L^2([0, T]; H^1(\mathbb{R}^d))$ continuously depends on $u_0$, i.e. $$ \lVert u(t, \cdot)\rVert_{L^2} \leq C \lVert u_0 \rVert_{L^2}~~~? $$
Thank you!