we consider the equation \begin{equation} \partial_t u (t , x) + v (t, x) \cdot \nabla u ( t, x) = \kappa \Delta u (t , x) + F ( t, x, u (t , x) ) \qquad \mbox{in} \ \ \mathbb{R}_+ \times \mathbb{R}^d...…(0.1) \end{equation}
with initial condition
\begin{equation} u (0 , x) = u_0 ( x) \qquad \mbox{dans} \ \ \mathbb{R}^d ….(2) \end{equation} In (0.1) $ v (t, x) $ and $ F ( t, x, u ) $ are function doned and $ \kappa $ is constant strictly positive.
We denote $u^{[\kappa]}(t,x)$ the solution of the equation (0.1) with the initial condition (0.2).
We also consider the equation of transport
\begin{equation} \partial_t u (t , x) + v (t, x) \cdot \nabla u ( t, x) = F ( t, x, u (t , x) ) \qquad \mbox{in} \ \ \mathbb{R}_+ \times \mathbb{R}^d ,.....(0.3) \end{equation} who correspond to equation (0.1) in the case $ \kappa = 0 $.
We denote by $ u^{[0]} (t,x) $ the solution of the equation (0.3) with the initial condition (0.2).
I have defined a family of solution $u^{[\kappa,n]}$ of (0.1) who converge to $u^{[\kappa]}(t,x)$ when $n \to +\infty$ and a family of solutions $u^{[0,n]}(t,x)$ of (0.3) who converge to $u^{[0]}(t,x)$.
and i proved that $u^{[\kappa]}(t,x)$ converge to $u^{[0]}(t,x)$ when $\kappa \to 0$.
My question is how we proved that $\nabla u^{[\kappa]}(t,x)$ converge to $u^{[0]}(t,x)$ when $\kappa \to 0$ and $\Delta u^{[\kappa]}(t,x)$ converge to $\Delta u^{[0]}(t,x)$ when $\kappa \to 0$? To conclude that the the solutio of (0.1) converge to the solution to (0.3) when $\kappa \to 0$?
Thank you in advance.
Best regards