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Set $ -\Delta: H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3) $. Then $ \mathcal{R}(-\Delta) $ is non-closed? Sorry if this question is trivial. I am not familiar with theory of ...

How does one show directly that the solution following parabolic partial differential equation (PDE) of $p(t,v)$ approaches its stationary solution which is a solution of an elliptic partial ...

This is a follow-up on a previous question. Now the parabolic PDE of $P(t,x,v)$ has two spatial dimensions. $$ \partial_t P = L^* P \tag1 $$ $$L^*P = \frac12\left(\kappa^2\frac{\partial^2}{\partial v^...