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Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$ \begin{align} L ...

I am studying Pazy's book "Semigroups of Linear Operators and Applications to Partial Differential Equations" and when considering an operator like: A linear differential operator, $$A : W^{...

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^...

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 ...