| Abstract: |
| The talk is about the so-called $H$-Theorem for a class of nonlinear Fokker--Planck equations which are of porous media type on the whole Euclidean space perturbed by a transport term.
We first construct a solution in the sense of mild solutions on $L^1$ through a nonlinear semigroup of contractions.
Then we study the asymptotic behavior of the solutions when time tends to infinity.
For a large class $M$ of initial conditions we show their relative compactness with respect to local $L^1$ convergence, while all limit points belong to $L^1$.
Under an additional assumption we obtain that we in fact have convergence in $L^1$, if the initial condition is a probability density.
The limit is then identified as the unique stationary solution in $M$ to the nonlinear Fokker--Planck equation.
This solution is thus an invariant measure of the solution to the corresponding distribution dependent SDE whose time marginals converge to it in $L^1$.
It turns out that under our conditions the underlying nonlinear Kolmogorov operator is a (both in the second and first order part) nonlinear analog of the generator of a distorted Brownian motion.
The solution of the above mentioned distribution dependent SDE can thus be interpreted as a ``nonlinear distorted Brownian motion``.
Our main technique for the proofs is to construct a suitable Lyapunov function acting nonlinearly on the path in $L^1$, which is given by the nonlinear contraction semigroup applied to the initial condition, and then adapt a classical technique of Pazy to our situation.
This Lyapunov function is given by a generalized entropy function (which in the linear case specializes to the usual Boltzmann--Gibbs entropy) plus a mean energy part.
Ref. arXiv: 1904.08291 |
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