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Long time behavior of a non linear Fokker-Planck equation with multiple steady states
Abstract
We consider a Fokker-Planck equation which is the mean-field limit of an interacting particles system with a temperature parameter $D>0$, a confinement potential $V$ and an interacting kernel $K$. The solution to this equation can be seen as the gradient flow of the "free energy" in the space of probability measure with Wasserstein 2 distance. Taking a specific potential of "self-propulsion" and an interacting kernel which tends to align the velocity of a particle with the average velocity, we observe a phase transition that is a limit temperature $D_*$ under which there is a continuum of steady states. We will answer the two following question:
-Do the free energy of the solution converges to the free energy of a steady state ?
-If the answer to the precedent question is yes, do the solution converges to a specific steady state ?
