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I'm currently reading Villani's notes on hypocoercivity and on pp87 it states that the differential inequality (14.10): $$ \frac{d}{dt}(\mathcal E (f) - \mathcal E(f_\infty)) = -\mathcal D(f) \leq -K_\epsilon(\mathcal E(f) - \mathcal E(f_\infty))^{1+\epsilon} $$ cannot in general be 'closed'. What does it mean for a differential inequality to be closed?

meaninglesspen
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  • I think the following link will help you a lot: https://math.stackexchange.com/questions/9199/what-does-closed-form-solution-usually-mean – nmasanta Mar 31 '21 at 10:11
  • Thanks for the comment, but I don't think this is what I'm looking for. I'm not interested in solving the equation and finding a closed form solution - I'm only interested in whether the differential inequality implies decay of $\mathcal E(f)$ to $\mathcal E(f_\infty)$. – meaninglesspen Mar 31 '21 at 10:40
  • Then your question should have been more specific than the current form. @meaninglesspen – nmasanta Mar 31 '21 at 11:05
  • On the contrary I think I've been very specific - I gave a link to the precise equation where Villani used 'closed' and I'm obviously asking for the meaning of 'closed' in the given context. – meaninglesspen Mar 31 '21 at 11:10

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Clearly, if we have a inequality of the form as you wrote: $$\frac{d}{dt}(\mathcal E (f) - \mathcal E(f_\infty)) = -\mathcal D(f) \leq -K_\epsilon(\mathcal E(f) - \mathcal E(f_\infty))^{1+\epsilon},$$ with $K_\epsilon$ being a constant depending only on $\epsilon$, this would imply that $\mathcal{E}(f)$ decays to $\mathcal E(f_\infty)$ polynomially fast in time. However, I think Villani's (14.10) stated instead that $$\frac{d}{dt}(\mathcal E (f) - \mathcal E(f_\infty)) = -\mathcal D(f) \leq -K_\epsilon\left(\mathcal E(f) - \mathcal E(\Pi_1 f)\right)^{1+\epsilon},$$ this inequality is clearly "not closed" (in the sense that no information regarding the decay of $\mathcal{E}(f)$ to $\mathcal E(f_\infty)$ can be inferred from this inequality).

Side remark: I have also read a bit Villani's "Hypocoercivity" so maybe we can discuss more on his monograph if you want, also I think my question (with a bounty) A possible error in Villani's monograph “Hypocoercivity” might interest you as well.

Fei Cao
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  • Sorry for the overdue acceptance, and thanks for the clarification. It turns out I did misread the inequality as you said. I'm not able to help with your question as well, sorry, I'm not reading the monograph in detail at the moment. – meaninglesspen Apr 15 '21 at 00:42