Timeline for Mathematical difference between entropy and energy
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 14, 2015 at 18:40 | vote | accept | guacho | ||
| Apr 14, 2015 at 17:32 | comment | added | Deane Yang | I think if you google "Moser iteration parabolic PDE", you'll find lots of presentations on it. They will also cite Moser's original papers. | |
| Apr 14, 2015 at 15:13 | comment | added | guacho | where can I see this computation? Can you give me a reference? | |
| Apr 14, 2015 at 12:53 | comment | added | Deane Yang | An aside: The $L^p$ inequality is used for Moser iteration, which establishes an a priori estimate for $u$. The basic idea is to rewrite the negative term (up to a constant factor depending on $p$) as $-\int |\nabla u^{p/2}|^2$ and apply the Sobolev inequality. This gives you an estimate for a higher $L^p$ norm of $u$ in terms of a lower one. Repeat and show that the accumulation of constant factors is bounded as $p \rightarrow \infty$. In the limit, you get an $L^\infty$ bound on $u$ in terms of an $L^p$ bound (where $p > 1$). This works for a variable coefficient heat equation. | |
| Apr 13, 2015 at 17:37 | comment | added | Deane Yang | $\int u^2$ decays in time if and only if $\frac{1}{2}\log \int u^2$ does. | |
| Apr 13, 2015 at 17:35 | history | edited | Deane Yang | CC BY-SA 3.0 |
edited body
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| Apr 13, 2015 at 15:58 | comment | added | guacho | Thank you for your answer! I was assuming $u_0$, to be positive, so there was no problem on difining the entropy. Why do you say the case $2=p$ is the usual energy inequality? Maybe is a stupid question, but, even if I see that this quotient has the same flavour, I don't see why it's the same thing. | |
| Apr 13, 2015 at 14:28 | history | edited | Deane Yang | CC BY-SA 3.0 |
added 70 characters in body
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| Apr 13, 2015 at 13:49 | history | answered | Deane Yang | CC BY-SA 3.0 |