Timeline for Entropy condition for quasi-linear evolution equations
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 1, 2021 at 9:31 | comment | added | Ma Joad | Thank you. Let me have a look. | |
| Apr 1, 2021 at 9:30 | vote | accept | Ma Joad | ||
| Apr 1, 2021 at 1:26 | comment | added | sharpend | (If this answer is helpful, you can consider accepting it.) | |
| Apr 1, 2021 at 1:24 | comment | added | sharpend | Sure, the proof uses a subclass of entropies, but (if I remember correctly) there is a way to recover all convex entropies from the subclass. Maybe you can have a look at, say, Bressan's book on conservation laws for more details. I'm sure his presentation is much better than the ad hoc one I give here. One way to see the importance of convexity is from the viscous equation I mentioned in (4). Multiply that equation by $\eta'(u^\nu)$. Notice that $\Delta \eta = \nabla \cdot (\eta' \nabla u^\nu) = \eta' \Delta u^\nu + \eta'' |\nabla u^\nu|^2$. What does $\eta'' \geq 0$ do in that equation? | |
| Mar 31, 2021 at 23:31 | comment | added | Ma Joad | And why are we requiring this for all convex $\eta$? Apparently the proof only uses a much smaller class of "entropy" $\eta$ for energy estimates. Also, are there any intuitive way to understand why it needs to be convex? | |
| Mar 31, 2021 at 23:25 | comment | added | Ma Joad | So $\psi(u)$ is the "flow" of entropy which is designed to be just right for the equation to hold? | |
| Mar 31, 2021 at 22:52 | comment | added | sharpend | When $u$ is smooth, the equation for $\eta(u)$ is $\partial_t \eta(u) + \partial_x \psi(u) = 0$, so when you integrate in $x$, you have $\frac{d}{dt} \int \eta(u) \, dx = 0$, that is, $\int \eta \, dx$ is conserved. When $u$ is not smooth, we require that $\partial_t \eta(u) + \partial_x \psi(u) \leq 0$ is satisfied, and the equation you wrote in (3) is the weak formulation of that. | |
| Mar 31, 2021 at 22:44 | comment | added | Ma Joad | By "equivalent" I just mean "one implies another", which is not the case here since we are working with different class of equations. And a priori, before proving the uniqueness, it is not easy to see the conditions actually implies the same thing. | |
| Mar 31, 2021 at 22:40 | comment | added | Ma Joad | Why does condition 3 mean "conservation of entropy"? It does not look like familiar conservation laws in physics. Could you explain it a bit more? Thank you. | |
| Mar 31, 2021 at 22:00 | history | answered | sharpend | CC BY-SA 4.0 |