Timeline for When a PDE add a Laplacian term
Current License: CC BY-SA 3.0
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| Apr 20, 2017 at 16:08 | comment | added | Delio Mugnolo | Similarly, if you take a linear transport equation, it has hyperbolic nature. If you add a diffusive term, say $\epsilon\Delta$, then no matter how small $\epsilon>0$ is, the corresponing PDE will have parabolic nature. Understanding how the solutions to this regularized PDE converge as $\epsilon\to 0$ is a subtle issue, and all the more so if boundary conditios play a role (think of the number of necessary boundary conditions). On the other hand, if you perturb a bi-Laplacian $\Delta^2$ by a Laplacian, you surely won't get anything essentially different. | |
| Apr 18, 2017 at 18:35 | comment | added | Michał Miśkiewicz | If you're looking for any example showing that adding a Laplacian term (even with a small coefficient) may change the game, see Burgers' equation. Googling "vanishing viscosity" might also be helpful. | |
| Apr 18, 2017 at 9:12 | comment | added | Ali Taghavi | @qiewen I am interested in your question. Some how I encountered this concept in the comment conversation of this post: mathoverflow.net/questions/182415/… | |
| Apr 18, 2017 at 3:19 | comment | added | qie wen | @WillieWong Thanks! I just updated the question. | |
| Apr 18, 2017 at 2:52 | history | edited | qie wen | CC BY-SA 3.0 |
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| Apr 18, 2017 at 2:38 | comment | added | Willie Wong | It would definitely depend on the context. Can you at the very least specify which equation you are talking about? For example, you can think of Navier-Stokes as "adding a Laplacian" to Euler's equations. That the regularity of Navier-Stokes is a Clay Problem is precisely because you cannot always say that the Laplacian will dominate! | |
| Apr 18, 2017 at 2:19 | review | Close votes | |||
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| Apr 18, 2017 at 1:52 | review | First posts | |||
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| Apr 18, 2017 at 1:49 | history | asked | qie wen | CC BY-SA 3.0 |