Timeline for Classification of a system of two second order PDEs with two dependent and two independent variables
Current License: CC BY-SA 4.0
27 events
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| Jul 9, 2018 at 13:12 | comment | added | SpaceChild | @IgorKhavkine yes, right with positive diagonal, I understand, it's Cholesky decomposition. | |
| Jul 9, 2018 at 13:00 | comment | added | Igor Khavkine | @DK13 Positive to be diagonalizable? No. Positive to be diagonalizable with positive diagonal? Yes. And I repeat: "If the quadratic form fails to be positive semidefinite with $\lambda=0$, it could still be positive semi-definite with another value of $\lambda$. So no, in general $\lambda$ cannot be avoided." | |
| Jul 9, 2018 at 12:40 | comment | added | SpaceChild | @IgorKhavkine I think that it is due to Cholesky decomposition. But as a final question, what about all those arguments are not necessary? I think that $\tau(\xi,\eta)$ is already a quadratic form which can be written in the form $x^T A x$. As stated in mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html $x^TAx$ is positive definite iff $A$ is. So perhaps the reference to the sum of squares and Hilbert's theorem and the inclusion of $\lambda$ are all redundant... what do you think? | |
| Jul 9, 2018 at 12:18 | comment | added | SpaceChild | @IgorKhavkine thank you very much. All these make sense. The only missing link to me, is why the quadratic form has to be positive definite in order to be diagonalizable... | |
| Jul 9, 2018 at 12:10 | comment | added | Igor Khavkine | @DK13, You're right. I've redefined $W$ and $Y$ suitably. If the quadratic form fails to be positive semidefinite with $\lambda=0$, it could still be positive semi-definite with another value of $\lambda$. So no, in general $\lambda$ cannot be avoided. No other parameters are needed, since $\lambda$ exhausts the possible quadratic form representations of $\tau(\xi,\eta)$. As for references, I found all of this information starting from the references and key words in the Wikipedia pages that I linked to. | |
| Jul 9, 2018 at 12:07 | history | edited | Igor Khavkine | CC BY-SA 4.0 |
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| Jul 9, 2018 at 10:36 | vote | accept | SpaceChild | ||
| Jul 9, 2018 at 10:36 | comment | added | SpaceChild | @IgorKhavkine Really interesting! but one minor correction, I think that in your matrix the entries W and Y should be replaced by W/2 and Y/2, and a question: why should we proceed with a one parameter representation introducing parameter λ? can we work without loss of generality with λ=0? Also is there some additional bibliography? | |
| Jul 9, 2018 at 9:01 | history | edited | Igor Khavkine | CC BY-SA 4.0 |
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| Jul 9, 2018 at 8:54 | history | edited | Igor Khavkine | CC BY-SA 4.0 |
Answered updated question about ellipticity.
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| Jul 8, 2018 at 12:12 | vote | accept | SpaceChild | ||
| Jul 8, 2018 at 18:54 | |||||
| Jul 8, 2018 at 12:12 | comment | added | SpaceChild | @DeaneYang although I couldn't find anything about the quasilinear case, which makes me suspicious, I believe you're right! | |
| Jul 7, 2018 at 18:06 | comment | added | Deane Yang | @DK13, the classification of quasi linear systems is the same. The only difference is that, since the coefficients depend on the unknown functions, the type might, too. | |
| Jul 7, 2018 at 17:27 | comment | added | SpaceChild | @IgorKhavkine Thank you very much for your answer, indeed helped me a lot to understand the main idea. I didn't indicate that it is useful because I'm not subscribed, I posted as a visitor. I will register in order to have the right to vote. By the way the book you suggested contains theory only for the classification of linear equations... | |
| Jul 7, 2018 at 17:21 | history | edited | Igor Khavkine | CC BY-SA 4.0 |
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| Jul 7, 2018 at 17:18 | comment | added | Igor Khavkine | @Dimitri, I was not trying to be rigorous, I only wanted to sketch the main idea. So far you haven't even indicated whether my answer is even the kind of answer that you are looking for. If it's not, then increasing the level of rigor would be a waste of time. FYI, that reference [7] in Italian also only treats linear systems, in case you were curious. | |
| Jul 7, 2018 at 16:58 | comment | added | SpaceChild | @DeaneYang I think no... I can't see any rigorous definition or condition... | |
| Jul 7, 2018 at 16:56 | comment | added | SpaceChild | @DeaneYang I found this paper for quasilinear pdes people.dm.unipi.it/~acquistp/pub19.pdf in which ellipticity is understood as in the references [4,7] therein, however ref [4] defines Douglis-Nirenberg ellipticity which holds for linear equations and [7] is written in italian... numdam.org/article/RSMUP_1967__38__121_0.pdf | |
| Jul 7, 2018 at 16:52 | comment | added | Deane Yang | Then Igor’s answer should suffice, no? | |
| Jul 7, 2018 at 16:48 | comment | added | SpaceChild | @DeaneYang I don't want to parallelize the work done by Douglis Nirenberg results for quasilinear systems. I just want to classify a quasilinear system of two pdes of second order, is it elliptic parabolic or hyperbolic? But I can't find the suitable theory... | |
| Jul 7, 2018 at 16:16 | comment | added | Deane Yang | All you have to do is study how the constants in linear estimates depend on the coefficients. | |
| Jul 7, 2018 at 16:11 | comment | added | Deane Yang | I have no doubt that quasi linear Douglis Nirenberg systems can be studied using the same tools as standard quasi linear elliptic systems. It’s just that someone needs to work through the details. Perhaps someone already has, but I don’t know where. | |
| Jul 7, 2018 at 15:45 | comment | added | SpaceChild | Thank you @IgorKhavkine , unfortunately I'm interested in quasilinear systems, only.... | |
| Jul 7, 2018 at 15:25 | comment | added | Igor Khavkine | @Dimitri, Douglis-Nirenberg slightly generalize what one means by the principal symbol, by allowing the "highest differential order" to vary depending on which component of the PDE system you are considering. Otherwise, the classification is the same. I'm sorry, I didn't notice that you referred to quasi-linear, not just linear, equations in your question. But not much changes in the definition $\mathcal{C}(x,u)$, except that it is allowed to depend on $u$ (possibly up to $\partial^{k-1} u$). But even if the same definitions apply, there might not be nice theorems to go along with them. | |
| Jul 7, 2018 at 14:58 | comment | added | SpaceChild | I'm afraid that the Douglis-Nirenberg approach works only for linear PDEs, not for quasilinear | |
| Jul 7, 2018 at 14:33 | comment | added | SpaceChild | Thank you @IgorKhavkine, but what about Douglis-Nirenberg ellipticity? see onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160080406 | |
| Jul 7, 2018 at 13:06 | history | answered | Igor Khavkine | CC BY-SA 4.0 |