Timeline for Convergence as measure vs in $H^{-2}$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jun 4, 2020 at 21:51 | comment | added | maria_c | @leomonsaingeon Ok got it, it seems one really need to use the particular structure of the flow, instead of something general. Thanks! | |
| Jun 4, 2020 at 20:16 | comment | added | leo monsaingeon | @Yemon Choi: fair enough! | |
| Jun 4, 2020 at 20:13 | comment | added | leo monsaingeon | I would bet that the lower semi-continuity cannot really work, at least not for the weak $H^1$ convergence: because of my previous comment, and because you cannot take products of weak limits, there is no hope for a full "weak continuity" argument to work (as you were trying to use). So really the only hope is for true lower semicontinuity (by that I mean a real inequality between $\int\leq \liminf\int $, not an actual equality in general). But note that changing $f$ to $-f$ should give the reversed inequality, if any, so I really don't think that such a "true" lower sc can work here. | |
| Jun 4, 2020 at 20:10 | comment | added | leo monsaingeon | well I'm no expert in Ladau de Gennes theory, but I can tell you right ahead that something really intricate is going on here. In your simplified setting, and if all you know is that $f\in H^1$, the term $\int f \partial_x f\partial_y f$ may not even be integrable. Indeed $2^*=\frac{2d}{d-2}=+\infty$ at least formally, but this is a borderline case for the Sobolev embedding and $H^1$ functions fail to be $L^\infty$ in general in dimension $d=2$. Since the best one can hope for the crossed term is $\partial_x f\partial_yf\in L^2L^2=L^1$ clearly your $u$ term may very well fail to be integrable. | |
| Jun 4, 2020 at 16:48 | comment | added | maria_c | @leomonsaingeon The reason I "hope" this holds is gut feeling: what I hope is actually the lsc of the elastic part of the Landau de Gennes energy $\|\nabla Q\|_{L^2}^2 + \epsilon\sum_{i,j,k,l} Q^{kl} \partial_{x_k}Q^{ij} \partial_{x_l}Q^{ij}$, where $Q$ is a 2x2 Q-tensor. I just wrote one of the terms in the cubic term here, since I don't see lots of simplifications coming from the cubic term. With an added bulk term in $Q$, it was shown for the gradient flow to exist, under small initial data (Xu, Iyer, Zarnescu 2015). So for a gradient flow to exist, I'd guess its leading part to be lsc... | |
| Jun 4, 2020 at 13:10 | comment | added | Yemon Choi | For what it's worth I think that the question may as well stay here, even though there is a problem with the reasoning as Leo Monsaingeon has pointed out. For instance, it is not immediately clear to me that the original question has a negative answer, merely that the OP's reasoning along the way is incorrect | |
| Jun 4, 2020 at 8:06 | review | Close votes | |||
| Jun 5, 2020 at 14:53 | |||||
| S Jun 4, 2020 at 7:55 | history | suggested | Alapan Das | CC BY-SA 4.0 |
Latex inclusion
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| Jun 4, 2020 at 7:50 | comment | added | leo monsaingeon | How on earth do you get that $f_n \partial_xf_n\partial_yf_n\to f\partial_xf\partial_y f$ "in a weak topology" (presumably $H^{-2}$, as you seem to believe)? you cannot simply take the product of weakly converging sequences and conclude that it is weakly converging to the product of the limits... I recommend cross-posting your question to Math.StackExchange, people there will certainly explain why your naive hope cannot hold | |
| Jun 4, 2020 at 4:38 | review | Suggested edits | |||
| S Jun 4, 2020 at 7:55 | |||||
| Jun 4, 2020 at 4:34 | review | First posts | |||
| Jun 4, 2020 at 6:02 | |||||
| Jun 4, 2020 at 4:33 | history | asked | maria_c | CC BY-SA 4.0 |