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 Yearling
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May
15
awarded  Yearling
May
1
answered logarithmic sobolev inequalities
Apr
26
comment Problem on Majda's Vorticity and Incompressible Flow
You can read all this in the book by Brezis "functional analysis".
Apr
26
comment Problem on Majda's Vorticity and Incompressible Flow
The dual of $H^m$ is $H^{-m}$, and the dual of $H^{m'}$ is $H^{-m'}$, thus they are not the same. Then the brakets are always different duality pairings. However, all them can be understood as merely the integral I wrote previously. Finally notice that $$ H^{-s}\subset H^{-t}\subset L^2\subset H^t\subset H^s, $$ for $0<t<s$. Let me know if you need more help with this paragraph.
Apr
25
comment Problem on Majda's Vorticity and Incompressible Flow
Well, I don't have access to this book right now. However, I think I rememeber this part. Is the part where they show the local existence, right? I think that $$ [\phi,v]=<\phi,v>=\int \phi v dx. $$ The dual of $H^m$ is $H^{-m}$ (use Fourier to convince yourself).
Apr
25
answered About fractional Sobolev space
Apr
14
comment Conservation of norms by the 2-d euler vorticity equation
Yes :-). Everything is clear now, right?
Apr
14
comment Conservation of norms by the 2-d euler vorticity equation
Mmm, I don't know if I'm understanding you correctly. What I'm saying is, that if you assume $w(0)\in L^1\cap L^\infty$, then definitely, you will have a global bound $w(t)\in L^2$. Then you have that $$ \|f\|_{L^p}\rightarrow \|f\|_{L^\infty}, $$ as $p\rightarrow\infty$ for $f\in L^q$ (see math.stackexchange.com/questions/242779/limit-of-lp-norm)
Apr
13
answered Conservation of norms by the 2-d euler vorticity equation
Apr
9
comment $L^{\infty}$ estimate on the boundary by means of Sobolev norms in the interior
jajajaja ouch, my bad. No, no, that was what I said. It's just that for some reason I though that you were using the Evans version of the Theorem with a full derivative. I'm sorry for the confusion.
Apr
7
comment $L^{\infty}$ estimate on the boundary by means of Sobolev norms in the interior
Well, maybe it's better if you, instead of applying the trace theorem with a full derivative, apply the trace theorem with half derivative: $$ \|u\|_{H^s(\partial \Omega)}\leq C\|u\|_{H^{s+0.5}(\Omega)}. $$ Does this help?
Mar
17
comment Regularity of a parabolic equation
I already did :-). Check chapter 7 in the Evans's book. By parabolic smoothing I mean that the solution will gain some space derivatives in some integral sense in time. Something like $$ \mu\in L^2_t H^1_x $$
Mar
17
comment Regularity of a parabolic equation
Well, you can read about this kind of parabolic pde in the book by Evans "Partial differential equations". Actually,if you assume $\nu$ to be smooth you can apply the standard energy methods straightforwardly to check that the solution is smooth. If the initial data is merely $L^1$, then you have to argue with parabolic smoothing.
Mar
10
comment $\|u\|_{L^{3}(\mathbb R)} \leq C \|Du\|_{L^{2}(\mathbb R)}^{\alpha} \|u\|_{L^{2}(\mathbb R)}^{1-\alpha}$?
In one dimension you have $$ \|u\|_{L^\infty}^2\leq C\|u\|_{L^2}\|D u\|_{L^2} $$
Mar
10
answered $\|u\|_{L^{3}(\mathbb R)} \leq C \|Du\|_{L^{2}(\mathbb R)}^{\alpha} \|u\|_{L^{2}(\mathbb R)}^{1-\alpha}$?
Mar
3
answered Gronwall's inequality and polynomial
Mar
3
answered Show that $2\nabla \sqrt f\,+\,x \sqrt f=0$ (a.e.). $\implies$ $\sqrt f\in \mathcal C_0$. (Derivatives are in weak sense)
Jun
12
comment Convergence weak* in $L^{\infty}([0,T];L^2(\mathbb{T}^2))$ implies convergence weak* a.e in $L^2(\mathbb{T}^2)$?
Why does $x_n$ tend weakly to zero? I see that this should be true, but I don't see the proof now. Please, can you elaborate?
Jun
4
accepted A bound in Sobolev spaces of negative order
Jun
3
asked A bound in Sobolev spaces of negative order