Timeline for A priori energy estimates for Burger's equation with dissipation
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Sep 15, 2010 at 7:11 | answer | added | Denis Serre | timeline score: 4 | |
| Sep 12, 2010 at 17:37 | comment | added | Dorian | I forgot that from the $L^2([0,T];H_0^1(\Omega)$ bounds (btw I'm assuming Dirichlet boundary conditions $u(t,0) = u(t,L)=0$) we get immediately uniform $1/2$ Holder continuity in time. That's enough to deal with the $\int u u_x u_t$ term when estimating $\int (u_t)^2$ since I can put an $L^{\infty}$ bound on $u$ and use Cauchy's inequality to take out the other two terms. | |
| Sep 12, 2010 at 13:48 | comment | added | fedja | Show spacial $C^\infty$. To this end just compute the derivative of the integral of the square of $\Delta_x^s u$ with large $s$, use appropriate Gagliardo-Nierenberg to estimate the trilinear term and conclude that the high $H^s$ norm decays (and very fast so) if it is large. This trick works perfectly well in the periodic setting (the key is that $\int ff_x=0$ for all $f$). You may need some adjustments for the interval case but to say more, I need to know how exactly you pose your question (the solution on the interval is far from unique unless you impose some boundary conditions). | |
| Sep 12, 2010 at 13:25 | history | edited | Dorian | CC BY-SA 2.5 |
clarification
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| Sep 12, 2010 at 13:24 | comment | added | Dorian | Well I've already establsihed control in $L^([0,T];H_0^1(\Omega))$ but this is just the standard energy estimate needed for existence. Usually one then shows that $u_t \in L^2([0,T];L^2(\Omega))$ and then uses that to reduce the problem to the elliptic problem: $uu_x - u_{xx} = -u_t$. I'm not sure what other method you had in mind. | |
| Sep 12, 2010 at 6:11 | history | edited | Charles Matthews | CC BY-SA 2.5 |
copy edit
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| Sep 12, 2010 at 5:30 | comment | added | fedja | Usually it is much easier to establish the space regularity first. This case is no exception. | |
| Sep 12, 2010 at 5:24 | history | asked | Dorian | CC BY-SA 2.5 |