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I've prove existence using the Galerkin method forBurger's equation with dissipation: $u_t + uu_x - u_{xx} = 0$ on $[0,L] \times [0,T]$ and now am trying to prove regularity.

Clarification: I have proven existence for $u \in L^2([0,T];H_0^1(\Omega))$, $u_t \in L^2([0,T];H^{-1}(\Omega))$.

I start by trying to show that $u_t \in L^2([0,T];L^2[0,L])$ but am having some trouble with this. I multiply by $u_t$ and I obtain after an integration,

$\int_0^T \int_0^L u_t^2dxdt + \int_0^L u_x(t,x)^2dx = \int_0^L u_x(0,x)^2dx + \int_0^T \int_0^L uu_xu_t dxdt$.

I can't see what I could possibly do with the second term on the right hand side of the equation. Perhaps this is not the correct way to proceed concerning $L^2$ regularity for inviscid Burger's equation? Or perhaps I'm missing something obvious?

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    $\begingroup$ Usually it is much easier to establish the space regularity first. This case is no exception. $\endgroup$
    – fedja
    Commented Sep 12, 2010 at 5:30
  • $\begingroup$ 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. $\endgroup$
    – Dorian
    Commented Sep 12, 2010 at 13:24
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    $\begingroup$ 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). $\endgroup$
    – fedja
    Commented Sep 12, 2010 at 13:48
  • $\begingroup$ 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. $\endgroup$
    – Dorian
    Commented Sep 12, 2010 at 17:37

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Since $u_x(0,\cdot)$ is $L^2$, $u(0,\cdot)$ is continuous over $[0,L]$. The Burgers equation satisfies the maximum principle, thus $\|u(t)\|_{L^\infty}\le\|u_0\|_{L^\infty}$. Then the last integral $I$ can be bounded by a use of the Young inequality: $$I\le\frac12\int_0^T\int_0^Lu_t^2dx dt+\frac{\|u_0\|_{L^\infty}^2}{2}\int_0^T\int_0^Lu_x^2dx dt.$$ The first term above is absorbed by your left-hand side. The second one is bounded because of $u\in L^2(0,T;H^1_0)$. Whence the required estimate.

I am not fond of the Galerkin method for proving the existence to the Cauchy problem for the Burgers equation and related ones. It does not give uniqueness, and it is hard to get the maximum principle that way. I prefer Picard fixed point iteration, applied to the mild formulation $$u(t)=K^t*u_0-\int_0^tK^{t-s}*(uu_x(s))ds,$$ where $K$ is the heat kernel. You may find estimates in Chapter 6 of my book Hyperbolic conservation laws I, Cambridge University Press (1999). This method has the advantage to provide regularization for $t>0$.

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  • $\begingroup$ Thanks Denis! I see that you're at ENS de Lyon? I was in fact there 2 years ago doing some work with Professor Villani. Small world after all. $\endgroup$
    – Dorian
    Commented Sep 15, 2010 at 20:23

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