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Is the energy $\| u \|^2_{L^2}$ a conserved quantity for the 3D Burgers' equation for smooth solutions that decay rapidly?

Finite time singularities can appear, but I am interested in the behavior BEFORE the the blow-ups.

The 3D Burgers' equation $ {\partial v \over \partial t} + (v \cdot \nabla) v =0 $ can be written as $Dv/Dt =0$ where $D/Dt$ is the material derivative. The energy density $v^2$ is thus advected, i.e. the energy is conserved "locally".

But is it conserved globally?

Specifically, is the norm $\| u \|_{L^2}$ a conserved quantity? In 1D it is easy to show that the equivalent quantity is conserved, but in 3D I am not so sure.

Either way, I would like a proof, or reference, etc. so that I can see it for myself.

(I am concerned only about smooth and rapidly decaying solutions...)

1 Answers 1

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  1. The answer is no in general. Here is an example. Let $f(r,t)$ be a classical solution to the 1-dimensional Burger's equation $f_t+ff_r=0$, defined for $r>0$ and some interval $0\le t\le T$, and $f$ having compact support in $0. Then denote $r=|x|$ as the radial distance in $R^3$, and set $ u(x,t) = \frac{x}{r}f(r,t).$ Then $u$ is a solution to the 3D Burger's equation, smooth during $0\le t\le T$. But the energy is not conserved, because $ \int_{R^3}|u|^2\,dx = 4\pi\int_0^\infty f^2(r,t)r^2\,dr, $ which is not known to be independent of $t$ (It is $\int f^2\,dr$ which is.)

  2. The answer is yes for an incompressible solution, i.e. if the divergence of $u$ is 0. The reason is that $\frac{\partial}{\partial t}\int_{R^3}|u|^2\,dx = -2\int_{R^3} u\cdot(u\cdot\nabla u)\,dx.$ The integrand on the right is equal to $ {\rm div\,}\left(\frac{1}{2}|u|^2u\right)-\frac{1}{2}|u|^2{\rm div\,}u. $ By the divergence theorem, the first term integrates to 0 for rapidly vanishing functions, and the second is 0 for incompressible $u$.

Edit: but I don't know whether there are any solutions in case 2, other than $u=0$. There are many of the type described in 1.

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    I am back and checked out (1). Excellent. I am also glad the community gave the bounty.2011-11-16