Global energy conservation in 3D Burgers' equation? 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...)
 A: *

*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 $0<r<\infty$.
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.)

*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.
