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...)