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On page 82 of Tao's Nonlinear dispersive equations: local and global analysis he talks about conservation laws. He introduces $E(t) = \int_{\mathbb{R}^d}{e_0(t, x)dx}$ and that this can be manifested in a local form via a pointwise conservation law

$$\partial_t e_0(t, x) + \partial_{x_j}e_j(t, x) = 0$$

then we can get a conservation law by "integrating the continuity equation in space and using the divergence theorem (assuming some suitable spatial decay...)". I believe the $\partial_{x_j}e_j(t, x)$ is using Einstein notation and the terms range over the $d$ dimensions in $\mathbb{R}^d$.

I am not understanding either: (a) the connection between $E$ and the $e_j$ for $j > 0$; or (b) how we can integrate this to get the conservation of $E$. Any helps in this is appreciated. Thanks!

roundsquare
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  • Thr continuity equation he refers to is most likely a different equation not involving $E$ that is not in this post. – Ninad Munshi Oct 18 '23 at 20:16
  • @NinadMunshi you might be right (though it doesn't seem so from the book). Maybe it is saying that we should be able to take $e_j$ which satisfy the equation and get a conservation law out of that? If so, I'm not seeing how. – roundsquare Oct 18 '23 at 20:35

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