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This is a bit fuzzy, but I've somewhere read or heard something like:

"For linear hyperbolic equations smoothing in time leads to smoothing in space"

Is this in any sense true?

References, elaborations or counterexamples would be much appreciated.

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    $\begingroup$ Very roughly speaking the intuition is true. The basic idea is that solutions to linear hyperbolic PDEs can be written as superpositions of plane waves, and the plane waves look like (taking for example the linear wave equation, but similar forms hold for general hyperbolic PDEs) $A e^{i(t \tau - x\cdot \xi)}$ where the dispersion relation $\tau = \pm|\xi|$ hold. So if you do a smoothing (e.g. a cut-off or a weight in $\tau$) you necessarily put in also a weight in $|\xi|$. $\endgroup$
    – Willie Wong
    Commented May 22, 2017 at 0:47
  • $\begingroup$ @WillieWong How dispersion comes into play in this view? $\endgroup$
    – F.M.R.
    Commented Jul 15, 2017 at 3:17
  • $\begingroup$ I don't understand your question. But it maybe helpful to be reminded that the "dispersion relation" merely describes the laws of physics (how frequency relates to momentum). Having a dispersion relation is not the same as exhibiting dispersive behavior. (Indeed one may say that the dispersion relation of the one dimensional linear wave equation shows that it is not dispersive.) $\endgroup$
    – Willie Wong
    Commented Jul 16, 2017 at 12:49

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