Is there any way how virial identity implies Strichartz estimates ( or some smoothing properties) for solutions to a) wave equation b) Schrodinger equation ( say in 3d)? To keep things clear I am interested in the most simple linear case. Or may be someone knows where I can read about this?
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A small number of Strichartz estimates can be proven by virial methods, see the paper of Planchon and Vega at http://arxiv.org/abs/0712.4076 . Unfortunately, despite some effort, it does not appear that the methods cover the majority of Strichartz estimates. See also the interaction Morawetz inequalities, which are proven by the Morawetz multiplier method (a variant of the virial method) and have some similarities with Strichartz estimates (but with some additional convolution kernels such as $1/|x-y|$ thrown in); these are also discussed in the paper above. On the other hand, the big advantage of the virial and Morawetz methods is that they apply directly to the nonlinear equation, whereas the Strichartz estimates only directly apply to the linear equation and can only be extended to the nonlinear equation in sufficiently perturbative situations.
answered Oct 13, 2014 at 15:50
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$\begingroup$ asked in Math SE: Terry Tao didn't list the "dozens and dozens of wave equations out there" in his talk, but is there such a list somewhere? $\endgroup$– uhohCommented Dec 12, 2023 at 23:49