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I am currently investigating an equation that is : $$\partial ^2_t u = \Delta _{3/2} u,$$ where $u : \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}$ and $\Delta _{3/2}$ stands for the $3/2$-laplacian i.e $\Delta _{3/2} u := \operatorname{div} (|\nabla u| ^{1/2} \nabla u)$.

I am looking for a Morawetz type inequality : in the setting of the linear wave equation, and if $u$ denotes a solution to this pde, it is known that : $$ \int _0^T \int_{\mathbb{R}^d} \frac{|\nabla \!\!\!\! / u|^2}{|x|}\, \mathrm{d}x \, \mathrm{d}t \leqslant C$$ which is the standard Morawetz inequality for the linear wave equation.

I tried to follows the method presented in Tao's book "Nonlinear Dispersive Equations; Local and Global analysis", page 161.

For the record it consists in finding the stress energy tensor $T^{ij}$ : $T^{00}$ must be the energy, and the other quantities satisfied : $$\partial _t T^{00} + \partial _ j T^{0j}=0,$$ and $$\partial T^{0j} + \partial _k T^{jk}=0.$$ Then the method consists in computing : $$\sum _j \int_{\mathbb{R}^d} \frac{x_j}{|x|} T^{0j} \, \mathrm{d}x ,$$ and hoping for good by parts integration, and finaly get a Morawetz type inequliaty.

In my setting, $T^{00}=\frac{(\partial _t u)^2}{2} + \frac{2}{5} |\nabla u|^{5/2}$ so that $T^{0j}=-\partial _t u |\nabla u|^{1/2} \partial _{x_j} u$ but we can not find the $T^{jk}$ corresponding and the the method of Tao doesn't give anything that I could use.

Is there a way to prove a Morawetz type inequality for this equation ?

Thank you.

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  • $\begingroup$ The method presented in Tao's book is really well adapted to linear equations; for your quasilinear equation you cannot hope for something so naive. One of the first things to observe is that for your equation at the points where $\nabla u = 0$ your equation is degenerate as a hyperbolic PDE (wave speed = 0); the local wellposedness of the equation is already tricky, and I am not even sure why you expect a Morawetz type inequality will hold true. The $p$-Laplacian improves your equation for high frequencies, but in the low frequency regime $\endgroup$
    – Willie Wong
    Commented Apr 21, 2017 at 12:48
  • $\begingroup$ it looks like it makes the matter worse. $\endgroup$
    – Willie Wong
    Commented Apr 21, 2017 at 12:49
  • $\begingroup$ @WillieWong I know that the method presented in Tao's book is well designed for linear (and semilinear) equations, but I do not know any other method in order to find Morawetz identities. Even if the local well posedness is not easy one may expect a morawetz identity because this equation really looks like a wave equation, for which Morawetz identities hold. Maybe this is a bad heuristic, but in my opinion finding a Morawetz identity was not fatally bounded to fail ! $\endgroup$
    – M.LTA
    Commented Apr 21, 2017 at 14:19
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    $\begingroup$ What I am worried about is whether a finite speed of propagation result holds true. In particular, it seems to me that there is a possibility a statement of the form "if $\Omega\subset\mathbb{R}^d$ is an open set such that $u(0,x) = u_t(0,x) = 0$ for all $x\in \Omega$, then $u(t,x) = 0$ for all $t$ and all $x\in \Omega$." (This is suggested by the fact that the wave speed vanishes when $\nabla u = 0$.) If such a statement is true (it is in fact true for higher degree nonlinearities by the paper I cited for you in my previous comment, I am not sure about your specific case), then solutions $\endgroup$
    – Willie Wong
    Commented Apr 21, 2017 at 15:06
  • $\begingroup$ cannot possibly disperse and conservation of energy will almost certainly gurantee that the correct analogous Morawetz cannot hold for any initial data of compact support. $\endgroup$
    – Willie Wong
    Commented Apr 21, 2017 at 15:07

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