I am interested in the following family of gradient flows on $\mathbb{R}^d$, $d\geq 1$:
$$\begin{cases}\partial_t u = \nabla\cdot(u\nabla|\nabla|^{-\alpha} u) \\ u\geq 0,\end{cases} \qquad d>\alpha>0. \tag{1}\label{eq}$$ Here, $|\nabla|^{-\alpha}$ is the Fourier multiplier with symbol $|\xi|^{-\alpha}$.
When $0<\alpha<2$, this equation seems referred to as a fractional porous medium equation. When $\alpha=2$ and $d=2$, the equation is related to the Chapman-Rubinstein-Schatzman mean-field model of superconductivity, as well as E's model of superfluidity. I am interested in the range $\alpha>2$, which to my knowledge, is not considered in the literature. Specifically, I am interested in temporal decay estimates for solutions $u$ to \eqref{eq} of the form $$\|u(t)\|_{L^\infty} \leq \frac{C(\alpha,d,\|u_0\|)}{t^{\gamma}}, \qquad \forall t>0 \tag{2}\label{dcay}$$ where $\gamma>0$ is some exponent presumably depending on $\alpha, d$ and $C$ is a constant depending on $\alpha, d$ and possibly some $L^p$ norm of the initial datum $u_0$.
Let me summarize what seems to be known in the cases $0<\alpha\leq 2$.
- If $\alpha=2$, then it is a result of Lin, Fanghua; Zhang, Ping, On the hydrodynamic limit of Ginzburg-Landau vortices., Discrete Contin. Dyn. Syst. 6, No. 1, 121-142 (2000). ZBL1034.35128. and Serfaty, Sylvia; Vázquez, Juan Luis, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators, Calc. Var. Partial Differ. Equ. 49, No. 3-4, 1091-1120 (2014). ZBL1290.35316. that the bound \eqref{dcay} holds with $\gamma=1$ and constant $C$ only depending on $\alpha,d$.
- If $0<\alpha<2$, then Caffarelli, Luis A.; Soria, Fernando; Vázquez, Juan Luis, Regularity of solutions of the fractional porous medium flow, J. Eur. Math. Soc. (JEMS) 15, No. 5, 1701-1746 (2013). ZBL1292.35312. and Biler, Piotr; Imbert, Cyril; Karch, Grzegorz, The nonlocal porous medium equation: Barenblatt profiles and other weak solutions, Arch. Ration. Mech. Anal. 215, No. 2, 497-529 (2015). ZBL1308.35197. have independently shown a bound of the form
$$\|u(t)\|_{L^\infty} \lesssim_{\alpha,d} \frac{\|u_0\|_{L^1}^{\frac{2-\alpha}{d+2-\alpha}}}{t^{\frac{d}{d+2-\alpha}}}, \qquad \forall t>0$$ for all suitable weak solutions to equation \eqref{eq}.
Question. Does a decay estimate of the form \eqref{dcay} hold for solutions of \eqref{eq} if $2<\alpha<d$?
I am not a specialist in this area, so forgive me if I am missing something, but looking at the proofs of the aforementioned works treating the range $0<\alpha\leq 2$, it seems that their arguments break down for the range in which I am interested. The work of Lin and Zhang exploits the fact that if $\alpha=2$, then equation \eqref{eq} can be written as transport plus an absorption term. Introducing a new unknown $\tilde{u}$ for which the transport has been removed, we have the equation $$\partial_t\tilde{u} = -\tilde{u}^2,$$ from which the decay bound follows. The proofs of Caffarelli et al. and Biler et al. are rather different: the former uses a de Georgi-type argument and the scaling invariance of equation \eqref{eq}, while the latter uses the Stroock–Varopoulos inequality and an iteration argument. But both results essentially rely on the elementary fact that $$-\Delta|\nabla|^{-\alpha} = |\nabla|^{2-\alpha}$$ is the usual fractional Laplacian if $0<\alpha<2$. While if $2<\alpha<d$, then we instead have a Riesz potential.
