Abstract
We derive \(L^2([0,T);H_{loc}^{\alpha /2}(\mathbb {R}^d))\), \(\alpha \in [1,2)\), apriori estimate for solutions to the fractional or anomalous diffusion equation using a generalization of the Leibnitz rule for the fractional Laplacean. The equation models a wide range of physical phenomena and, in particular, it is a linearized variant of the fractional porous media equation. The apriori estimates can be further used to rate convergence of corresponding numerical schemes, in the control and optimization theory and for various non-linear fractional PDEs. We use them here to prove existence of solution to a Cauchy problem for the fractional porous media equationas well as a result concerning optimal control.
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This work was partially supported by project P30233 of the Austrian Science Fund FWF.
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Burazin, K., Mitrovic, D. Apriori estimates for fractional diffusion equation. Optim Lett 13, 1793–1801 (2019). https://doi.org/10.1007/s11590-018-1332-0
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DOI: https://doi.org/10.1007/s11590-018-1332-0