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Dissipation estimates of the Fisher information for the Landau equation
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abstract
We establish an a priori estimate for the dissipation of the Fisher information for the space-homogeneous Landau equation with very soft potentials. This work is motivated by the recent breakthrough by Guillen and Silvestre, which proves that the Fisher information is monotone decreasing. As a direct consequence, we show that the Fisher information becomes instantaneously bounded, even if it is not initially bounded. This leads to a proof of the global existence of smooth solutions for the space-homogeneous Landau equation with very soft potentials, given initial data $f_0 \in L^1_{2-\gamma} \cap L \log L$. This result includes the case of the Coulomb potential.
Forward citations
Cited by 3 Pith papers
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Weak-strong uniqueness for the Landau equation by a relative entropy method
For the Landau equation with soft potentials (including Coulomb), this paper proves that all H-solutions with enough moments coincide with the smooth solution, via a relative entropy estimate.
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The fuzzy Landau equation: global well-posedness and Fisher information
The fuzzy Landau equation has unique global smooth solutions for moderately soft potentials, and its spatial Fisher information decreases monotonically over time.
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