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.
Entropy dissipation estimates for the Landau equation with Coulomb potentials
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abstract
We prove a lower bound for the entropy dissipation of the Landau equation with Coulomb potentials by a weighted Lebesgue norm $L^3_{-5/3}$. In particular, we enhance the weight exponent from $-5$, which was established by Desvillettes, to $-5/3$. Moreover, we prove that the weighted Lebesgue norm $L^3_{-5/3}$ is optimal for both exponents.
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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.