A new monotonicity formula for the spatially homogeneous Landau equation with Coulomb potential and its applications. (English) Zbl 1541.35109
Summary: We describe a time-dependent functional involving the relative entropy and \(\dot{H}^1 \) the seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functional sheds light on the competition between dissipation and nonlinearity for this equation. It enables us to obtain new results concerning regularity/blowup issues for the Landau equation with Coulomb potential.
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35K67 | Singular parabolic equations |
| 35R09 | Integro-partial differential equations |
| 45G05 | Singular nonlinear integral equations |
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 82D10 | Statistical mechanics of plasmas |
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