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**Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential.**
*(English)*
Zbl 1378.35325

Summary: Motivated by the question of existence of global solutions, we obtain pointwise upper bounds for radially symmetric and monotone solutions to the homogeneous Landau equation with Coulomb potential. The estimates say that blow-up in the \(L^\infty\) norm at some finite time \(T\) occurs only if a certain quotient involving \(f\) and its Newtonian potential concentrates near zero, which implies blow-up in more standard norms, such as the \(L^{3/2}\) norm. This quotient is shown to be always less than a universal constant, suggesting that the problem of regularity for the Landau equation is in some sense critical. The bounds are obtained using the comparison principle both for the Landau equation and for the associated mass function. In particular, the method provides long-time existence results for a modified version of the Landau equation with Coulomb potential, recently introduced by J. Krieger and R. M. Strain [Commun. Partial Differ. Equations 37, No. 4–6, 647–689 (2012; Zbl 1247.35087)].

### MSC:

35R11 | Fractional partial differential equations |

35B65 | Smoothness and regularity of solutions to PDEs |

35K57 | Reaction-diffusion equations |

35B44 | Blow-up in context of PDEs |

35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |

35Q20 | Boltzmann equations |

35Q70 | PDEs in connection with mechanics of particles and systems of particles |