论文标题
局部规律性的，具有非常柔软电势的空间均匀的兰道方程
Local regularity for the space-homogeneous Landau equation with very soft potentials
论文作者
论文摘要
本文介绍了带有非常柔软的潜力的空间源性Landau方程，包括库仑案。该非线性方程是抛物线类型的，由矩阵$ a_ {ij}（z）= | z | |^γ（| z | |^2Δ_{ij} - z_iz_jj）$ in [-3，-2）$ in [-3，2）$ for solix $ a_ {ij}（z）= | z |^2Δ__{ij}-2Δ_{ij}Δ_{ij {ij} - Z__IS_JJ J \我们得出局部截断的熵估计，并使用它们来建立两个事实。首先，我们证明了像[C. C.维拉尼，拱门。理性的机甲。肛门。 143（1998），273-307]具有零$ \ mathscr {p}^{m_ \ ast} $ parrabolic hausdorff测量$ m_ \ ast：= \ frac72 | 2+γ| $。其次，我们证明，如果这样的弱溶液是轴对称的，那么它远离对称轴。特别是，径向对称的弱解远离原点。
This paper deals with the space-homogenous Landau equation with very soft potentials, including the Coulomb case. This nonlinear equation is of parabolic type with diffusion matrix given by the convolution product of the solution with the matrix $a_{ij} (z)=|z|^γ(|z|^2 δ_{ij} - z_iz_j)$ for $γ\in [-3,-2)$. We derive local truncated entropy estimates and use them to establish two facts. Firstly, we prove that the set of singular points (in time and velocity) for the weak solutions constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has zero $\mathscr{P}^{m_\ast}$ parabolic Hausdorff measure with $m_\ast:= \frac72 |2+γ|$. Secondly, we prove that if such a weak solution is axisymmetric, then it is smooth away from the symmetry axis. In particular, radially symmetric weak solutions are smooth away from the origin.
