The inhomogeneous Landau equation with hard potentials
The inhomogeneous Landau equation with hard potentials
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发表时间：
2018-05
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通讯作者：
Stanley Snelson
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作者：
Stanley Snelson
我们考虑具有硬势（\(\gamma\in(0,1]\)）的空间非齐次朗道方程的弱解，假设质量、能量和熵密度是可控的。在这种情形下，对于任意初始数据，我们表明解在速度变量中满足逐点的高斯上下界。这与软势情形（\(\gamma<0\)）中的行为不同，在软势情形中，已知如果对初始数据没有相应假设，高斯估计不成立。我们的上界意味着弱解在所有三个变量中都是\(C^\infty\)的，并且解的延拓仅由质量、能量和熵控制。
We consider weak solutions of the spatially inhomogeneous Landau equation with hard potentials ($\gamma \in (0,1]$), under the assumption that mass, energy, and entropy densities are under control. In this regime, with arbitrary initial data, we show that solutions satisfy pointwise Gaussian upper and lower bounds in the velocity variable. This is different from the behavior in the soft potentials case ($\gamma <0$), where Gaussian estimates are known not to hold without corresponding assumptions on the initial data. Our upper bounds imply weak solutions are $C^\infty$ in all three variables, and that continuation of solutions is governed only by the mass, energy, and entropy.