Local existence, lower mass bounds, and a new continuation criterion for the Landau equation
Local existence, lower mass bounds, and a new continuation criterion for the Landau equation
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DOI：
10.1016/j.jde.2018.08.005
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发表时间：
2017-12
影响因子：
2.4
通讯作者：
Christopher Henderson;Stanley Snelson;Andrei Tarfulea
中科院分区：
文献类型：
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作者：
Christopher Henderson;Stanley Snelson;Andrei Tarfulea
我们考虑具有软势的空间非齐次朗道方程。首先，我们建立解的短时存在性，假设初始数据在速度变量和规律性方面具有足够的衰减（在空间变量中没有做出衰减假设）。接下来，我们展示了进化瞬间将质量传播到整个域。由此产生的下界是亚高斯的，我们证明这是最优的。大规模传播的证明基于随机过程，并充分利用了非定域性。通过将该定理与先前的结果相结合，我们得出了两个重要的应用：C∞-平滑，即使对于具有真空区域的初始数据也是如此，以及连续准则（只要质量和能量密度保持在上面的范围内，解就可以扩展）。这是已知的防止爆炸的最弱条件。特别是，它不需要质量密度的下限或熵密度的上限。
We consider the spatially inhomogeneous Landau equation with soft potentials. First, we establish the short-time existence of solutions, assuming the initial data has sufficient decay in the velocity variable and regularity (no decay assumptions are made in the spatial variable). Next, we show that the evolution instantaneously spreads mass throughout the domain. The resulting lower bounds are sub-Gaussian, which we show is optimal. The proof of mass-spreading is based on a stochastic process, and makes essential use of nonlocality. By combining this theorem with prior results, we derive two important applications: C∞-smoothing, even for initial data with vacuum regions, and a continuation criterion (the solution can be extended as long as the mass and energy densities stay bounded from above). This is the weakest condition known to prevent blow-up. In particular, it does not require a lower bound on the mass density or an upper bound on the entropy density.