朗道-库仑方程的麦基恩猜想的一个反例
A counterexample to McKean's conjecture for the Landau-Coulomb equation
- University of Rennes, Inria, CNRS, IRMAR – UMR 6625（雷恩大学）
- Cardiff University（卡迪夫大学）
- Delft University of Technology（代尔夫特理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对空间均匀朗道-库仑方程，研究人员构造出显式反例否定麦基恩关于熵耗散单调非增的猜想，且该反例也适用于玻尔兹曼方程，验证其猜想不成立。
AI中文摘要：
我们否定了麦基恩（McKean）的猜想，该猜想断言：对于空间均匀的朗道-库仑方程，熵耗散沿解单调非增，等价于熵关于时间是凸的。我们构造了一个显式反例，该反例由麦克斯韦平衡态及尺度R≫1的环带上的径向对称光滑扰动构成。将熵耗散的导数按R的幂展开后发现，对于适当选取的扰动，主导阶项可为正。值得注意的是，这些反例在相对熵和标准 Sobolev 范数下都接近平衡态。由于朗道方程是玻尔兹曼方程在 grazing 碰撞极限下的形式，我们的反例还表明麦基恩猜想对玻尔兹曼方程一般而言不成立。
英文摘要：
We disprove McKean's conjecture, which asserts that the entropy dissipation is monotone nonincreasing along solutions, or equivalently that the entropy is convex in time, for the spatially homogeneous Landau-Coulomb equation. We provide an explicit counterexample which consists of a Maxwellian equilibrium under radially symmetric, smooth perturbation on an annulus of scale $R\gg 1$. Developing the derivative of the entropy dissipation in powers of $R$ reveals that the leading order terms can be positive for a suitably chosen perturbation. Remarkably, the counterexamples are close to equilibrium in relative entropy and standard Sobolev norms. Since the Landau equation arises from the Boltzmann equation in the grazing collisions limit, our counterexample also shows that McKean's conjecture is not true in general for the Boltzmann equation.
