Onsager-type conjecture and renormalized solutions for the relativistic Vlasov–Maxwell system
Onsager-type conjecture and renormalized solutions for the relativistic Vlasov–Maxwell system
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DOI：
10.1090/qam/1549
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发表时间：
2019-03
影响因子：
0.8
通讯作者：
C. Bardos;N. Besse;Toan T. Nguyen
中科院分区：
文献类型：
--
作者：
C. Bardos;N. Besse;Toan T. Nguyen
本文证明了关于相对论性Vlasov-麦克斯韦方程组弱解的能量守恒和熵的Onsager型猜想。关于弱解的正则性，比如在Sobolev空间W^{\alpha，p}$中，我们确定了保证所有熵守恒的Onsager型指数。特别是，Onsager指数$\alpha$小于为流体模型建立的$\alpha = 1/3$。熵守恒等价于DiPerna-Lions为研究被动输运方程和无碰撞动力学方程的适定性而引入的重整化性质。对于光滑解，重整化性质或熵守恒仅仅是链式法则的结果。对于弱解，链式法则的使用并不总是合理的。然后出现的问题，弱解，以保证这些性质所需的最小正则性。在DiPerna-Lions和Bouchut-Ambrosio理论中，重整化性质在平流场正则性的充分条件下成立，这些条件大致是某些Lebesgue空间中的整导数（DiPerna-Lions）或有限全变差测度空间中的整导数（Bouchut-Ambrosio）.作为回报，平流密度没有平滑性要求，除了一些自然的先验界限。在这里，我们表明，重整化性质持有的电磁场只有一个分数空间导数在某些勒贝格空间。为了补偿电磁场导数的这种损失，分布函数需要额外的平滑度，通常是相空间中的分数Sobolev可微性。关于总能量守恒，如果宏观动能是L^2，那么总能量守恒。
In this paper we give a proof of an Onsager type conjecture on conservation of energy and entropies of weak solutions to the relativistic Vlasov--Maxwell equations. As concerns the regularity of weak solutions, say in Sobolev spaces $W^{\alpha,p}$, we determine Onsager type exponents $\alpha$ that guarantee the conservation of all entropies. In particular, the Onsager exponent $\alpha$ is smaller than $\alpha = 1/3$ established for fluid models. Entropies conservation is equivalent to the renormalization property, which have been introduced by DiPerna--Lions for studying well-posedness of passive transport equations and collisionless kinetic equations. For smooth solutions renormalization property or entropies conservation are simply the consequence of the chain rule. For weak solutions the use of the chain rule is not always justified. Then arises the question about the minimal regularity needed for weak solutions to guarantee such properties. In the DiPerna--Lions and Bouchut--Ambrosio theories, renormalization property holds under sufficient conditions in terms of the regularity of the advection field, which are roughly speaking an entire derivative in some Lebesgue spaces (DiPerna--Lions) or an entire derivative in the space of measures with finite total variation (Bouchut--Ambrosio). In return there is no smoothness requirement for the advected density, except some natural a priori bounds. Here we show that the renormalization property holds for an electromagnetic field with only a fractional space derivative in some Lebesgue spaces. To compensate this loss of derivative for the electromagnetic field, the distribution function requires an additional smoothness, typically fractional Sobolev differentiability in phase-space. As concerns the conservation of total energy, if the macroscopic kinetic energy is in $L^2$, then total energy is preserved.