理想磁流体力学中的Onsager猜想
Onsager's Conjecture for Ideal Magnetohydrodynamics
中文总结 AI 辅助
该研究构造了理想MHD方程在$\gamma<1/3$下的弱解，不守恒能量和交叉螺旋度，建立各向异性Hölder界，扩展了近期结果并简化了证明。
中文摘要 AI 辅助
对于任意$0\le\gamma<1/3$，我们构造了理想MHD方程的弱解$(v,B,p)$，其中$v,B\in C^\gamma(\mathbb T^3\times\mathbb R)$，这些解既不守恒总能量也不守恒交叉螺旋度，且具有非平凡的磁螺旋度。我们还建立了具有不同速度和磁场指数的各向异性Hölder界，并证明了沿每条磁场线具有更强的正则性。可容许的指数将Goldreich--Sridhar \cite{GoldreichSridhar1995}的空间和并行对$1/3, 1/2$与Iroshnikov--Kraichnan弱湍流图景\cite{Iroshnikov1963,Kraichnan1965}的$1/4$空间标度联系起来。秉承Arnold对理想流体动力学的表述精神，解被视为体积保持微分同胚的路径；证明则基于经典凸积分技术与该李群李代数层面几何构造之间的相互作用。我们的工作实质性地扩展了Enciso、Peñafiel-Tomás和Peralta-Salas \cite{EnPePe}的最新结果，并可用于在不使用Newton迭代的情况下重新证明Giri和Radu的\cite{GiRa}。
英文摘要
For any $0\leγ<1/3$ we construct weak solutions $(v,B,p)$ of the ideal MHD equations with $v,B\in C^γ(\mathbb T^3\times\mathbb R)$, which conserve neither the total energy nor the cross-helicity and have nontrivial magnetic helicity. We also establish anisotropic Hölder bounds with distinct velocity and magnetic exponents and stronger regularity along every magnetic field line. The admissible exponents connect the Goldreich--Sridhar \cite{GoldreichSridhar1995} spatial and parallel pair $1/3, 1/2$ with the $1/4$ spatial scaling of the Iroshnikov--Kraichnan weak-turbulence picture \cite{Iroshnikov1963,Kraichnan1965}. In the spirit of Arnold's formulation of ideal hydrodynamics, a solution is regarded as a path of volume-preserving diffeomorphisms; the proof is then based on the interplay between classical convex integration techniques and geometric constructions at the level of the Lie algebra of this Lie group. Our work substantially extends the recent result of Enciso, Peñafiel-Tomás and Peralta-Salas \cite{EnPePe} and can be used to reprove \cite{GiRa} of Giri and Radu without a Newton iteration.
