发表机构
Koç University（科奇大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究二维不可压缩 Euler-Maxwell 系统，发现投影无电荷系统在亚光速条件下具有局部适定性，而未投影系统中电荷密度导致涡量范数膨胀，揭示了高斯约束对正则性与膨胀的区分作用。
AI 中文摘要
我们研究了法向几何下的二维不可压缩 Euler-Maxwell 系统，其中速度和电场为平面场，磁场垂直于平面。我们识别了投影的无电荷公式与未投影系统之间的结构差异。对于投影系统，我们在速度满足严格亚光速条件下，建立了初始速度在 $H^1(\mathbb R^2)$ 中且初始涡量有界、电磁数据在 $H^{\frac 32}(\mathbb R^2)$ 中的局部适定性理论。主要工具是半波的端点类时迹估计，该估计沿每条流体轨迹在 $L_t^2$ 中控制磁场梯度。这在不要求电磁场具有欧拉 Lipschitz 界的情况下闭合了 Yudovich 涡量估计。对于未投影系统，纵向电模式产生一个包含电荷密度 $\operatorname{div }E$ 的额外涡量源项。在这种情况下，我们构造了光滑、紧支撑的径向电磁数据，这些数据在 $H^{\frac32}(\mathbb R^2)$ 中收敛到零，且初始速度为零，其唯一全局光滑解在 $L^\infty_x$ 中表现出涡量范数膨胀。因此，在相同的电磁 Sobolev 正则性下，高斯约束将端点正则性传播与电荷驱动的涡量范数膨胀区分开来。
英文摘要
We study the two-dimensional incompressible Euler-Maxwell system under the normal geometry in which the velocity and electric field are planar and the magnetic field is normal to the plane. We identify a structural distinction between the projected, charge-free formulation and the unprojected system. For the projected system, we establish a local well-posedness theory for initial velocities in $H^1(\mathbb R^2)$ with bounded initial vorticity and electromagnetic data in $H^{\frac 32}(\mathbb R^2)$, under a strict sub-luminal condition on the velocity. The main ingredient is an endpoint time-like trace estimate for half-waves, which controls the magnetic gradient in $L_t^2$ along every fluid trajectory. This closes the Yudovich vorticity estimate without requiring an Eulerian Lipschitz bound on the electromagnetic field. For the unprojected system, the longitudinal electric mode produces an additional vorticity source term containing the charge density $\operatorname{div }E$. In this case, we construct smooth, compactly supported radial electromagnetic data converging to zero in $H^{\frac32}(\mathbb R^2)$, with zero initial velocity, whose unique global smooth solution exhibits a vorticity norm inflation in $L^\infty_x$. Thus, at the same electromagnetic Sobolev regularity, the Gauss constraint separates endpoint regularity propagation from charge-driven vorticity norm inflation.
Comments22 pages
