AI 中文总结
该研究证明了-3<γ≤1范围内带角截断势的一维二元气体混合物玻尔兹曼方程小振幅行波激波剖面的存在，扩展了经典构造，采用多种方法克服软势相关问题，得到激波剖面的渐近行为。
AI 中文摘要
我们证明了在-3<γ≤1的整个范围内，具有角截断势的一维二元气体混合物玻尔兹曼方程存在小振幅行波激波剖面。该结果将Caflisch和Nicolaenko的经典构造从硬势扩展到了截断软势区域。证明过程结合了宏观分量的Lyapunov-Schmidt约化（约化为伯格斯方程）与加速反向双特征线方法、加权L²-L∞迭代。加速过程恢复了一致正的碰撞频率，弥补了软势缺乏谱间隙的问题；L²-L∞框架则适配了截断导致的速度平滑缺失，包括沿掠射特征线v₁=s的可能奇点。当|x|→∞时，激波剖面以混合指数速率趋近于兰金-于戈尼奥双麦克斯韦分布，带有阶为|εx|^(2/(3−γ))的次指数余项。
英文摘要
We prove the existence of small-amplitude traveling shock profiles for the one-dimensional Boltzmann equation of a binary gas mixture with angular cutoff potentials in the full range $-3<γ\le 1$. The result extends the classical construction of Caflisch and Nicolaenko from hard potentials to the cutoff soft-potential regime. Indeed, the argument of proofs combines a Lyapunov--Schmidt reduction of the macroscopic component to a Burgers equation with an accelerated backward bi-characteristic method and a weighted $L^2$--$L^\infty$ iteration. Acceleration restores a uniformly positive collision frequency, compensating for the lack of a spectral gap for soft potentials, while the $L^2$--$L^\infty$ framework accommodates the absence of velocity smoothing induced by the cutoff, including a possible singularity along the grazing characteristic $v_1=s$. The shock profile tends to the Rankine--Hugoniot bi-Maxwellians at a mixed exponential rate as $|x|\to \infty$, with a sub-exponential remainder of order $|\varepsilon x|^{2/(3-γ)}$.
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