AI 中文总结
研究可压缩流一维信息几何正则化中激波解的相关问题，通过行波假设将方程简化，证明了跨声速压缩IGR激波剖面的存在性、唯一性和平移模下正则性，分析了正则化消失极限，表明激波宽度缩放规律及IGR剖面收敛情况。
AI 中文摘要
信息几何正则化（IGR）是可压缩欧拉方程的无粘正则化，它改变拉格朗日特征的几何结构以防止轨迹在有限时间内交叉。先前关于IGR的工作建立了一维全局强解，探索了模型的热力学效应并实现了可压缩流的大规模模拟。然而，一个基本问题仍然存在，即这种正则化如何改变类激波解的结构和正则性。我们证明了一维跨声速压缩IGR激波剖面的存在性、平移模下的唯一性和正则性。该分析适用于具有一般状态方程的完整热力学可压缩欧拉 - IGR模型，且满足温和的凸性假设。行波假设将欧拉 - IGR方程简化为密度剖面的退化二阶标量方程。在声速交叉处，椭圆系数退化：密度剖面保持连续，但其导数发散。该剖面在远离这个单点处是经典解，而在退化处它保持量化的赫尔德和索伯列夫正则性。我们还分析了正则化消失极限，表明激波宽度像$\sqrt{\alpha}$一样缩放，并且IGR剖面收敛到熵容许的欧拉激波。
英文摘要
The information geometric regularization (IGR) is an inviscid regularization of the compressible Euler equations that alters the geometry of Lagrangian characteristics to prevent trajectories from crossing in finite time. Previous work on IGR established global strong solutions in one dimension, explored thermodynamic effects of the model, and enabled large-scale simulations of compressible flow. However, a fundamental question that remains is how this regularization alters the structure and regularity of a shock-like solution. We prove existence, uniqueness modulo translation, and regularity of transonic compressive IGR shock profiles in one spatial dimension. The analysis applies to the full thermodynamic compressible Euler--IGR model with a general equation of state, subject to mild convexity hypotheses. A traveling-wave ansatz reduces the Euler--IGR equations to a degenerate second-order scalar equation for the density profile. At the sonic crossing, the elliptic coefficient degenerates: the density profile remains continuous, but its derivative diverges. The profile is a classical solution away from this single point, while at the degeneracy it retains quantified Hölder and Sobolev regularity. We also analyze the vanishing-regularization limit, showing that the shock width scales like $\sqrt{α}$ and that the IGR profiles converge to the entropy-admissible Euler shock.
