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We consider the Cauchy problem for the equations of pressureless gases in two space dimensions. For a generic set of smooth initial data (density and velocity), it is known that the solution loses regularity at a finite time $t_0$, where both the the density and the velocity gradient become unbounded. Aim of this paper is to provide an asymptotic description of the solution beyond the time of singularity formation. For $t>t_0$ we show that a singular curve is formed, where the mass has positive density w.r.t.1-dimensional Hausdorff measure. The system of equations describing the behavior of the singular curve is not hyperbolic. Working within a class of analytic data, local solutions can be constructed using a version of the Cauchy-Kovalevskaya theorem. For this purpose, by a suitable change of variables we rewrite the evolution equations as a first order system of Briot-Bouquet type, to which a general existence-uniqueness theorem can then be applied., Comment: This paper contains 5 figures

Automatic measurements of tear meniscus height (TMH) have been achieved by using deep learning techniques; however, annotation is significantly influenced by subjective factors and is both time-consuming and labor-intensive. In this paper, we introduce an automatic TMH measurement technique based on edge detection-assisted annotation within a deep learning framework. This method generates mask labels less affected by subjective factors with enhanced efficiency compared to previous annotation approaches. For improved segmentation of the pupil and tear meniscus areas, the convolutional neural network Inceptionv3 was first implemented as an image quality assessment model, effectively identifying higher-quality images with an accuracy of 98.224%. Subsequently, by using the generated labels, various algorithms, including Unet, ResUnet, Deeplabv3+FcnResnet101, Deeplabv3+FcnResnet50, FcnResnet50, and FcnResnet101 were trained, with Unet demonstrating the best performance. Finally, Unet was used for automatic pupil and tear meniscus segmentation to locate the center of the pupil and calculate TMH,respectively. An evaluation of the mask quality predicted by Unet indicated a Mean Intersection over Union of 0.9362, a recall of 0.9261, a precision of 0.9423, and an F1-Score of 0.9326. Additionally, the TMH predicted by the model was assessed, with the fitting curve represented as y= 0.982x-0.862, an overall correlation coefficient of r^2=0.961 , and an accuracy of 94.80% (237/250). In summary, the algorithm can automatically screen images based on their quality,segment the pupil and tear meniscus areas, and automatically measure TMH. Measurement results using the AI algorithm demonstrate a high level of consistency with manual measurements, offering significant support to clinical doctors in diagnosing dry eye disease., Comment: 22 pages, 5 figures

In this paper, we consider the Cauchy problem for pressureless gases in two space dimensions with generic smooth initial data (density and velocity). These equations give rise to singular curves, where the mass has positive density w.r.t.~1-dimensional Hausdorff measure. We observe that the system of equations describing these singular curves is not hyperbolic. For analytic data, local solutions are constructed using a version of the Cauchy-Kovalevskaya theorem. We then study the interaction of two singular curves, in generic position. Finally, for a generic initial velocity field, we investigate the asymptotic structure of the smooth solution up to the first time when a singularity is formed., Comment: The manuscript contains 4 figures