2024/11/22 by Ruijia Cao, Florian Schäfer, Cao, Ruijia +1 · 3 citations
Engineering · Mathematics · #35L65 #49J45 #49Q22 #58B20 #76L05 #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #Differential Geometry (math.DG) #FOS: Mathematics #Navier-Stokes equation solutions #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2411.15121
openalex publication_date 2024/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Partial differential equations describing compressible fluids are prone to the formation of shock singularities, arising from faster upstream fluid particles catching up to slower, downstream ones. In geometric terms, this causes the deformation map to leave the manifold of diffeomorphisms. Information geometric regularization addresses this issue by changing the manifold geometry to make it geodesically complete. Empirical evidence suggests that this results in smooth solutions without adding artificial viscosity. This work makes a first step towards understanding this phenomenon rigorously, in the setting of the unidimensional pressureless Euler equations. It shows that their information geometric regularization has smooth global solutions. By establishing Γ-convergence of its variational description, it proves convergence of these solutions to entropy solutions of the nominal problem, in the limit of vanishing regularization parameter. A consequence of these results is that manifolds of unidimensional diffeomorphisms with information geometric regularization are geodesically complete.