A quantitative version of the transversality theorem

2021/12/14 by Andrew Murdza, Murdza, Andrew, Khai T. Nguyen +1
Mathematics · #35L67 #46T20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2112.07107

openalex publication_date 2021/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The present paper studies a quantitative version of the transversality theorem. More precisely, given a continuous function g∈ C([0,1]^d,ℝ^m) and a global smooth manifold W⊂ ℝ^m of dimension p, we establish a quantitative estimate on the (d+p-m)-dimensional Hausdorff measure of the set Z_W^g=\x∈ [0,1]^d: g(x)∈ W\. The obtained result is applied to quantify the total number of shock curves in weak entropy solutions to scalar conservation laws with uniformly convex fluxes in one space dimension.

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