Communications in Mathematical Sciences

Volume 21 (2023)

Number 5

A quantitative version of the transversality theorem

Pages: 1303 – 1320

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n5.a5

Authors

Andrew Murdza (Department of Mathematics, North Carolina State University, Raleigh, N.C., U.S.A.)

Khai T. Nguyen (Department of Mathematics, North Carolina State University, Raleigh, N.C., U.S.A.)

Abstract

The present paper studies a quantitative version of the transversality theorem. More precisely, given a continuous function $g \in \mathcal{C} {(\lbrace 0,1 \rbrace}^d , \mathbb{R}^m)$ and a global smooth manifold $W \subset \mathbb{R}^m)$ of dimension $p$, we establish a quantitative estimate on the $(d+p-m)$-dimensional Hausdorff measure of the set $\mathcal{Z}^g_W = {\lbrace x \in {(\lbrace 0,1 \rbrace}^d : g(x) \in W \rbrace}$. 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.

Keywords

transversality lemma, quantitative estimates, conservation laws

2010 Mathematics Subject Classification

35L67, 46T20

Received 21 December 2021

Received revised 26 August 2022

Accepted 30 September 2022

Published 30 August 2023