Quantitative Stability and Regularity of Large Data for Conservation Laws

The proposed project will focus on the well-posedness theory for hyperbolic systems of conservation laws in multiple space dimensions.

The project will consider the stability and well-posedness theory for such systems, in the cases with and without source.In his 2023 survey of the field, Dafermos writes that ""in regard to systems [of conservation laws], in one spatial dimension, the fundamental question whether the Cauchy problem in the BV setting is well-posed for initial data with large total variation remains wide open.'' Our program to study quantitative stability will allow us to consider not only large BV solutions, but also large L^2 solutions with ""infinite BV.""Classically, the best theory of well-posedness for hyperbolic systems in one space dimension is the L^1 theory of Bressan and coworkers, which considers small-BV solutions.

Recent results of the researcher, Chen, and Vasseur go significantly beyond the classical small-BV theory, and are able to treat even large L^2 data.

The theory uses the technique of a-contraction, and the key assumption is a strong trace condition, a regularity assumption strictly weaker than BV_loc.

Our first main objective in this proposal is to quantify the stability in the a-contraction theory, which hasn't been done so far.

More precisely, Objective 1 (Quantitative stability): In the setting of large data, derive quantitative stability estimates between L^2 and BV solutions for a large class of systems in one space dimension.The strong trace condition is the key boundary between general weak solutions and the solutions we can show uniqueness for.

This brings us our Objective 2 (Regularity): For scalar conservation laws, possibly with nonlocal or unbounded source, under mild technical assumptions, show the existence of the strong traces.This is open even in the one dimensional scalar case with unbounded source. Showing the existence of strong traces would be a significant step towards the program of Dafermos.