arXiv subjects

Cooper Faile

Publications and source records attributed to Cooper Faile.

4 recordsLinked to original sources

arXiv2026

The unique limit of the Glimm-Lax construction for Sobolev data and obstructions to 1-d convex integration

We consider a genuinely nonlinear $1$-d system of hyperbolic conservation laws with two unknowns. A famous construction of Glimm & Lax shows that global-in-time "Glimm-Lax" weak entropy solutions exist in this setting for any initial data with small $L^\infty$ norm [Mem. Amer. Math. Soc. (1970), no. 101]. Recent work in the $L^1$-stability theory by Bressan, Marconi & Vaidya has given the first partial uniqueness and stability results for these solutions [Arch. Ration. Mech. Anal. (2025), vol. 249]. In this paper, we build on these results by combining them with recent advances in the $L^2$-theory. We show that solutions with initial data in the Sobolev space $H^s$ for $s>0$ are unique in the full class of Glimm--Lax solutions that decay in total variation at a rate of $1/t$. As a secondary result, our techniques are also used to show the recent non-uniqueness result of Chen, Vasseur & Yu for continuous solutions (arxiv:2407.02927) cannot extend to $C^\alpha$ solutions for $\alpha > 1/2$, alongside some appropriate fractional Sobolev spaces $W^{s,p}$. An auxiliary result of independent interest is the development of a weighted relative entropy contraction for perturbations of rarefaction waves.

math.AP↗

arXiv2025

Quantitative weak-BV stability of "wild'' solutions to compressible Euler equations, with a view towards higher systems

For hyperbolic systems of conservation laws, including important physical models from continuum mechanics, the question of stability for large data solutions remains a challenging open problem. In recent work (arXiv:2507.23645) the authors introduce a framework for showing H\"older stability of potentially "wild" large data solutions, relative to a class of BV solutions, for systems with two conserved quantities. This is referred to as "weak-BV" stability. In this paper, we give a short introduction to the methods while applying them to the "full" Euler system with three conserved quantities. We discuss applications to future work for higher systems with additional conserved quantities.

math.AP↗

arXiv2025

Solutions to conservation laws are H\"older-stable in $L^2$ in the weak-BV setting

We consider hyperbolic systems of conservation laws in one spatial dimension. For any limit of front tracking solutions $v$, and for a general weak solution $u\in L^\infty$ with no BV assumption, we prove the following H\"older-type stability estimate in $L^2$: $$||u(\cdot,\tau)-v(\cdot,\tau)||_{L^2} \leq K \sqrt{||u( \cdot,0)-v( \cdot,0)||_{L^2}}$$ for all $\tau$ without smallness and for a universal constant $K$. Our result holds for all limits of front tracking solutions $v$ with BV bound, either for general systems with small-BV data, or for special systems (isothermal Euler, Temple-class systems) with large-BV data. Our results apply to physical systems such as isentropic Euler. The stability estimate is completely independent of the BV norm of the potentially very wild solution $u$. We use the $L^2$ theory of shock stability modulo an artificial shift of position (Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 -- 376, 2008]) but our stability results do not depend on an artificial shift. Moreover, we give the first result within this framework which can show uniqueness of some solutions with large $L^\infty$ and infinite BV initial data. We apply these techniques to isothermal Euler.

math.AP↗

arXiv2025

Necessary and sufficient conditions for $a$-contraction

In this paper we investigate the theory of $a$-contraction with shifts with the intention of extending it to intermediate families. The theory of $a$-contraction with shifts is used to prove orbital $L^2$ stability to shock solutions of conservation laws. In this setting there are strong results for scalar laws and the extremal families of $n\times n$ systems of conservation laws. The only known results showing contraction of interior families are for the contact family the full Euler system and the case of rich systems due to Serre and Vasseur '16. This investigation culminates in finding necessary and sufficient conditions for which small shocks of general systems are local attractors with respect to the $a$-contraction theory.

math.AP↗