arXiv subjects

Sam G. Krupa

Publications and source records attributed to Sam G. Krupa.

14 recordsLinked to original sources

arXiv2026

The computational ansatz for convex integration of hyperbolic systems and a resolution of the Strong Trace Conjecture

In this paper, we consider $2\times2$ hyperbolic systems of conservation laws in one spatial dimension. We use the computational ansatz introduced in the hyperbolic theory by the author and Székelyhidi to study the constitutive set corresponding to the PDE. The rank-one convex geometry of this set relates to non-uniqueness and the existence of low-regularity solutions. Through a computer-assisted search, we find a pressure law $p$ such that the $p$-system with this pressure law, and its natural strictly convex entropy, verifies all of the conditions necessary for the large data $L^2$ stability and the technique of ``$a$-contraction with shifts'' and thus we have uniqueness of certain Riemann solutions in the class of solutions verifying the Strong Trace Property. At the same time, the constitutive set contains a $T_6$ configuration and we use it to construct non-unique solutions without the Strong Trace Property. This resolves the question of sharpness of strong traces. We also present proofs which show nonexistence of $T_\infty$ structures for all genuinely nonlinear systems and nonexistence of $T_N$ structures for all $N$ for the $p$-system with $p''>0$, thus blocking these routes towards convex integration.

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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 $C^α$ solutions for $α> 1/2$, 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 Hölder 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 Hölder-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 Hölder-type stability estimate in $L^2$: $$||u(\cdot,τ)-v(\cdot,τ)||_{L^2} \leq K \sqrt{||u( \cdot,0)-v( \cdot,0)||_{L^2}}$$ for all $τ$ 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↗

arXiv2024

Contact discontinuities for 2-D isentropic Euler are unique in 1-D but wildly non-unique otherwise

We develop a general framework for studying non-uniqueness of the Riemann problem for the isentropic compressible Euler system in two spatial dimensions, and in this paper we present the most delicate result of our method: non-uniqueness of the contact discontinuity. Our approach is computational, and uses the pressure law as an additional degree of freedom. The stability of the contact discontinuities for this system is a major open problem (see Gui-Qiang Chen and Ya-Guang Wang [Nonlinear partial differential equations, volume 7 of Abel Symposia. Springer, Heidelberg, 2012.]). We find a smooth pressure law $p$, verifying the physically relevant condition $p'>0$, such that for the isentropic compressible Euler system with this pressure law, contact discontinuity initial data is wildly non-unique in the class of bounded, admissible weak solutions. This result resolves the question of uniqueness for contact discontinuity solutions in the compressible regime. Moreover, in the same regularity class in which we have non-uniqueness of the contact discontinuity, i.e. $L^\infty$, with no $BV$ regularity or self-similarity, we show that the classical contact discontinuity solution to the two-dimensional isentropic compressible Euler system is in fact unique in the class of bounded, admissible weak solutions if we restrict to 1-D solutions.

math.AP↗

arXiv2024

Finite time BV blowup for Liu-admissible solutions to $p$-system via computer-assisted proof

In this paper, we consider finite time blowup of the $BV$-norm for exact solutions to genuinely nonlinear hyperbolic systems in one space dimension, in particular the $p$-system. We consider solutions verifying shock admissibility criteria such as the Lax E-condition and the Liu E-condition. In particular, we present Riemann initial data which admits infinitely many bounded solutions, each of which experience, not just finite time, but in fact instantaneous blowup of the $BV$ norm. The Riemann initial data is allowed to come from an open set in state space. Our method provably does not admit a strictly convex entropy. The main results in this article compare to Jenssen [SIAM J. Math. Anal., 31(4):894--908, 2000], who shows $BV$ blowup for bounded solutions, or alternatively, blowup in $L^\infty$, for an artificial $3\times 3$ system which is not genuinely nonlinear. Baiti-Jenssen [Discrete Contin. Dynam. Systems, 7(4):837--853, 2001] improves upon this Jenssen result and can consider a genuinely nonlinear system, but then the blowup is only in $L^\infty$ and they cannot construct bounded solutions which blowup in $BV$. Moreover, their system is non-physical and provably does not admit a global, strictly convex entropy. Our result also shows sharpness of the recent Bressan-De Lellis result [Arch. Ration. Mech. Anal., 247(6):Paper No. 106, 12, 2023] concerning well-posedness via the Liu E-condition. The proof of our theorem is computer-assisted, following the framework of Székelyhidi [Arch. Ration. Mech. Anal., 172(1):133--152, 2004]. Our code is available on the GitHub.

math.AP↗

arXiv2023

Theory of shifts, shocks, and the intimate connections to $L^2$-type a posteriori error analysis of numerical schemes for hyperbolic problems

In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension. Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates. We work entirely within the context of the theory of shifts and $a$-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for $2\times 2$ conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022]. Our theoretical findings are complemented by a numerical implementation in MATLAB and numerical experiments.

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arXiv2022

Nonexistence of $T_4$ configurations for hyperbolic systems and the Liu entropy condition

We study the constitutive set $\mathcal{K}$ arising from a $2\times 2$ system of conservation laws in one space dimension, endowed with one entropy and entropy-flux pair. The convexity properties of the set $\mathcal{K}$ relate to the well-posedness of the underlying system and the ability to construct solutions via convex integration. Relating to the convexity of $\mathcal{K}$, in the particular case of the $p$-system, Lorent and Peng [Calc. Var. Partial Differential Equations, 59(5):Paper No. 156, 36, 2020] show that $\mathcal{K}$ does not contain $T_4$ configurations. Recently, Johansson and Tione [arXiv e-prints, page arXiv:2208.10979, August 2022] showed that $\mathcal{K}$ does not contain $T_5$ configurations. In this paper, we provide a substantial generalization of these results, based on a careful analysis of the shock curves for a large class of $2\times 2$ systems. In particular, we provide several sets of hypothesis on general systems which can be used to rule out the existence of $T_4$ configurations in the constitutive set $\mathcal{K}$. In particular, our results show the nonexistence of $T_4$ configurations for every well-known $2\times 2$ hyperbolic system of conservation laws which verifies the Liu entropy condition.

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arXiv2022

Numerical Analysis of Target Enumeration via Euler Characteristic Integrals

Given a continuous sensor field, we can apply the Euler characteristic integral approach to count the number of targets in the sensor field. If the sensor field is discrete, the Euler integral approach introduces errors into our target count. In this paper, we study the behavior of the Euler integral when applied to discrete sensor fields. Under precise assumptions, we count the number of first- and second-order errors in target count, and discover a formula proportional to much higher order errors. This allows us to derive a point estimator for the number of targets in a discrete sensor field. Finally we derive an asymptotic result, providing insight into how the discrete Euler integral behaves for a large number of targets.

math.PR↗

arXiv2020

Uniqueness and weak-BV stability for $2\times 2$ conservation laws

Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of small $BV$ functions which are global solutions of this equation. For any small $BV$ initial data, such global solutions are known to exist. Moreover, they are known to be unique among $BV$ solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. In this paper, we show that these solutions are stable in a larger class of weak (and possibly not even $BV$) solutions of the system. This result extends the classical weak-strong uniqueness results which allow comparison to a smooth solution. Indeed our result extends these results to a weak-$BV$ uniqueness result, where only one of the solutions is supposed to be small $BV$, and the other solution can come from a large class. As a consequence of our result, the Tame Oscillation Condition, and the Bounded Variation Condition on space-like curves are not necessary for the uniqueness of solutions in the $BV$ theory, in the case of systems with 2 unknowns. The method is $L^2$ based. It builds up from the theory of a-contraction with shifts, where suitable weight functions $a$ are generated via the front tracking method.

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arXiv2019

Finite time stability for the Riemann problem with extremal shocks for a large class of hyperbolic systems

In this paper on hyperbolic systems of conservation laws in one space dimension, we give a complete picture of stability for all solutions to the Riemann problem which contain only extremal shocks. We study stability of the Riemann problem amongst a large class of solutions. We show stability among the family of solutions with shocks from any family. We assume solutions verify at least one entropy condition. We have no small data assumptions. The solutions we consider are bounded and satisfy a strong trace condition weaker than $BV_{\text{loc}}$. We make only mild assumptions on the system. In particular, our work applies to gas dynamics, including the isentropic Euler system and the full Euler system for a polytropic gas. We use the theory of a-contraction (see Kang and Vasseur [Arch. Ration. Mech. Anal., 222(1):343--391, 2016]), and introduce new ideas in this direction to allow for two shocks from different shock families to be controlled simultaneously. This paper shows $L^2$ stability for the Riemann problem for all time. Our results compare to Chen, Frid, and Li [Comm. Math. Phys., 228(2):201--217, 2002] and Chen and Li [J. Differential Equations, 202(2):332--353, 2004], which give uniqueness and long-time stability for perturbations of the Riemann problem -- amongst a large class of solutions without smallness assumptions and which are locally $BV$. Although, these results lack global $L^2$ stability.

math.AP↗

arXiv2019

Stability and uniqueness for piecewise smooth solutions to Burgers-Hilbert among a large class of solutions

In this paper, we show uniqueness and stability for the piecewise-smooth solutions to the Burgers--Hilbert equation constructed in Bressan and Zhang [Commun. Math. Sci., 15(1):165--184, 2017]. The Burgers--Hilbert equation is $u_t+(\frac{u^2}{2})_x=\mathbf{H}[u]$ where $\mathbf{H}$ is the Hilbert transform, a nonlocal operator. We show stability and uniqueness for solutions amongst a larger class than the uniqueness result in Bressan and Zhang. The solutions we consider are measurable and bounded, satisfy at least one entropy condition, and verify a strong trace condition. We do not have smallness assumptions. We use the relative entropy method and theory of shifts (see Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323 -- 376, 2008]).

math.AP↗

arXiv2019

Criteria for the a-contraction and stability for the piecewise-smooth solutions to hyperbolic balance laws

We show uniqueness and stability in $L^2$ and for all time for piecewise-smooth solutions to hyperbolic balance laws. We have in mind applications to gas dynamics, the isentropic Euler system and the full Euler system for a polytropic gas in particular. We assume the discontinuity in the piecewise smooth solution is an extremal shock. We use only mild hypotheses on the system. Our techniques and result hold without smallness assumptions on the solutions. We can handle shocks of any size. We work in the class of bounded, measurable solutions satisfying a single entropy condition. We also assume a strong trace condition on the solutions, but this is weaker than $BV_{\text{loc}}$. We use the theory of a-contraction (see Kang and Vasseur [Arch. Ration. Mech. Anal., 222(1):343--391, 2016]) developed for the stability of pure shocks in the case without source.

math.AP↗

arXiv2018

On Uniqueness of Solutions to Conservation Laws Verifying a Single Entropy Condition

For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see Panov [Mat. Zametki, 55(5):116--129, 159, 1994]). This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Quart. Appl. Math., 62(4):687--700, 2004]. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In this paper, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case.

math.AP↗