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  1. ; ; (, Communications in Mathematical Analysis and Applications)
    We investigate $L^2$-contraction and time-asymptotic stability of large shock for scalar viscous conservation laws with polynomial flux. For the flux $f(u) = u^p (2 ≤ p ≤ 4)$ in the regime of its strict convexity, we can prove $L^2$-contraction and time-asymptotic stability of arbitrarily large viscous shock profile in $H^1$-framework by using $$a$$-contraction method with time-dependent shift and suitable weight function, which answers a question in [Blochas and Cheng, arXiv2501.01537, 2025]. Additionally, if the initial perturbation belongs to $L^1$ , then $L^2$ time-asymptotic decay rate $$t^{−1/4}$$ can be obtained. 
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    Free, publicly-accessible full text available February 25, 2027
  2. (, Quarterly of Applied Mathematics)
    In this paper, we study stability properties of solutions to scalar conservation laws with a class of nonconvex fluxes. Using the theory of a a -contraction with shifts, we show L 2 L^2 -stability for shocks among a class of large perturbations and give estimates on the weight coefficient a a in regimes where the shock amplitude is both large and small. Then, we use these estimates as a building block to show a uniqueness theorem under minimal entropy conditions for weak solutions to the conservation law via a modified front tracking algorithm. The proof is inspired by an analogous program carried out in the 2 ×<#comment/> 2 2 \times 2 system setting by Chen, Golding, Krupa, and Vasseur [Arch. Ration. Mech. Anal. 246 (2022), no. 1, 299–332 and J. Hyperbolic Differ. Equ. 20 (2023), no. 3, 541–602]. 
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    Free, publicly-accessible full text available December 1, 2026
  3. (, Journal of Differential Equations)
    Free, publicly-accessible full text available November 1, 2026
  4. ; ; (, Archive for Rational Mechanics and Analysis)
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  5. ; (, SIAM Journal on Mathematical Analysis)
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  6. ; (, Communications in Partial Differential Equations)
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  7. ; (, Communications in Mathematical Analysis and Applications)
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  8. (, SIAM Journal on Mathematical Analysis)
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  9. (, Quarterly of Applied Mathematics)
    Dafermos [Arch. Rational Mech. Anal. 70 (1979), pp. 167–179] proved the weak/strong principle for conservation laws. It states that Lipschitz solutions to conservation laws endowed with convex entropies are unique and stable among weak solutions. The method, based on relative entropy, was extended by Di Perna [Indiana Univ. Math. J. 28 (1979), pp. 137–188] to show the uniqueness of shocks among weak solutions with strong traces. This theory has been recently revisited with the notion of weighted contractions up to shifts. We review in this paper recent applications of this method, including the weak/BV principle and the stability of discontinuous solutions among inviscid double limits of Navier-Stokes systems. 
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