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1. Uniqueness and Weak-BV Stability for $$2\times 2$$ Conservation Laws

   https://doi.org/10.1007/s00205-022-01813-0  

   Chen, Geng ; Krupa, Sam G. ; Vasseur, Alexis F. ( September 2022 , Archive for Rational Mechanics and Analysis)

   Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of smallBVfunctions which are global solutions of this equation. For any smallBVinitial data, such global solutions are known to exist. Moreover, they are known to be unique amongBVsolutions 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 evenBV) 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-BVuniqueness result, where only one of the solutions is supposed to be smallBV, 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 theBVtheory, in the case of systems with 2 unknowns. The method is$$L^2$$$L^2$based, and builds up from the theory of a-contraction with shifts, where suitable weight functionsaare generated via the front tracking method.

    

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2. On uniqueness of solutions to conservation laws verifying a single entropy condition

   https://doi.org/10.1142/S0219891619500061  

   Krupa, Sam G. ; Vasseur, Alexis F. ( March 2019 , Journal of Hyperbolic Differential Equations)

   null (Ed.)

   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 [E. Yu. Panov, Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mat. Z. 55(5) (1994) 116–129 (in Russian), Math. Notes 55(5) (1994) 517–525]. This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Minimal entropy conditions for Burgers equation, Quart. Appl. Math. 62(4) (2004) 687–700]. 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. 

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3. Lp time asymptotic decay for general hyperbolic–parabolic balance laws with applications

   https://doi.org/10.1142/S021989161950022X  

   Zeng, Yanni ( December 2019 , Journal of Hyperbolic Differential Equations)

   We study the time asymptotic decay of solutions for a general system of hyperbolic–parabolic balance laws in one space dimension. The system has a physical viscosity matrix and a lower-order term for relaxation, damping or chemical reaction. The viscosity matrix and the Jacobian matrix of the lower-order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper, we obtain optimal [Formula: see text] decay rates for [Formula: see text]. Our result is general and applies to models such as Keller–Segel equations with logarithmic chemotactic sensitivity and logistic growth, and gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic–parabolic conservation laws and hyperbolic balance laws, respectively. 

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4. The Littlewood–Paley decomposition for periodic functions and applications to the Boussinesq equations

   https://doi.org/10.1142/s0219530519500234  

   Dai, Yichen ; Hu, Weiwei ; Wu, Jiahong ; Xiao, Bei ( July 2020 , Analysis and Applications)

   The Littlewood–Paley decomposition for functions defined on the whole space [Formula: see text] and related Besov space techniques have become indispensable tools in the study of many partial differential equations (PDEs) with [Formula: see text] as the spatial domain. This paper intends to develop parallel tools for the periodic domain [Formula: see text]. Taking advantage of the boundedness and convergence theory on the square-cutoff Fourier partial sum, we define the Littlewood–Paley decomposition for periodic functions via the square cutoff. We remark that the Littlewood–Paley projections defined via the circular cutoff in [Formula: see text] with [Formula: see text] in the literature do not behave well on the Lebesgue space [Formula: see text] except for [Formula: see text]. We develop a complete set of tools associated with this decomposition, which would be very useful in the study of PDEs defined on [Formula: see text]. As an application of the tools developed here, we study the periodic weak solutions of the [Formula: see text]-dimensional Boussinesq equations with the fractional dissipation [Formula: see text] and without thermal diffusion. We obtain two main results. The first assesses the global existence of [Formula: see text]-weak solutions for any [Formula: see text] and the existence and uniqueness of the [Formula: see text]-weak solutions when [Formula: see text] for [Formula: see text]. The second establishes the zero thermal diffusion limit with an explicit convergence rate. 

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5. Harmonic spinors on the Davis hyperbolic 4-manifold

   https://doi.org/10.1142/S1793525320500247  

   Ratcliffe, John G. ; Ruberman, Daniel ; Tschantz, Steven T. ( September 2021 , Journal of Topology and Analysis)

   In this paper, we use the [Formula: see text]-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic [Formula: see text]-manifold that admits harmonic spinors. We also explicitly describe the spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the [Formula: see text]-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold. 

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