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1. Stability and Instability of Self-Gravitating Relativistic Matter Distributions

   https://doi.org/10.1007/s00205-021-01647-2  

   Hadžić, Mahir ; Lin, Zhiwu ; Rein, Gerhard ( May 2021 , Archive for Rational Mechanics and Analysis)

   We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein–Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein–Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift$$\kappa >0$$$\kappa >0$. We prove their linear instability when$$\kappa $$$\kappa$is sufficiently large, i.e., when they are strongly relativistic, and prove that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel’dovich & Podurets (for the Einstein–Vlasov system) and by Harrison, Thorne, Wakano, & Wheeler (for the Einstein–Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein–Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.

    

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2. Infinite families of manifolds of positive $$k\mathrm{th}$$-intermediate Ricci curvature with k small

   https://doi.org/10.1007/s00208-022-02420-w  

   Domínguez-Vázquez, Miguel ; González-Álvaro, David ; Mouillé, Lawrence ( August 2022 , Mathematische Annalen)

   Positive$$k\mathrm{th}$$$k\mathrm{th}$-intermediate Ricci curvature on a Riemanniann-manifold, to be denoted by$${{\,\mathrm{Ric}\,}}_k>0$$${\mathrm{Ric}}_k>0$, is a condition that interpolates between positive sectional and positive Ricci curvature (when$$k =1$$$k=1$and$$k=n-1$$$k=n-1$respectively). In this work, we produce many examples of manifolds of$${{\,\mathrm{Ric}\,}}_k>0$$${\mathrm{Ric}}_k>0$withksmall by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension$$n\ge 7$$$n\ge 7$congruent to$$3\,{{\,\mathrm{mod}\,}}4$$$3\mathrm{mod}4$supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of$${{\,\mathrm{Ric}\,}}_k>0$$${\mathrm{Ric}}_k>0$for some$$k$k<n/2$. We also prove that each dimension$$n\ge 4$$$n\ge 4$congruent to 0 or$$1\,{{\,\mathrm{mod}\,}}4$$$1\mathrm{mod}4$supports closed manifolds which carry metrics of$${{\,\mathrm{Ric}\,}}_k>0$$${\mathrm{Ric}}_k>0$with$$k\le n/2$$$k\le n/2$, but do not admit metrics of positive sectional curvature.

    

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3. Local Well-Posedness of the Skew Mean Curvature Flow for Small Data in $$d\geqq 2$$ Dimensions

   https://doi.org/10.1007/s00205-023-01952-y  

   Huang, Jiaxi ; Tataru, Daniel ( January 2024 , Archive for Rational Mechanics and Analysis)

   The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${\mathbb {R}}^{d+2}$$$R^{d+2}$(or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In an earlier paper, the authors introduced a harmonic/Coulomb gauge formulation of the problem, and used it to prove small data local well-posedness in dimensions$$d \geqq 4$$$d\geqq 4$. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\geqq 2$$$d\geqq 2$. This is achieved by introducing a new, heat gauge formulation of the equations, which turns out to be more robust in low dimensions.

    

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4. Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space

   https://doi.org/10.1007/s10959-021-01152-y  

   Yin, Xiuwei ; Wu, Jiang-Lun ; Shen, Guangjun ( January 2022 , Journal of Theoretical Probability)

   The well-posedness of stochastic Navier–Stokes equations with various noises is a hot topic in the area of stochastic partial differential equations. Recently, the consideration of stochastic Navier–Stokes equations involving fractional Laplacian has received more and more attention. Due to the scaling-invariant property of the fractional stochastic equations concerned, it is natural and also very important to study the well-posedness of stochastic fractional Navier–Stokes equations in the associated critical Fourier–Besov spaces. In this paper, we are concerned with the three-dimensional stochastic fractional Navier–Stokes equation driven by multiplicative noise. We aim to establish the well-posedness of solutions of the concerned equation. To this end, by utilising the Fourier localisation technique, we first establish the local existence and uniqueness of the solutions in the critical Fourier–Besov space$$\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$$${\dot{B}}_{p,r}^{4-2\alpha -\frac{3}{p}}$. Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense.

    

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5. Stochastic dual dynamic programming for multistage stochastic mixed-integer nonlinear optimization

   https://doi.org/10.1007/s10107-022-01875-8  

   Zhang, Shixuan ; Sun, Xu Andy ( August 2022 , Mathematical Programming)

   In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization withnon-Lipschitzianvalue functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a$$(T+1)$$$(T+1)$-stage stochastic MINLP satisfyingL-exact Lipschitz regularization withd-dimensional state spaces, to obtain an$$\varepsilon $$$\epsilon$-optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by$${\mathcal {O}}((\frac{2LT}{\varepsilon })^d)$$$O\left({\left(\frac{2LT}{\epsilon }\right)}^d\right)$, and is lower bounded by$${\mathcal {O}}((\frac{LT}{4\varepsilon })^d)$$$O\left({\left(\frac{\mathrm{LT}}{4\epsilon }\right)}^d\right)$for the general case or by$${\mathcal {O}}((\frac{LT}{8\varepsilon })^{d/2-1})$$$O\left({\left(\frac{\mathrm{LT}}{8\epsilon }\right)}^{d/2-1}\right)$for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity dependspolynomiallyon the number of stages. We further show that the iteration complexity dependslinearlyonT, if all the state spaces are finite sets, or if we seek a$$(T\varepsilon )$$$\left(T\epsilon \right)$-optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale withT. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization.

    

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