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1. Global Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics

   https://doi.org/10.1007/s00220-022-04522-7  

   Laurens, Thierry (November 2022, Communications in Mathematical Physics)

   Abstract Given a suitable solutionV(t, x) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data$$u(0,x) \in V(0,x) + H^{-1}(\mathbb {R})$$ $u(0,x)\in V(0,x)+H^{-1}\left(R\right)$. Our conditions onVdo include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles$$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$ $V(0,x)\in H^5(R/Z)$satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022.https://doi.org/10.1088/1361-6544/ac37f5) we show that smooth step-like initial data also satisfy our hypotheses. We employ the method of commuting flows introduced in Killip and Vişan (Ann. Math. (2) 190(1):249–305, 2019.https://doi.org/10.4007/annals.2019.190.1.4) where$$V\equiv 0$$ $V\equiv 0$. In that setting, it is known that$$H^{-1}(\mathbb {R})$$ $H^{-1}\left(R\right)$is sharp in the class of$$H^s(\mathbb {R})$$ $H^s\left(R\right)$spaces. 

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2. Vector-Valued Maximal Inequalities and Multiparameter Oscillation Inequalities for the Polynomial Ergodic Averages Along Multi-dimensional Subsets of Primes

   https://doi.org/10.1007/s00041-024-10119-6  

   Mehlhop, Nathan (December 2024, Journal of Fourier Analysis and Applications)

   Abstract We prove uniform$$\ell ^2$$ ${\ell }^2$-valued maximal inequalities for polynomial ergodic averages and truncated singular operators of Cotlar type modeled over multidimensional subsets of primes. In the averages case, we combine this with earlier one-parameter oscillation estimates (Mehlhop and Słomian in Math Ann, 2023,https://doi.org/10.1007/s00208-023-02597-8) to prove corresponding multiparameter oscillation estimates. This provides a fuller quantitative description of the pointwise convergence of the mentioned averages and is a generalization of the polynomial Dunford–Zygmund ergodic theorem attributed to Bourgain (Mirek et al. in Rev Mat Iberoam 38:2249–2284, 2022). 

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3. Global well-posedness for 2D generalized parabolic Anderson model via paracontrolled calculus

   https://doi.org/10.1007/s40072-025-00358-z  

   Shen, Hao; Zhu, Rongchan; Zhu, Xiangchan (May 2025, Stochastics and Partial Differential Equations: Analysis and Computations)

   Abstract This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on$$\mathbb {R}^+\times \mathbb {T}^2$$ $R^{+}\times T^2$within the framework of paracontrolled calculus (Gubinelli et al. in Forum Math, 2015). The model is given by the equation:$$\begin{aligned} (\partial _t-\Delta ) u=F(u)\eta \end{aligned}$$ $\begin{array}{c}({\partial }_t-\Delta )u=F\left(u\right)\eta \end{array}$where$$\eta \in C^{-1-\kappa }$$ $\eta \in C^{-1-\kappa }$with$$1/6>\kappa >0$$ $1/6>\kappa >0$, and$$F\in C_b^2(\mathbb {R})$$ $F\in C_b^2\left(R\right)$. Assume that$$\eta \in C^{-1-\kappa }$$ $\eta \in C^{-1-\kappa }$and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows from the recent work by Chandra et al. (A priori bounds for 2-d generalised Parabolic Anderson Model,,2024), to represent the leading error term as a transport type term, and our techniques encompass the paracontrolled calculus, the maximum principle, and the localization approach (i.e. high-low frequency argument). 

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4. Scaling laws and exact results in turbulence ^*

   https://doi.org/10.1088/1361-6544/ad6057  

   Novack, Matthew (July 2024, Nonlinearity)

   Abstract In this note, we address the validity of certain exact results from turbulence theory in the deterministic setting. The main tools, inspired by the work of Duchon and Robert (2000Nonlinearity13249–55) and Eyink (2003Nonlinearity16137), are a number of energy balance identities for weak solutions of the incompressible Euler and Navier–Stokes equations. As a consequence, we show that certain weak solutions of the Euler and Navier–Stokes equations satisfy deterministic versions of Kolmogorov’s $\frac{4}{5}$, $\frac{4}{3}$, $\frac{4}{15}$laws. We apply these computations to improve a recent result of Hofmanovaet al(2023 arXiv:2304.14470), which shows that a construction of solutions of forced Navier–Stokes due to Bruèet al(2023Commun. Pure Appl. Anal.) and exhibiting a form of anomalous dissipation satisfies asymptotic versions of Kolmogorov’s laws. In addition, we show that the globally dissipative 3D Euler flows recently constructed by Giriet al(2023 arXiv:2305.18509) satisfy the local versions of Kolmogorov’s laws. 

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5. An almost linear time algorithm testing whether the Markoff graph modulo p is connected

   https://doi.org/10.1007/s40993-024-00592-9  

   Brown, Colby Austin (March 2025, Research in Number Theory)

   Abstract The Markoff graphs modulopwere proven by Chen (Ann Math 199(1), 2024) to be connected for all but finitely many primes, and Baragar (The Markoff equation and equations of Hurwitz. Brown University, 1991) conjectured that they are connected for all primes, equivalently that every solution to the Markoff equation moduloplifts to a solution over$$\mathbb {Z}$$ $Z$. In this paper, we provide an algorithmic realization of the process introduced by Bourgain et al. [arXiv:1607.01530] to test whether the Markoff graph modulopis connected for arbitrary primes. Our algorithm runs in$$o(p^{1 + \epsilon })$$ $o\left(p^{1+ϵ}\right)$time for every$$\epsilon > 0$$ $ϵ>0$. We demonstrate this algorithm by confirming that the Markoff graph modulopis connected for all primes less than one million. 

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