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1. Global regularity for critical SQG in bounded domains

   https://doi.org/10.1002/cpa.22221  

   Constantin, Peter; Ignatova, Mihaela; Nguyen, Quoc‐Hung (January 2025, Communications on Pure and Applied Mathematics)

   Abstract We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in . We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system. 

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2. Bathysciadium pacificum Dall, 1908 from the bathyal of Oregon: range extension and anatomy (Gastropoda: Cocculinida: Bathysciadiidae).

   Geiger, Daniel L.; Thacker, Christine E. (August 2023, The Nautilus)

   Leal, José (Ed.)

   Three new records of Bathysciadium pacificum Dall, 1908 from off Oregon and Washington extend the range of the species from Peru by ~7,300 km to the temperate northeastern Pacific. The preserved and relaxed animal shows that the mantle exten-sions are not sensory papillae, but mantle folds forming the periostracum rays on the shell. Examination of radula, mantle, and mantle cavity by scanning electron microscopy (SEM), including environmental SEM (ESEM) of the fluid-preserved animal, adds a novel dimension to the understanding of the anatomy of these rare deep-sea snails. ESEM is a suitable tool for the morphological investigations of rare natural history specimens. The monotypic genera Bathypelta Moskalev, 1971 and Bonus Moskalev, 1973 are re-synonymized under Bathy-sciadium Dautzenberg and H. Fischer, 1900. No mineralization of the radula was detected by SEM-Energy Dispersive Spec-troscopy (EDS/EDAX) mapping. 

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3. Bounded Geometry and p-Harmonic Functions Under Uniformization and Hyperbolization

   https://doi.org/10.1007/s12220-020-00477-0  

   Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari (May 2021, The Journal of Geometric Analysis)

   null (Ed.)

   Abstract The uniformization and hyperbolization transformations formulated by Bonk et al. in “Uniformizing Gromov Hyperbolic Spaces” , Astérisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local p -Poincaré inequality, then the transformed measure is globally doubling and supports a global p -Poincaré inequality on the corresponding uniformized space. In the opposite direction, we show that such global properties on bounded locally compact uniform spaces yield similar uniformly local properties for the transformed measures on the corresponding hyperbolized spaces. We use the above results on uniformization of measures to characterize when a Gromov hyperbolic space, equipped with a uniformly locally doubling measure supporting a uniformly local p -Poincaré inequality, carries nonconstant globally defined p -harmonic functions with finite p -energy. We also study some geometric properties of Gromov hyperbolic and uniform spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be Gromov hyperbolic, we construct an indirect product of such spaces that does result in a Gromov hyperbolic space. This is done by first showing that the Cartesian product of two bounded uniform domains is a uniform domain. 

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4. Proper mappings between indefinite hyperbolic spaces and type I classical domains

   https://doi.org/10.1090/tran/8618  

   Huang, Xiaojun; Lu, Jin; Tang, Xiaomin; Xiao, Ming (December 2022, Transactions of the American Mathematical Society)

   In this paper, we first study a mapping problem between indefinite hyperbolic spaces by employing the work established earlier by the authors. In particular, we generalize certain theorems proved by Baouendi-Ebenfelt-Huang [Amer. J. Math. 133 (2011), pp. 1633–1661] and Ng [Michigan Math. J. 62 (2013), pp. 769–777; Int. Math. Res. Not. IMRN 2 (2015), pp. 291–324]. Then we use these results to prove a rigidity result for proper holomorphic mappings between type I classical domains, which confirms a conjecture formulated by Chan [Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373] after the work of Zaitsev-Kim [Math. Ann. 362 (2015), pp. 639-677], Kim [ Proper holomorphic maps between bounded symmetric domains , Springer, Tokyo, 2015, pp. 207–219] and himself. 

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5. Conditional $$L^{\infty }$$ Estimates for the Non-cutoff Boltzmann Equation in a Bounded Domain

   https://doi.org/10.1007/s00205-024-02002-x  

   Ouyang, Zhimeng; Silvestre, Luis (August 2024, Archive for Rational Mechanics and Analysis)

   We consider weak solutions of the inhomogeneous non-cutoff Boltzmann equa- tion in a bounded domain with any of the usual physical boundary conditions: in-flow, bounce-back, specular-reflection and diffuse-reflection. When the mass, energy and entropy densities are bounded above, and the mass density is bounded away from a vacuum, we obtain an estimate of the L∞ norm of the solution de- pending on the macroscopic bounds on these hydrodynamic quantities only. This is a regularization effect in the sense that the initial data is not required to be bounded. We present a proof based on variational ideas, which is fundamentally different to the proof that was previously known for the equation in periodic spatial domains 

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