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1. Small scale creation in active scalars

   https://doi.org/10.1007/978-3-030-54899  

   Alexander Kiselev (January 2020, Progress in mathematical fluid dynamics, Lecture Notes in Mathematics)

   Berselli, Luigi G; Ruzicka, Michael (Ed.)

   The focus of the course is on small scale formation in solutions of the incompressible Euler equation of fluid dynamics and associated models. We first review the regularity results and examples of small scale growth in two dimensions. Then we discuss a specific singular scenario for the three-dimensional Euler equation discovered by Hou and Luo, and analyze some associated models. Finally, we will also talk about the surface quasi-geostrophic (SQG) equation, and construct an example of singularity formation in the modified SQG patch solutions as well as examples of unbounded growth of derivatives for the smooth solutions. 

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2. Hölder Parameterization of Iterated Function Systems and a Self-Aflne Phenomenon

   https://doi.org/10.1515/agms-2020-0125  

   Badger, Matthew; Vellis, Vyron (January 2021, Analysis and Geometry in Metric Spaces)

   null (Ed.)

   Abstract We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/ s )-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s . At the endpoint, α = s , a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/ s )-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s -dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n , the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n . 

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3. Rough solutions of the relativistic Euler equations

   https://doi.org/10.1142/S0219891624500127  

   Yu, Sifan (June 2024, Journal of Hyperbolic Differential Equations)

   We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm [Formula: see text] of the variables in the “wave-part” and the Hölder norm [Formula: see text] of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption [Formula: see text] is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory. 

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4. Realization of spaces of commutative rings

   https://doi.org/10.1112/jlms.70175  

   Cossu, Laura; Olberding, Bruce (May 2025, Journal of the London Mathematical Society)

   Abstract Motivated by recent work on the use of topological methods to study collections of rings between an integral domain and its quotient field, we examine spaces of subrings of a commutative ring, endowed with the Zariski or patch topologies. We introduce three notions to study such a space : patch bundles, patch presheaves and patch algebras. When is compact and Hausdorff, patch bundles give a way to approximate with topologically more tractable spaces, namely Stone spaces. Patch presheaves encode the space into stalks of a presheaf of rings over a Boolean algebra, thus giving a more geometrical setting for studying . To both objects, a patch bundle and a patch presheaf, we associate what we call a patch algebra, a commutative ring that efficiently realizes the rings in as factor rings, or even localizations, and whose structure reflects various properties of the rings in . 

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5. Assouad–Nagata dimension of minor‐closed metrics

   https://doi.org/10.1112/plms.70032  

   Liu, Chun‐Hung (March 2025, Proceedings of the London Mathematical Society)

   Abstract Assouad–Nagata dimension addresses both large‐ and small‐scale behaviors of metric spaces and is a refinement of Gromov's asymptotic dimension. A metric space is a minor‐closed metric if there exists an (edge‐)weighted graph satisfying a fixed minor‐closed property such that the underlying space of is the vertex‐set of , and the metric of is the distance function in . Minor‐closed metrics naturally arise when removing redundant edges of the underlying graphs by using edge‐deletion and edge‐contraction. In this paper, we determine the Assouad–Nagata dimension of every minor‐closed metric. Our main theorem simultaneously generalizes known results about the asymptotic dimension of ‐minor free unweighted graphs and about the Assouad–Nagata dimension of complete Riemannian surfaces with finite Euler genus (Bonamy et al., Asymptotic dimension of minor‐closed families and Assouad–Nagata dimension of surfaces,JEMS(2024)). 

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