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1. Small-data global existence of solutions for the Pitaevskii model of superfluidity

   https://doi.org/10.1088/1361-6544/ad3cae  

   Jang, Juhi; Chaitanya_Jayanti, Pranava; Kukavica, Igor (April 2024, Nonlinearity)

   Abstract We investigate a micro-scale model of superfluidity derived by Pitaevskii (1959Sov. Phys. JETP8282–7) to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model involves the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in ${\mathbb{T}}^2$—strong in wavefunction and velocity, and weak in density. 

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2. Affine isoperimetric inequalities for higher-order projection and centroid bodies

   https://doi.org/10.1007/s00208-025-03271-x  

   Haddad, Julián; Langharst, Dylan; Putterman, Eli; Roysdon, Michael; Ye, Deping (August 2025, Mathematische Annalen)

   Abstract In 1970, Schneider introduced the mth order difference body of a convex body, and also established the mth-order Rogers–Shephard inequality. In this paper, we extend this idea to the projection body, centroid body, and radial mean bodies, as well as prove the associated inequalities (analogues of Zhang’s projection inequality, Petty’s projection inequality, the Busemann–Petty centroid inequality and Busemann’s random simplex inequality). We also establish a new proof of Schneider’s mth-order Rogers–Shephard inequality. As an application, a mth-order affine Sobolev inequality for functions of bounded variation is provided. 

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3. On the order of the classical Erdős–Rogers functions

   https://doi.org/10.1112/blms.13214  

   Mubayi, Dhruv; Verstraete, Jacques (February 2025, Bulletin of the London Mathematical Society)

   Abstract For an integer , the Erdős–Rogers function is the maximum integer such that every ‐vertex ‐free graph has a ‐free induced subgraph with vertices. It is known that for all , as . In this paper, we show that for all , there exists a constant such thatThis improves previous bounds of order by Dudek, Retter and Rödl and answers a question of Warnke. 

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4. On the Maximum F‐Free Induced Subgraphs in Kt‐Free Graphs

   https://doi.org/10.1002/rsa.21273  

   Balogh, József; Chen, Ce; Luo, Haoran (January 2025, Random Structures & Algorithms)

   ABSTRACT For graphs and , let be the minimum possible size of a maximum ‐free induced subgraph in an ‐vertex ‐free graph. This notion generalizes the Ramsey function and the Erdős–Rogers function. Establishing a container lemma for the ‐free subgraphs, we give a general upper bound on , assuming the existence of certain locally dense ‐free graphs. In particular, we prove that for every graph with , where , we have For the cases where is a complete multipartite graph, letting , we prove that We also make an observation which improves the bounds of by a polylogarithmic factor. 

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5. 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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