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1. Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects

   https://doi.org/10.1515/acv-2020-0114  

   Gao, Yuan; Lu, Xin Yang; Wang, Chong (May 2021, Advances in Calculus of Variations)

   Abstract We study the following parabolic nonlocal 4-th order degenerate equation arising from the epitaxial growth on crystalline materials. Here a>0 is a given parameter. By relying on the theory of gradient flows,we first prove the global existence of a variational inequality solution with a general initial datum.Furthermore, to obtain a global strong solution, the main difficulty is the singularity of the logarithmic term when {u_{xx}+a} approaches zero. Thus we show that,if the initial datum is uniformly bounded away from zero,then such property is preserved for all positive times.Finally, we will prove several higher regularity results for this global strong solution.These finer properties provide a rigorous justification for the global-in-time monotone solution to the epitaxial growth model with nonlocal elastic effects on vicinal surface. 

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2. Potential Singularity of the 3D Euler Equations in the Interior Domain

   https://doi.org/10.1007/s10208-022-09585-5  

   Hou, Thomas Y. (September 2022, Foundations of Computational Mathematics)

   Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blow-up scenario revealed by Luo and Hou (111:12968–12973, 2014) and (12:1722–1776, 2014), which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou and Huang in (arXiv:2102.06663, 2021) and (435:133257, 2022). One important difference between these two blow-up scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in Hou and Huang (arXiv:2102.06663, 2021) and (435:133257, 2022). More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier–Stokes equations. We will present numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data. 

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3. Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha}$$ Velocity and Boundary

   https://doi.org/https://doi.org/10.1007/s00220-021-04067-1  

   Chen, Jiajie; Hou, Thomas Y (April 2021, Communications in mathematical physics)

   null (Ed.)

   Inspired by the numerical evidence of a potential 3D Euler singularity by Luo- Hou [30,31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with $$C^{1,\alpha}$$ initial data for the velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with $$C^{1,\alpha}$$ initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [30,31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use $$C^{1,\alpha}$$ initial data for the velocity field. We use a dynamic rescaling formulation and follow the general framework of analysis developed by Elgindi in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler equations or the 2D Boussinesq equations with $$C^{1,\alpha}$$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time. 

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4. Taylor dispersion and phase mixing in the non‐cutoff Boltzmann equation on the whole space

   https://doi.org/10.1112/plms.12616  

   Bedrossian, Jacob; Coti_Zelati, Michele; Dolce, Michele (July 2024, Proceedings of the London Mathematical Society)

   Abstract In this paper we describe the long‐time behavior of the non‐cutoff Boltzmann equation with soft potentials near a global Maxwellian background on the whole space in the weakly collisional limit (that is, infinite Knudsen number ). Specifically, we prove that for initial data sufficiently small (independent of the Knudsen number), the solution displays several dynamics caused by the phase mixing/dispersive effects of the transport operator and its interplay with the singular collision operator. For ‐wavenumbers with , one sees anenhanced dissipationeffect wherein the characteristic decay time‐scale is accelerated to , where is the singularity of the kernel ( being the Landau collision operator, which is also included in our analysis); for , one seesTaylor dispersion, wherein the decay time‐scale is accelerated to . Additionally, we prove almost uniform phase mixing estimates. For macroscopic quantities such as the density , these bounds imply almost uniform‐in‐ decay of in due to phase mixing and dispersive decay. 

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5. Support for Integrable Hopf Algebras via Noncommutative Hypersurfaces

   https://doi.org/10.1093/imrn/rnab264  

   Negron, Cris; Pevtsova, Julia (October 2021, International Mathematics Research Notices)

   Abstract We consider finite-dimensional Hopf algebras $$u$$ that admit a smooth deformation $$U\to u$$ by a Noetherian Hopf algebra $$U$$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, restricted enveloping algebras in finite characteristic, and Drinfeld doubles of height $$1$$ group schemes. We provide a means of analyzing (cohomological) support for representations over such $$u$$, via the singularity categories of the hypersurfaces $U/(f)$ associated with functions $$f$$ on the corresponding parametrization space. We use this hypersurface approach to establish the tensor product property for cohomological support, for the following examples: functions on a finite group scheme, Drinfeld doubles of certain height 1 solvable finite group schemes, bosonized quantum complete intersections, and the small quantum Borel in type $$A$$. 

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