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1. Geometry of turbulent dissipation and the Navier–Stokes regularity problem

   https://doi.org/10.1038/s41598-021-87774-y  

   Rafner, Janet; Grujić, Zoran; Bach, Christian; Bærentzen, Jakob Andreas; Gervang, Bo; Jia, Ruo; Leinweber, Scott; Misztal, Marek; Sherson, Jacob (December 2021, Scientific Reports)

   null (Ed.)

   Abstract The question of whether a singularity can form in an initially regular flow, described by the 3D incompressible Navier–Stokes (NS) equations, is a fundamental problem in mathematical physics. The NS regularity problem is super-critical, i.e., there is a ‘scaling gap’ between what can be established by mathematical analysis and what is needed to rule out a singularity. A recently introduced mathematical framework—based on a suitably defined ‘scale of sparseness’ of the regions of intense vorticity—brought the first scaling reduction of the NS super-criticality since the 1960s. Here, we put this framework to the first numerical test using a spatially highly resolved computational simulation performed near a ‘burst’ of the vorticity magnitude. The results confirm that the scale is well suited to detect the onset of dissipation and provide numerical evidence that ongoing mathematical efforts may succeed in closing the scaling gap. 

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2. A problem in control of elastodynamics with piezoelectric effects

   https://doi.org/10.1093/imanum/drz047  

   Antil, Harbir; Brown, Thomas S.; Sayas, Francisco-Javier (December 2019, IMA Journal of Numerical Analysis)

   Abstract We consider an optimal control problem where the state equations are a coupled hyperbolic–elliptic system. This system arises in elastodynamics with piezoelectric effects—the elastic stress tensor is a function of elastic displacement and electric potential. The electric flux acts as the control variable and bound constraints on the control are considered. We develop a complete analysis for the state equations and the control problem. The requisite regularity on the control, to show the well-posedness of the state equations, is enforced using the cost functional. We rigorously derive the first-order necessary and sufficient conditions using adjoint equations and further study their well-posedness. For spatially discrete (time-continuous) problems, we show the convergence of our numerical scheme. Three-dimensional numerical experiments are provided showing convergence properties of a fully discrete method and the practical applicability of our approach. 

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3. A regularity result for the free boundary compressible Euler equations of a liquid

   https://doi.org/10.1088/1361-6544/ad1f9d  

   Li, Linfeng (February 2024, Nonlinearity)

   Abstract We derivea prioriestimates for the compressible free boundary Euler equations in the case of a liquid without surface tension. We provide a new weighted functional framework which leads to the improved regularity of the flow map by using the Hardy inequality. One of main ideas is to decompose the initial density function. It is worth mentioning that in our analysis we do not need the higher order wave equation for the density. 

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4. Higher-order error estimates for physics-informed neural networks approximating the primitive equations

   https://doi.org/10.1007/s42985-023-00254-y  

   Hu, Ruimeng; Lin, Quyuan; Raydan, Alan; Tang, Sui (July 2023, Partial Differential Equations and Applications)

   Abstract Large-scale dynamics of the oceans and the atmosphere are governed by primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is generally challenging. Neural networks have been shown to be a promising machine learning tool to tackle this challenge. In this work, we employ physics-informed neural networks (PINNs) to approximate the solutions to the PEs and study the error estimates. We first establish the higher-order regularity for the global solutions to the PEs with either full viscosity and diffusivity, or with only the horizontal ones. Such a result for the case with only the horizontal ones is new and required in the analysis under the PINNs framework. Then we prove the existence of two-layer tanh PINNs of which the corresponding training error can be arbitrarily small by taking the width of PINNs to be sufficiently wide, and the error between the true solution and its approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, all the estimates area priori, and our analysis includes higher-order (in spatial Sobolev norm) error estimates. Numerical results on prototype systems are presented to further illustrate the advantage of using the$$H^s$$ $H^s$norm during the training. 

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5. Recent developments in mathematical aspects of relativistic fluids

   https://doi.org/10.1007/s41114-024-00052-x  

   Disconzi, Marcelo (October 2024, Living Reviews in Relativity)

   Abstract We review some recent developments in mathematical aspects of relativistic fluids. The goal is to provide a quick entry point to some research topics of current interest that is accessible to graduate students and researchers from adjacent fields, as well as to researches working on broader aspects of relativistic fluid dynamics interested in its mathematical formalism. Instead of complete proofs, which can be found in the published literature, here we focus on the proofs’ main ideas and key concepts. After an introduction to the relativistic Euler equations, we cover the following topics: a new wave-transport formulation of the relativistic Euler equations tailored to applications; the problem of shock formation for relativistic Euler; rough (i.e., low-regularity) solutions to the relativistic Euler equations; the relativistic Euler equations with a physical vacuum boundary; relativistic fluids with viscosity. We finish with a discussion of open problems and future directions of research. 

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