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1. Super-Exponential Convergence Rate of a Nonlinear Continuous Data Assimilation Algorithm: The 2D Navier–Stokes Equation Paradigm

   https://doi.org/10.1007/s00332-024-10014-w  

   Carlson, Elizabeth; Larios, Adam; Titi, Edriss S (April 2024, Journal of Nonlinear Science)

   We study a nonlinear-nudging modification of the Azouani–Olson–Titi continuous data assimilation (downscaling) algorithm for the 2D incompressible Navier–Stokes equations. We give a rigorous proof that the nonlinear-nudging system is globally well posed and, moreover, that its solutions converge to the true solution exponentially fast in time. Furthermore, we also prove that once the error has decreased below a certain order one threshold, the convergence becomes double exponentially fast in time, up until a precision determined by the sparsity of the observed data. In addition, we demonstrate the applicability of the analytical and sharpness of the results computationally. 

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2. Fluctuations Driven by Multicomponent Pickup Ion Distributions in the Outer Heliosheath

   https://doi.org/10.3847/1538-4357/adaaf4  

   Mousavi, Ameneh; Roytershteyn, Vadim; Fraternale, Federico; Pogorelov, Nikolai (February 2025, The Astrophysical Journal)

   Abstract The stability of a realistic multicomponent pickup ion (PUI) velocity distribution derived from a global model of neutral atoms in the heliosphere, which treats hydrogen and helium atoms self-consistently and includes equations for electrons and helium ions, is investigated using linear instability analysis and hybrid simulations. Linear instability analysis shows that the excited oblique mirror waves and the parallel/quasi-parallel Alfvén-cyclotron (AC) waves have lower growth rates than those obtained previously by A. Mousavi et al. for the PUI velocity distributions given by J. Heerikhuisen et al. The PUI scattering by each of the two modes alone is studied. In contrast to the previous investigations, our current simulations using the updated realistic distributions indicate that mirror waves alone do not effectively scatter PUIs in pitch angle. Instead, they primarily contribute to reducing the thermal spread anisotropy of the PUIs originating from the neutral solar wind. The unstable AC waves exhibit lower growth rates but higher saturation levels than the mirror waves. Two-dimensional (2D) simulation results show that when all unstable waves are present, the predominant contributor to the fluctuating magnetic field energy is the AC mode. The AC waves quickly scatter the PUIs with pitch angles away from 90^∘toward isotropy, while the PUIs near 90^∘pitch angle maintain a degree of anisotropy within our simulation timeframe. Moreover, several 1D and 2D hybrid simulations with different numbers of particles per cell are performed to examine the impact of numerical noise on PUI scattering. Finally, the implications of these results for the Interstellar Boundary Explorer energetic neutral atom ribbon are discussed. 

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3. Flexibility and rigidity of free boundary MHD equilibria

   https://doi.org/10.1088/1361-6544/ac5d6a  

   Constantin, Peter; Drivas, Theodore D; Ginsberg, Daniel (April 2022, Nonlinearity)

   Abstract We study stationary free boundary configurations of an ideal incompressible magnetohydrodynamic fluid possessing nested flux surfaces. In 2D simply connected domains, we prove that if the magnetic field and velocity field are never commensurate, the only possible domain for any such equilibria is a disk, and the velocity and magnetic field are circular. We give examples of non-symmetric equilibria occupying a domain of any shape by imposing an external magnetic field generated by a singular current sheet charge distribution (external coils). Some results carry over to 3D axisymmetric solutions. These results highlight the importance of external magnetic fields for the existence of asymmetric equilibria. 

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4. 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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5. A divergence-free and $H\left(div\right)$ -conforming embedded-hybridized DG method for the incompressible resistive MHD equations

   https://doi.org/10.1016/j.cma.2024.117415  

   Chen, Jau-Uei; Horváth, Tamás L; Bui-Thanh, Tan (December 2024, Computer Methods in Applied Mechanics and Engineering)

   We present a divergence-free and $$\Hsp\LRp{div}$$-conforming hybridized discontinuous Galerkin (HDG) method and a computationally efficient variant called embedded-HDG (E-HDG) for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. The proposed E-HDG approach uses continuous facet unknowns for the vector-valued solutions (velocity and magnetic fields) while it uses discontinuous facet unknowns for the scalar variable (pressure and magnetic pressure). This choice of function spaces makes E-HDG computationally far more advantageous, due to the much smaller number of degrees of freedom, compared to the HDG counterpart. The benefit is even more significant for three-dimensional/high-order/fine mesh scenarios. On simplicial meshes, the proposed methods with a specific choice of approximation spaces are well-posed for linear(ized) MHD equations. For nonlinear MHD problems, we present a simple approach exploiting the proposed linear discretizations by using a Picard iteration. The beauty of this approach is that the divergence-free and $$\Hsp\LRp{div}$$-conforming properties of the velocity and magnetic fields are automatically carried over for nonlinear MHD equations. We study the accuracy and convergence of our E-HDG method for both linear and nonlinear MHD cases through various numerical experiments, including two- and three-dimensional problems with smooth and singular solutions. The numerical examples show that the proposed methods are pressure robust, and the divergence of the resulting velocity and magnetic fields is machine zero for both smooth and singular problems. 

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