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1. Electron precipitation caused by intense whistler-mode waves: combined effects of anomalous scattering and phase bunching

   https://doi.org/10.3389/fspas.2023.1322934  

   Gan, Longzhi; Li, Wen; Hanzelka, Miroslav; Ma, Qianli; Albert, Jay M.; Artemyev, Anton V. (December 2023, Frontiers in Astronomy and Space Sciences)

   Resonant interactions with whistler-mode waves are a crucial mechanism that drives the precipitation of energetic electrons. Using test particle simulations, we investigated the impact of nonlinear interactions of whistler-mode waves on electron precipitation across a broad energy range (10 keV- 1 MeV). Specifically, we focused on the combined effects of conventional phase bunching and anomalous scattering, which includes anomalous trapping and positive bunching. It is shown that anomalous scattering transports electrons away from the loss cone and the only process directly causing precipitation in the nonlinear regime is the phase bunching. We further show that their combined effects result in a precipitation-to-trapped flux ratio lower than the quasilinear expectations in a quasi-equilibrium state. Additionally, we calculated the diffusion and advection coefficients associated with the nonlinear trapping and bunching processes, which are vital for understanding the underlying mechanisms of the precipitation. Based on these coefficients, we characterized the phase bunching boundary, representing the innermost pitch angle boundary where phase bunching can occur. A further analysis revealed that electrons just outside this boundary, rather than near the loss cone, are directly precipitated, while electrons within the boundary are prevented from precipitation due to anomalous scattering. Moreover, we demonstrated that the regime of dominant nonlinear precipitation is determined by the combination of the phase bunching boundary and the inhomogeneity ratio. This comprehensive analysis provides insights into the nonlinear effects of whistler-mode waves on electron precipitation, which are essential for understanding physical processes related to precipitation, such as microbursts, characterized by intense and bursty electron precipitation. 

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2. Directed mean curvature flow in noisy environment

   https://doi.org/10.1002/cpa.22158  

   Gerasimovičs, Andris; Hairer, Martin; Matetski, Konstantin (October 2023, Communications on Pure and Applied Mathematics)

   Abstract We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite‐dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation. 

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3. Hölder regularity of the Boltzmann equation past an obstacle

   https://doi.org/10.1002/cpa.22167  

   Kim, Chanwoo; Lee, Donghyun (April 2024, Communications on Pure and Applied Mathematics)

   Abstract Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in for the Boltzmann equation of the hard‐sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in‐flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)]. 

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4. Quantum scattering states in a nonlinear coherent medium

   https://doi.org/10.1103/PhysRevA.108.023314  

   Brattley, Allison; Huang, Hongyi; Das, Kunal K. (August 2023, Physical Review A)

   We present a comprehensive study of stationary states in a coherent medium with a quadratic or Kerr nonlinearity in the presence of localized potentials in one dimension for both positive and negative signs of the nonlinear term as well as for barriers and wells. The description is in terms of the nonlinear Schrödinger equation and hence applicable to a variety of systems, including interacting ultracold atoms in the mean field regime and light propagation in optical fibers. We determine the full landscape of solutions in terms of a potential step and build solutions for rectangular barrier and well potentials. It is shown that all the solutions can be expressed in terms of a Jacobi elliptic function with the inclusion of a complex-valued phase shift. Our solution method relies on the roots of a cubic polynomial associated with a hydrodynamic picture, which provides a simple classification of all the solutions, both bounded and unbounded, while the boundary conditions are intuitively visualized as intersections of phase space curves. We compare solutions for open boundary conditions with those for a barrier potential on a ring, and also show that numerically computed solutions for smooth barriers agree qualitatively with analytical solutions for rectangular barriers. A stability analysis of solutions based on the Bogoliubov equations for fluctuations shows that persistent instabilities are localized at sharp boundaries and are predicated by the relation of the mean density change across the boundary to the value of the derivative of the density at the edge. We examine the scattering of a wave packet by a barrier potential and show that at any instant the scattered states are well described by the stationary solutions we obtain, indicating applications of our results and methods to nonlinear scattering problems. 

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5. The regular free boundary in the thin obstacle problem for degenerate parabolic equations

   Banerjee, A.; Danielli, D.; Garofalo, N.; Petrosyan, A. (June 2020, Algebra i analiz)

   In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator $$(\partial_t - \Delta_x)^s$$ for $$s \in (0,1)$$. Our regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. Our approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. Using several results from the elliptic theory, including the epiperimetric inequality, we establish the optimal regularity of solutions as well as $$H^{1+\gamma,\frac{1+\gamma}{2}}$$ regularity of the free boundary near such regular points. 

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