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1. Existence and rigidity of the vectorial Peierls–Nabarro model for dislocations in high dimensions

   https://doi.org/10.1088/1361-6544/ac24e3  

   Gao, Yuan; Liu, Jian-Guo; Liu, Zibu (October 2021, Nonlinearity)

   Abstract We focus on the existence and rigidity problems of the vectorial Peierls–Nabarro (PN) model for dislocations. Under the assumption that the misfit potential on the slip plane only depends on the shear displacement along the Burgers vector, a reduced non-local scalar Ginzburg–Landau equation with an anisotropic positive (if Poisson ratio belongs to (−1/2, 1/3)) singular kernel is derived on the slip plane. We first prove that minimizers of the PN energy for this reduced scalar problem exist. Starting from H 1/2 regularity, we prove that these minimizers are smooth 1D profiles only depending on the shear direction, monotonically and uniformly converge to two stable states at far fields in the direction of the Burgers vector. Then a De Giorgi-type conjecture of single-variable symmetry for both minimizers and layer solutions is established. As a direct corollary, minimizers and layer solutions are unique up to translations. The proof of this De Giorgi-type conjecture relies on a delicate spectral analysis which is especially powerful for nonlocal pseudo-differential operators with strong maximal principle. All these results hold in any dimension since we work on the domain periodic in the transverse directions of the slip plane. The physical interpretation of this rigidity result is that the equilibrium dislocation on the slip plane only admits shear displacements and is a strictly monotonic 1D profile provided exclusive dependence of the misfit potential on the shear displacement. 

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2. Dynamic nonlocal passive scalar subgrid-scale turbulence modeling

   https://doi.org/10.1063/5.0106733  

   Seyedi, S. Hadi; Akhavan-Safaei, Ali; Zayernouri, Mohsen (October 2022, Physics of Fluids)

   Extensive experimental evidence highlights that scalar turbulence exhibits anomalous diffusion and stronger intermittency levels at small scales compared to that in fluid turbulence. This renders the corresponding subgrid-scale dynamics modeling for scalar turbulence a greater challenge to date. We develop a new large eddy simulation (LES) paradigm for efficiently and dynamically nonlocal LES modeling of the scalar turbulence. To this end, we formulate the underlying nonlocal model starting from the filtered Boltzmann kinetic transport equation, where the divergence of subgrid-scale scalar fluxes emerges as a fractional-order Laplacian term in the filtered advection–diffusion model, coding the corresponding superdiffusive nature of scalar turbulence. Subsequently, we develop a robust data-driven algorithm for estimation of the fractional (noninteger) Laplacian exponent, where we, on the fly, calculate the corresponding model coefficient employing a new dynamic procedure. Our a priori tests show that our new dynamically nonlocal LES paradigm provides better agreement with the ground-truth filtered direct numerical simulation data in comparison to the conventional static and dynamic Prandtl–Smagorinsky models. Moreover, in order to analyze the numerical stability and assessing the model's performance, we carry out comprehensive a posteriori tests. They unanimously illustrate that our new model considerably outperforms other existing functional models, correctly predicting the backscattering phenomena and, at the same time, providing higher correlations at small-to-large filter sizes. We conclude that our proposed nonlocal subgrid-scale model for scalar turbulence is amenable for coarse LES and very large eddy simulation frameworks even with strong anisotropies, applicable to environmental applications. 

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3. On the nonlinear stability and the existence of selective decay states of 3D quasi‐geostrophic potential vorticity equation

   https://doi.org/10.1002/mma.5962  

   Esen, Oğul; Han, Daozhi; Şengül, Taylan; Wang, Quan (November 2019, Mathematical Methods in the Applied Sciences)

   In this article, we study the dynamics of large‐scale motion in atmosphere and ocean governed by the 3D quasi‐geostrophic potential vorticity (QGPV) equation with a constant stratification. It is shown that for a Kolmogorov forcing on the first energy shell, there exist a family of exact solutions that are dissipative Rossby waves. The nonlinear stability of these exact solutions are analyzed based on the assumptions on the growth rate of the forcing. In the absence of forcing, we show the existence of selective decay states for the 3D QGPV equation. The selective decay states are the 3D Rossby waves traveling horizontally at a constant speed. All these results can be regarded as the expansion of that of the 2D QGPV system and in the case of 3D QGPV system with isotropic viscosity. Finally, we present a geometric foundation for the model as a general equation for nonequilibrium reversible‐irreversible coupling. 

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4. Inverse scattering transform for the complex coupled short‐pulse equation

   https://doi.org/10.1111/sapm.12463  

   Gkogkou, Aikaterini; Prinari, Barbara; Feng, Bao‐Feng; Trubatch, A. David (October 2021, Studies in Applied Mathematics)

   Abstract In this paper, we develop the Riemann–Hilbert approach to the inverse scattering transform (IST) for the complex coupled short‐pulse equation on the line with zero boundary conditions at space infinity, which is a generalization of recent work on the scalar real short‐pulse equation (SPE) and complex short‐pulse equation (cSPE). As a byproduct of the IST, soliton solutions are also obtained. As is often the case, the zoology of soliton solutions for the coupled system is richer than in the scalar case, and it includes both fundamental solitons (the natural, vector generalization of the scalar case), and fundamental breathers (a superposition of orthogonally polarized fundamental solitons, with the same amplitude and velocity but having different carrier frequencies), as well as composite breathers, which still correspond to a minimal set of discrete eigenvalues but cannot be reduced to a simple superposition of fundamental solitons. Moreover, it is found that the same constraint on the discrete eigenvalues which leads to regular, smooth one‐soliton solutions in the complex SPE, also holds in the coupled case, for both a single fundamental soliton and a single fundamental breather, but not, in general, in the case of a composite breather. 

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5. Nonlocal Wasserstein distance: metric and asymptotic properties

   https://doi.org/10.1007/s00526-023-02576-6  

   Slepčev, Dejan; Warren, Andrew (September 2023, Calculus of Variations and Partial Differential Equations)

   Abstract The seminal result of Benamou and Brenier provides a characterization of the Wasserstein distance as the path of the minimal action in the space of probability measures, where paths are solutions of the continuity equation and the action is the kinetic energy. Here we consider a fundamental modification of the framework where the paths are solutions of nonlocal (jump) continuity equations and the action is a nonlocal kinetic energy. The resulting nonlocal Wasserstein distances are relevant to fractional diffusions and Wasserstein distances on graphs. We characterize the basic properties of the distance and obtain sharp conditions on the (jump) kernel specifying the nonlocal transport that determine whether the topology metrized is the weak or the strong topology. A key result of the paper are the quantitative comparisons between the nonlocal and local Wasserstein distance. 

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