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In this paper, we study the existence of ground state solutions for an equation that involves a perturbation of a biharmonic operator with a non-local term. More precisely, we study the equation L u = τ f (x , u) + β | u | 2 ∗ ∗ - 2 u in Ω and u = ∂ u ∂ η = 0 on ∂ Ω , where Ω ⊂ R N is a bounded smooth domain, L (·) it is the biharmonic operator perturbed by the non-local operator that we will later define, τ > 0 ; here, 2 ∗ ∗ = 2 N N - 4 with N ≥ 5 . We show the existence of a ground state solution using variational methods considering the subcritical case, i.e., β = 0 and the critical case, i.e., β = 1 . [ABSTRACT FROM AUTHOR]

This paper is concerned with recovering the solution of a final value problem associated with a parabolic equation involving a non linear source and a non-local term, which to the best of our knowledge has not been studied earlier. It is shown that the considered problem is ill-posed, and thus, some regularization method has to be employed in order to obtain stable approximations. In this regard, we obtain regularized approximations by solving some non linear integral equations which is derived by considering a truncated version of the Fourier expansion of the sought solution. Under different Gevrey smoothness assumptions on the exact solution, we provide parameter choice strategies and obtain the error estimates. A key tool in deriving such estimates is a version of Grönwalls' inequality for iterated integrals, which perhaps, is proposed and analysed for the first time. [ABSTRACT FROM AUTHOR]

Abstract The representation of turbulent fluxes during oceanic convective events is important to capture the evolution of the oceanic mixed layer. To improve the accuracy of turbulent fluxes, we examine the possibility of adding a non‐local component in their expression in addition to the usual downgradient part. To do so, we extend the k–ε algebraic second‐moment closure by relaxing the assumption on the equilibrium of the temperature variance θ′2‾. With this additional evolution equation for the temperature variance, we obtain a k–ε–θ′2‾ model (the “kεt” model) which includes a non‐local term for the temperature flux. We validate this new model against Large Eddy Simulations (LES) in three test cases: free convection (FC), wind‐driven mixing, and diurnal cycle (DC). For wind‐driven mixing, kεt is equivalent to k–ε. However, in the presence of a buoyancy flux (FC and DC), we find that the vertical profile of temperature of the LES is better captured by kεt than k–ε. Particularly, the non‐local term increases the fraction of the mixed layer that is stably stratified. For FC, this fraction is near 50% for both kεt and the LES, whereas the k–ε value is 20%. We show that this improvement is due to a better representation of the temperature variance in the inner part of the mixed layer. This better representation is mainly caused by the diffusion of temperature variance, which is described by kεt and not by k–ε.

We review and extend some results for the fractional Schrödinger equation by considering nonlocal terms or potential given in terms of delta functions. For each case, we have obtained the solution in terms of the Green function approach. [ABSTRACT FROM AUTHOR]

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. We prove the existence of a blow-up solution, and give its blow-up profile. Our proof relies on the following method: we linearize the equation (in similarity variables) around the expected profile, then, we control the nonpositive directions of the spectrum thanks to the decreasing properties of the kernel. Finally, we use a topological argument to control the positive directions of the spectrum. [ABSTRACT FROM AUTHOR]

Estimating optical flow from successive video frames is one of the fundamental problems in computer vision and image processing. In the era of deep learning, many methods have been proposed to use convolutional neural networks (CNNs) for optical flow estimation in an unsupervised manner. However, the performance of unsupervised optical flow approaches is still unsatisfactory and often lagging far behind their supervised counterparts, primarily due to over-smoothing across motion boundaries and occlusion. To address these issues, in this paper, we propose a novel method with a new post-processing term and an effective loss function to estimate optical flow in an unsupervised, end-to-end learning manner. Specifically, we first exploit a CNN-based non-local term to refine the estimated optical flow by removing noise and decreasing blur around motion boundaries. This is implemented via automatically learning weights of dependencies over a large spatial neighborhood. Because of its learning ability, the method is effective for various complicated image sequences. Secondly, to reduce the influence of occlusion, a symmetrical energy formulation is introduced to detect the occlusion map from refined bi-directional optical flows. Then the occlusion map is integrated to the loss function. Extensive experiments are conducted on challenging datasets, i.e. FlyingChairs, MPI-Sintel and KITTI to evaluate the performance of the proposed method. The state-of-the-art results demonstrate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

A logistic equation in the whole space is considered. In this problem, a non-local perturbation is included. We establish a new sub–supersolution method for general nonlocal elliptic equations and, consequently, we obtain the existence of positive solutions of a nonlocal logistic equation. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new existence and uniqueness result of a continuous viscosity solution for integro-partial differential equation (IPDE in short). The novelty is that we relax the so-called monotonicity assumption on the driver which is classically assumed in the literature of viscosity solutions of equation with non-local terms. Our method strongly relies on the link between IPDEs and backward stochastic differential equations with jumps for which we already know that the solution exists and is unique for general drivers. In the second part of the paper, we deal with the IPDE with obstacle and we obtain similar results. [ABSTRACT FROM AUTHOR]

The accuracy of optical flow estimation algorithms has been improving steadily as evidenced by results on the Middlebury optical flow benchmark. The typical formulation, however, has changed little since the work of Horn and Schunck. We attempt to uncover what has made recent advances possible through a thorough analysis of how the objective function, the optimization method, and modern implementation practices influence accuracy. We discover that 'classical' flow formulations perform surprisingly well when combined with modern optimization and implementation techniques. One key implementation detail is the median filtering of intermediate flow fields during optimization. While this improves the robustness of classical methods it actually leads to higher energy solutions, meaning that these methods are not optimizing the original objective function. To understand the principles behind this phenomenon, we derive a new objective function that formalizes the median filtering heuristic. This objective function includes a non-local smoothness term that robustly integrates flow estimates over large spatial neighborhoods. By modifying this new term to include information about flow and image boundaries we develop a method that can better preserve motion details. To take advantage of the trend towards video in wide-screen format, we further introduce an asymmetric pyramid downsampling scheme that enables the estimation of longer range horizontal motions. The methods are evaluated on the Middlebury, MPI Sintel, and KITTI datasets using the same parameter settings. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study a simple non-local semilinear parabolic equation in a bounded domain with Neumann boundary conditions. We obtain a global existence result for initial data whose -norm is less than a constant depending explicitly on the geometry of the domain. A natural energy is associated to the equation and we establish a relationship between the finite-time blow up of solutions and the negativity of their energy. The proof of this result is based on a Gamma-convergence technique. [Copyright &y& Elsevier]