Compact Support Research Articles

On thermal conduction in the solar atmosphere: An analytical solution for nonlinear diffusivity without compact support

Context. The scientific community employs complicated multiphysics simulations to understand the physics in solar, stellar, and interstellar media. These must be tested against known solutions to ensure their validity. Several well-known tests exist, such as the Sod shock tube test. However, a test for nonlinear diffusivity is missing. This problem is highly relevant in the solar atmosphere, where various events release energy that subsequently diffuses by Spitzer thermal conductivity. Aims. The aim is to derive an analytical solution for nonlinear diffusivity in 1D, 2D, and 3D, which allows for a nonzero background value. The solution is used to design a test for numerical solvers and study Spitzer conductivity in the solar atmosphere. Methods. There exists an ideal solution assuming zero background value. We performed an analytical first-order perturbation of this solution. The first-order solution was first tested against a dedicated nonlinear diffusion solver, whereupon it was used to benchmark the single- and multifluid radiative magnetohydrodynamics code Ebysus, used to study the Sun. The theory and numerical modeling were used to investigate the role of Spitzer conductivity in the transport of energy released in a nanoflare. Results. The derived analytical solution models nonlinear diffusivity accurately within its region of validity and approximately beyond. Various numerical schemes implemented in the Ebysus code is found to model Spitzer conductivity correctly. The energy from a representative nanoflare is found to diffuse 9 Mm within the first second of its lifetime due to Spitzer conductivity alone, strongly dependent on the electron density. Conclusions. The analytical first-order solution is a step forward in ensuring the physical validity of intricate simulations of the Sun. Additionally, since the derivation and argumentation are general, they can easily be followed to treat other nonlinear diffusion problems.

Blow‐up of smooth solutions to the relaxed compressible Navier‐Stokes equations

In this paper, the full compressible Navier‐Stokes equations with Cattaneo's heat transfer law or revised Maxwell's law are considered. We present a loss of initial smoothness of smooth solutions within a finite time and give an upper bound of such time. It is not required that the initial data has compact support or contains vacuum in any finite regions. Moreover, the blow‐up result has no restriction about the specific heat ratio. The main approach is motivated by previous studies and constructing a differential inequality.

Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions

Abstract Compressible Euler equations with space-dependent damping in high dimensions R n ( n = 2 , 3 ) {{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3) are considered in this article. Assuming that the small initial velocity and small perturbation of the initial density have compact support, we establish finite-time blow-up results for the Euler system, by combining energy estimate and new test functions constructed by the solutions of the following linear elliptic partial differential equations system: − G 1 ( x ) + ∇ ⋅ G 2 → ( x ) = 0 , − G 2 → ( x ) + ∇ G 1 ( x ) = μ G 2 → ( x ) ( 1 + ∣ x ∣ ) λ . \left\{\begin{array}{l}-{G}_{1}\left(x)+\nabla \cdot \overrightarrow{{G}_{2}}\left(x)=0,\\ -\overrightarrow{{G}_{2}}\left(x)+\nabla {G}_{1}\left(x)=\frac{\mu \overrightarrow{{G}_{2}}\left(x)}{{(1+| x| )}^{\lambda }}.\end{array}\right. This result generalizes the one in the literature from 1 − D 1-D to high dimension R n ( n = 2 , 3 ) {{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3) .

Kolmogorov equations on the space of probability measures associated to the nonlinear filtering equation: the viscosity approach

. We study the backward Kolmogorov equation on the space of probability measures associated to the Kushner-Stratonovich equation of nonlinear filtering. We prove existence and uniqueness in the viscosity sense and, in particular, we provide a comparison theorem. In the context of stochastic filtering, it is natural to consider measure-valued processes that satisfy stochastic differential equations. In the literature, a classical way to address this problem is by assuming that these measure-valued processes admit a density. Our approach is different and we work directly with measures. Thus, the backward Kolmogorov equation we study is a second-order partial differential equation of parabolic type on the space of probability measures with compact support. In the literature, only few results are available on viscosity solutions for this kind of problems and in particular the uniqueness is a very challenging issue. Here we find a viscosity solution and then we prove that it is unique via comparison theorem.

Numerical investigation of the late-time tails of the solutions of the Fackerell–Ipser equation

The late-time behaviour of the solutions of the Fackerell–Ipser equation (which is a wave equation for the spin-zero component of the electromagnetic field strength tensor) on the closure of the domain of outer communication of sub-extremal Kerr spacetime is studied numerically. Within the Kerr family, the case of Schwarzschild background is also considered. Horizon-penetrating compactified hyperboloidal coordinates are used, which allow the behaviour of the solutions to be observed at the event horizon and at future null infinity as well. For the initial data, pure multipole configurations that have compact support and are either stationary or non-stationary are taken. It is found that with such initial data the solutions of the Fackerell–Ipser equation converge at late times either to a known static solution (up to a constant factor) or to zero. As the limit is approached, the solutions exhibit a quasinormal ringdown and finally a power-law decay. The exponents characterizing the power-law decay of the spherical harmonic components of the field variable are extracted from the numerical data for various values of the parameters of the initial data, and based on the results a proposal for a Price’s law relevant to the Fackerell–Ipser equation is made. Certain conserved energy and angular momentum currents are used to verify the numerical implementation of the underlying mathematical model. In the construction of these currents a discrete symmetry of the Fackerell–Ipser equation, which is the product of an equatorial reflection and a complex conjugation, is also taken into account.

Novel method to determine bedrock from a single sensor based seismic reflection signal using Haar wavelet transform

The determination of bedrock depth is crucial across various earth sciences and related fields. Geophysical techniques, notably the continuous wavelet transform, are increasingly employed to map subsurface bedrock structures. This method allows the transformation of 1D time domain seismic reflection signals into a 2D image, providing time–frequency representations for feature extraction. Leveraging the advantages of wavelet theory, particularly the Haar wavelet known for its simplicity and compact support, this study utilizes continuous Haar wavelet local maxima lines to enhance bedrock analysis. Seismic reflection data is acquired using an extremely sensitive geophone as a receiver along with a high dynamic range data logger. The Common Midpoint (CMP) approach, coupled with optimized offset distances, is utilized to ensure robust data quality and signal strength. Simple sources, such as a sledgehammer, generate sound waves for data collection prior to borehole drilling, correlating acquired data with lithogeological information. This research emphasizes the significance of adjusting parameters like sampling frequency, choice of wavelet, and optimal offset distances. The methodology employed, involving a single geophone, a one-channel data recorder, and a basic sledgehammer as a sound source, offers an inexpensive, straightforward, and non-invasive approach. The study demonstrates the effectiveness of Haar wavelet local maxima lines in accurately determining bedrock depth, producing results closely aligned with both predicted and actual depths.

Warp drives and closed timelike curves

Abstract It is commonly accepted that superluminal travel may be used to facilitate time travel. This is a purely special-relativistic argument, using the fact that for observers in two frames of reference, separated by a spacelike interval, the non-causal (spacelike) future of one observer includes part of the causal past of the other. In this paper we provide a concrete realization of this argument in a curved general-relativistic spacetime, using warp drives as the means of faster-than-light travel. By generalizing the usual warp drive metric to allow for a non-unit lapse function, we allow the warp drive to switch between reference frames in a purely geometric way. With an additional modification allowing the warp drive to have compact support, this permits us to glue two warp drives together to construct a closed timelike geodesic, such that a test particle following the geodesics of the two warp drives travels back to its own past. This provides a precise mathematical model for the connection between faster-than-light travel and time travel in general relativity, and the first such model to be explicitly formulated using two warp drives. We also give a detailed discussion of weak energy condition violations in the non-unit-lapse warp drive.

Radiation of the energy-critical wave equation with compact support

We prove exterior energy lower bounds for (nonradial) solutions to the energy-critical nonlinear wave equation in space dimensions 3≤d≤5, with compactly supported initial data. In particular, it is shown that nontrivial global solutions with compact spatial support must be radiative in a sharp sense. In space dimensions 3 and 4, a nontrivial soliton background is also considered. As an application, we obtain partial results on the rigidity conjecture concerning solutions with the compactness property, including a new proof for the global existence of such solutions.

Convolution equation and operators on the euclidean motion group

Let $G = \mathbb{R}^2\rtimes SO(2)$ be the Euclidean motion group, let g be the Lie algebra of G and let U(g) be the universal enveloping algebra of g. Then U(g) is an infinite dimensional, linear associative and non-commutative algebra consisting of invariant differential operators on G. The Dirac measure on G is represented by $\delta_G$, while the convolution product of functions or measures on G is represented by $\ast$. Among other notable results, it is demonstrated that for each u in U(g), there is a distribution E on G such that the convolution equation $u\ast E = \delta_G$ is solved by method of convolution. Further more, it is established that the (convolution) operator $A^\prime : C^\infty_c(G)\rightarrow C^\infty(G),$, which is defined as $A^\prime f = f\ast T^n\delta(t)$ extends to a bounded linear operator on $L^2(G)$, for $f\in C^\infty_c(G)$, the space of infinitely differentiable functions on G with compact support. Furthermore, we demonstrate that the left convolution operator LT denoted as $L_Tf = T\ast f$ commutes with left translation, for $T\in D^\prime(G)$.

A fourth-order kernel for improving numerical accuracy and stability in Eulerian SPH for fluids and total Lagrangian SPH for solids

The error of smoothed particle hydrodynamics (SPH) using a kernel for particle-based approximation mainly arises from smoothing and integration errors. The choice of the kernel significantly impacts numerical accuracy, stability and computational efficiency. Currently, the most popular kernels, such as B-spline, truncated Gaussian (for compact support), and Wendland kernels, have 2nd-order smoothing errors, and the Wendland kernel has become mainstream in the SPH community due to its stability and accuracy. Since the particle distribution after relaxation can achieve fast convergence of integration error with respect to support radius, it is logical to choose kernels with higher-order smoothing error to improve the numerical accuracy. In this paper, the error of the 4th-order Laguerre-Wendland kernel proposed by Litvinov et al. [1] is revisited, and another 4th-order truncated Laguerre-Gauss kernel is further analyzed and considered to replace the widely used Wendland kernel. The proposed kernel has the following three properties: One is that it avoids the pair-instability problem during the relaxation process, unlike the original truncated Gaussian kernel, and achieves much less relaxation residue than the Wendland and Laguerre-Wendland kernels; One is the truncated compact support size is the same as the non-truncated one of the Wendland kernel, which leads to both kernels' comparably computational efficiency; Another is that the truncation error of this kernel is much less than that of the Wendland kernel. Furthermore, a comprehensive set of 2D and 3D benchmark cases on Eulerian SPH for fluid dynamics and total Lagrangian SPH for solid dynamics validate the considerably improved numerical accuracy of the truncated Laguerre-Gauss kernel without introducing extra computational effort.

The generalized Stieltjes–Poisson transform over Lebesgue spaces and distributions of compact support

The generalized Stieltjes–Poisson transform over Lebesgue spaces and distributions of compact support

Semi-wavefront for a Belousov–Zhabotinskii reaction–diffusion system with spatio-temporal delay

This paper considers a Belousov–Zhabotinskii reaction–diffusion system with spatio-temporal delay. The spatio-temporal delay is modeled as the convolution of u(t,x) with a kernel function g(t,x)≥0, where ∫R∫0+∞g(s,y)dsdy=1. By constructing an auxiliary system, applying Schauder’s fixed point theorem, and using a limiting argument, we demonstrate that the model admits non-negative traveling wave solutions connecting the equilibrium (0,0) to some unknown positive steady states (also referred to as a semi-wavefront) when the wave speed satisfies c≥c∗=21−r. Moreover, no such semi-wavefront exists if c<c∗. It is noteworthy that the kernel function is not required to have compact support in this analysis.

Pointwise estimates on the fundamental solution to nonlocal diffusion equation

In this article we study the pointwise behavior of the fundamental solution S ( t , x ) to the nonlocal diffusion model u t = D ( J ∗ u − u ) on R , S ( t , x ) can be written as e − Dt δ ( x ) + S 0 ( t , x ) , where S 0 ( t , x ) is called the regularizing part. Under some conditions on J and its Fourier transform J ^ , we obtain some new pointwise estimates on S 0 ( t , x ) . Then we apply our results to study the spatial dynamics of the nonlocal diffusion population model in a shifting environment considered in Li et al. [Spatial dynamics of a nonlocal dispersal population model in a shifting environment. J Nonlinear Sci. 2018;28:1189–1219]. In our study the convolution kernel J can be assumed to be more general, i.e. not necessarily of compact support, also the proof is greatly simplified. It is our believe that these pointwise estimates possess their own interest in further studying nonlocal diffusion models.

The revised Riemann–Hilbert approach to the Kaup–Newell equation with a non-vanishing boundary condition: Simple poles and higher-order poles

With a non-vanishing boundary condition, we study the Kaup–Newell (KN) equation (or the derivative nonlinear Schrödinger equation) using the Riemann–Hilbert approach. Our study yields four types of Nth order solutions of the KN equation that corresponding to simple poles on or not on the ρ circle (ρ related to the non-vanishing boundary condition), and higher-order poles on or not on the ρ circle of the Riemann–Hilbert problem (RHP). We make revisions to the usual RHP by introducing an integral factor that ensures the RHP satisfies the normalization condition. This is important because the Jost solutions go to an integral factor rather than the unit matrix when the spectral parameter goes to infinity. To consider the cases of higher-order poles, we study the parallelization conditions between the Jost solutions without assuming that the potential has compact support, and present the generalizations of residue conditions of the RHP, which play crucial roles in solving the RHP with higher-order poles. We provide explicit closed-form formulae for four types of Nth order solutions, display the explicit first-order and double-pole solitons as examples and study their properties in more detail, including amplitude, width, and exciting collisions.

Porous medium type reaction-diffusion equation: Large time behaviors and regularity of free boundary

We consider the Cauchy problem of the porous medium type reaction-diffusion equation∂tρ=Δρm+ρg(ρ),(x,t)∈Rn×R+,n≥2,m>1, where g is the given monotonic decreasing function with the density critical threshold ρM>0 satisfying g(ρM)=0. We prove that the pressure P:=mm−1ρm−1 in Lloc∞(Rn) tends to the pressure critical threshold PM:=mm−1(ρM)m−1 at the time decay rate (1+t)−1. If the initial density ρ(x,0) is compactly supported, we justify that the support {x:ρ(x,t)>0} of the density ρ expands exponentially in time. Furthermore, we show that there exists a time T0>0 such that the pressure P is Lipschitz continuous for t>T0, which is the optimal (sharp) regularity of the pressure, and the free surface ∂{(x,t):ρ(x,t)>0}∩{t>T0} is locally Lipschitz continuous. In addition, under the same initial assumptions of compact support, we verify that the free boundary ∂{(x,t):ρ(x,t)>0}∩{t>T0} is a local C1,α surface.

Optimality Conditions in Control Problems with Random State Constraints in Probabilistic or Almost Sure Form

In this paper, we discuss optimality conditions for optimization problems involving random state constraints, which are modeled in probabilistic or almost sure form. Although the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them with a model based on robust constraints with respect to the (compact) support of the given distribution. Funding: The authors thank the Deutsche Forschungsgemeinschaft [Projects B02 and B04 in the “Sonderforschungsbereich/Transregio 154 Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks”] for support. C. Geiersbach acknowledges support from the Deutsche Forschungsgemeinschaft [Germany’s Excellence Strategy–the Berlin Mathematics Research Center MATH+ Grant EXC-2046/1, Project 390685689]. R. Henrion acknowledges support from the Fondation Mathématique Jacques Hadamard [Program Gaspard Monge in Optimization and Operations Research, including support to this program by Electricité de France].

Non-separable multidimensional multiresolution wavelets: A Douglas-Rachford approach

After re-casting the wavelet construction problem as a feasibility problem with constraints arising from the requirements of compact support, smoothness and orthogonality, the Douglas–Rachford algorithm is employed in the search for multi-dimensional, non-separable, compactly supported, smooth, orthogonal, multiresolution wavelets in the case of translations along the integer lattice and isotropic dyadic dilations. An algorithm for the numerical construction of such wavelets is described. By applying the algorithm, new one-dimensional wavelets are produced as well as genuinely non-separable two-dimensional wavelets.

Learning intrinsic shape representations via spectral mesh convolutions

We introduce spectral-based convolutional operators embedded within Generalized Graph Neural Networks (G-GNNs). These operators enable deep learning on graphs through a learnable, energy-driven evolution process. This approach empowers us to impose specific properties on the graph convolutional kernel directly derived from the corresponding variational formulations. Our model incorporates both parameterized and non-parameterized graph Laplacian-based energies within the generalized graph convolutional layer to address features like smoothness, sharpness, and compact support. By making appropriate choices within our G-GNN framework, we pave the way for designing novel paradigms for 3D shape representation, reconstruction, and processing, while also enabling effective feature embeddings for intrinsic neural fields.

PT -symmetry-enabled stable modes in a multicore fiber

Open systems with balanced gain and loss, described by parity-time (PT-symmetric) Hamiltonians have been deeply explored over the past decade. Most explorations are limited to finite discrete models (in real or reciprocal spaces) or continuum problems in one dimension. As a result, these models do not leverage the complexity and variability of two-dimensional continuum problems on a compact support. Here, we investigate eigenvalues of the Schrödinger equation on a disk with zero boundary condition, in the presence of constant, PT-symmetric, gain-loss potential that is confined to two mirror-symmetric disks. We find a rich variety of exceptional points, re-entrant PT-symmetric phases, and a nonmonotonic dependence of the PT-symmetry breaking threshold on the system parameters. By comparing results of two model variations, we show that this simple model of a multicore fiber supports propagating modes in the presence of gain and loss. Published by the American Physical Society 2024

Nonparametric estimation of densities on the hypersphere using a parametric guide

AbstractHyperspherical kernel density estimators (KDE), which use a parametric distribution as a guide, are studied in this paper. The main benefit is that these estimators improve the bias of nonguided kernel density estimators when the parametric guiding distribution is not too far from the true density, while preserving the variance. When using a von Mises‐Fisher density as guide, the proposal performs as well as the classical KDE, even when the guiding model is incorrect, and far from the true distribution. This benefit is particular for the hyperspherical setting given its compact support, and is in contrast to similar methods for real valued data. Moreover, we deal with the important issue of data‐driven selection of the smoothing parameter. Simulations and real data examples illustrate the finite‐sample performance of the proposed method, also in comparison with other recently proposed estimation methods.