Range Of Exponents Research Articles

A porous medium equation with spatially inhomogeneous absorption. Part I: Self-similar solutions

This is the first of a two-parts work on the qualitative properties and large time behavior for the following quasilinear equation involving a spatially inhomogeneous absorption∂tu=Δum−|x|σup, posed for (x,t)∈RN×(0,∞), N≥1, and in the range of exponents 1<m<p<∞, σ>0. We give a complete classification of (singular) self-similar solutions of the formu(x,t)=t−αf(|x|t−β),α=σ+2σ(m−1)+2(p−1),β=p−mσ(m−1)+2(p−1), showing that their form and behavior strongly depends on the critical exponentpF(σ)=m+σ+2N. For p≥pF(σ), we prove that all self-similar solutions have a tail as |x|→∞ of one of the formsu(x,t)∼C|x|−(σ+2)/(p−m)oru(x,t)∼(1p−1)1/(p−1)|x|−σ/(p−1), while for m<p<pF(σ) we add to the previous the existence and uniqueness of a compactly supported very singular solution. These solutions will be employed in describing the large time behavior of general solutions in a forthcoming paper.

On the trapped magnetic moment in type-II superconductors

Abstract Measurements of the trapped (remanent) magnetic moment, $M_{trap}\left(H\right)$, when a small magnetic field $H$ is turned off after cooling below the superconducting transition temperature, $T_c$, or ramping a magnetic field up and down after cooling in a zero magnetic field, offer substantial advantages in difficult cases of small samples and large field-dependent backgrounds, which is relevant for hydrogen-based ultra-high-$T_{c}$ superconductors (UHTS). Until recently, there was no need for a separate paper on the trapped magnetic flux for well-known critical-state models due to the simplicity of the physics involved. However, recent publications showed the need for such an analysis. This note summarizes the expectations for the Bean model with constant critical current density and the Kim model with field-dependent critical currents. It is shown that if the trapped moment is fitted to the power law, $M_{trap}\propto H^{\alpha}$, the fixed exponent $\alpha=2$ is exact for the Bean model, while Kim models show a wide interval of possible values, $2\leq\alpha\leq4$. Furthermore, accounting for reversible magnetization expands the range of possible exponents to $1\leq\alpha\leq4$. In addition, demagnetizing factors are essential and make the trapped moment orientation dependent even in isotropic materials.&amp;#xD;&amp;#xD;As a concrete application, it is shown that flux trapping experiments performed on H$_{3}$S UHTS compounds can be well described using this generalized approach, lending further support to the type-II superconducting nature of H$_{3}$S under ultra-high pressure.

Weakly coupled systems of semilinear σ$$ \sigma $$‐evolution equations with friction and visco‐elastic type damping

In the present paper, we are interested to study global (in time) existence of small data Sobolev solutions to the Cauchy problem for a weakly coupled system of two semilinear ‐evolution equations with friction and visco‐elastic type damping, where for . We study model (1.1) (see below) in several cases with respect to the regularity for the data: First, we assume data . By using estimates of Sobolev solutions to related linear models with vanishing right‐hand side, we explain the admissible range of exponents in (1.1) which allow to prove the global (in time) existence of small data Sobolev solutions. Then we suppose that the data belong to , where now , . So the data have additional regularity to the first assumed integrability. We restrict ourselves to data from energy space (on the basis of ) and from energy space with additional suitable higher regularity. We compare in our statements the admissible set of exponents and in the power nonlinearities with the so‐called modified Fujita exponent. Finally, some blow‐up results complete the paper.

AC Electric Conductivity of High Pressure and High Temperature Formed NaFePO4 Glassy Nanocomposite.

Olivine-like NaFePO4 glasses and nanocomposites are promising materials for cathodes in sodium batteries. Our previous studies focused on the preparation of NaFePO4 glass, transforming it into a nanocomposite using high-pressure-high-temperature treatment, and comparing both materials' structural, thermal, and DC electric conductivity. This work focuses on specific features of AC electric conductivity, containing messages on the dynamics of translational processes. Conductivity spectra measured at various temperatures are scaled by apparent DC conductivity and plotted against frequency scaled by DC conductivity and temperature in a so-called master curve representation. Both glass and nanocomposite conductivity spectra are used to test the (effective) exponent using Jonscher's scaling law. In both materials, the values of exponent range from 0.3 to 0.9, with different relation to temperature. It corresponds to the electronic conduction mechanism change from low-temperature Mott's variable range hopping (between Fe2+/Fe3+ centers) to phonon-assisted hopping, which was suggested by previous DC measurements. Following the pressure treatment, AC conductivity activation energies were reduced from EAC≈0.40 eV for glass to EAC≈0.18 eV for nanocomposite and are lower than their DC counterpart, following a typical empirical relation with the value of the exponent. While pressure treatment leads to a 2-3-orders-of-magnitude rise in the AC and apparent DC conductivity due to the reduced distance between the hopping centers, a nonmonotonic relation of AC power exponent and temperature is observed. It occurs due to the disturbance of polaron interactions with Na+ mobile ions.

Characteristics of concurrent flame spread over convex charring materials

In real forest fire scenarios, there are a large number of convex and concave terrains at the junction of flat land and slopes. In this paper, a typical charring material, kraft paper, is selected as the experimental sample, and the variation law of key characteristic parameters of flame spread over convex solid surface are investigated. The experimental results show that all the flame spread processes have obvious deceleration behaviors, and the influence mechanism of flame tangent angle on the flame spread deceleration of convex surfaces is revealed. The characteristic length of the pyrolysis zone shows a tendency of increasing and then decreasing with time. The variation curves of radiant heat flux present two peaks. Combining with the characteristic length, the reason for the occurrence of two peaks is explained. The exponential relationship between the instantaneous mass loss rate and the characteristic length of the pyrolysis zone is obtained. Based on the range of power exponents, the heat transfer controlling mechanism in the pyrolysis zone is classified into three categories for all the working conditions. Moreover, a piecewise power-law relationship between the dimensionless maximum flame height and the dimensionless heat release rate is established. The results of this study will fill the research gap of flame spread of solid materials with curved surfaces, and will be of great academic value for the improvement and development of theories on building and forest fire safety.

Local existence and uniqueness of solutions to the time-dependent Kohn–Sham equations coupled with classical nuclear dynamics

We prove the short-time existence and uniqueness of solutions to the initial-value problem associated with a class of time-dependent Kohn–Sham equations coupled with Newtonian nuclear dynamics, combining Yajima's theory for time-dependent Hamiltonians with Duhamel's principle, based on suitable Lipschitz estimates. We consider a pure power exchange term within a generalisation of the so-called local-density approximation, identifying a range of exponents for the existence and uniqueness of H2 solutions to the Kohn–Sham equations.

Global Existence of Small Data Solutions to Weakly Coupled Systems of Semi-Linear Fractional σ–Evolution Equations with Mass and Different Nonlinear Memory terms

We study in this paper the long-term existence of solutions to the system of weakly coupled equations with fractional evolution and various nonlinearities. Our objective is to determine the connection between the regularity assumptions on the initial data, the memory terms, and the permissible range of exponents in a specific equation . Using Lp−Lq estimates for solutions to the corresponding linear fractional σ–evolution equations with vanishing right-hand sides, and applying a fixed-point argument, the existence of small data solutions is established for some admissible range of powers (p1,p2,…,pk).

An n-dimensional polynomial modulo chaotic map with controllable range of Lyapunov exponents and its application in color image encryption

In chaos-based image encryption, better performance of the chaotic system represents higher security of the encryption algorithm, and the Lyapunov exponent (LE) is an important measure to evaluate the chaotic performance. In this paper, an n-dimensional(n-D) polynomial modulo chaotic map (nD-PMCM) is proposed as a general model, whose range of Lyapunov exponent is controlled by the system parameters and the robust chaotic behavior with the desired complexity can be obtained. And an instantiated 2D-PMCM is constructed by setting the corresponding parameters in nD-PMCM, which exhibits superior chaotic performance compared to other chaotic maps. Then a novel color image encryption scheme based on the 2D-PMCM is also constructed, which mainly uses cross-plane cyclic shift permutation and cross-plane XOR diffusion. Different from the traditional encryption scheme with separate processing of color planes, the cross-plane operation disrupts the correlation between color image planes and has higher encryption efficiency due to the ability to disrupt the pixel position while also changing the pixel value. The experimental simulation results show that this encryption scheme has better security and encryption effects.

A note on energy and cross‐helicity conservation in the ideal magnetohydrodynamic equations

In this paper, we are concerned with the conservation of total energy and cross‐helicity for the weak solutions in the ideal magnetohydrodynamic (MHD) equations. In the spirit of recent works of Berselli (J. Differ. Equ. 368 (2023), 350–375.) and Berselli‐Georgiadis (NoDEA Nonlinear Differ. Equ. Appl. 31 (2024), 33), by establishing a new generalized Constantin‐E‐Titi type commutator estimate to allow us to make full use of the total energy, we extend the previous classical results to a wider range of exponents. These results indicate the role of the time integrability, spatial integrability, and differential regularity of the velocity (magnetic field) in the conserved quantities of weak solutions in the ideal MHD equations.

A unified theory of the self-similar supersonic Marshak wave problem

We present a systematic study of the similarity solutions for the Marshak wave problem in the local thermodynamic equilibrium (LTE) diffusion approximation and in the supersonic regime. Self-similar solutions exist for a temporal power law surface temperature drive and a material model with power law temperature dependent opacity and energy density. The properties of the solutions in both linear and nonlinear conduction regimes are studied as a function of the temporal drive, opacity, and energy density exponents. We show that there exists a range of the temporal exponent for which the total energy in the system decreases, and the solution has a local maxima. For nonlinear conduction, we specify the conditions on the opacity and energy density exponents under which the heat front is linear or even flat and does possess its common sharp characteristic; this characteristic is independent of the drive exponent. We specify the values of the temporal exponents for which analytical solutions exist and employ the Hammer–Rosen perturbation theory to obtain highly accurate approximate solutions, which are parameterized using only two numerically fitted quantities. The solutions are used to construct a set of benchmarks for supersonic LTE radiative heat transfer, including some with unusual and interesting properties such as local maxima and non-sharp fronts. The solutions are compared in detail to implicit Monte Carlo and discrete-ordinate transport simulations as well gray diffusion simulations, showing a good agreement, which highlights their usefulness as a verification test problem for radiative transfer simulations.

The very singular solution for the Anisotropic Fast Diffusion Equation and its consequences

We construct the Very Singular Solution (VSS) for the Anisotropic Fast Diffusion Equation (AFDE) in the suitably good exponent range. VSS is a solution that, starting from an infinite mass located at one point as initial datum, evolves as an admissible solution of the corresponding equation everywhere away from the point singularity. It is expected to represent important properties of the fundamental solutions when the initial mass is very big. Here we work in the whole Euclidean space.In this setting we show how the diffusion process distributes mass from the initial infinite singularity with different rates along the different space directions. Indeed, and up to constant factors, there is a simple partition formula for the anisotropic mass expansion, given approximately as the minimum of separate 1-D VSS solutions. This striking fact is a consequence of the improved scaling properties of the special solution, and it has strong consequences.If we consider the family of fundamental solutions for different masses, we prove that they all share the same universal tail behaviour (i.e., for large |x|) as the VSS. Namely, their tail is asymptotically convergent to the unique VSS tail. This means that the VSS partition formula holds also for the fundamental solutions at spatial infinity. With the help of this analysis we study the behaviour of the class of nonnegative finite-mass solutions of the Anisotropic FDE, and prove the Global Harnack Principle (GHP) and the Asymptotic Convergence in Relative Error (ACRE) under a natural assumption on the decay of the initial tail.

Intrinsic Harnack’s Inequality for a General Nonlinear Parabolic Equation in Non-divergence Form

AbstractWe prove the intrinsic Harnack’s inequality for a general form of a parabolic equation that generalizes both the standard parabolic p-Laplace equation and the normalized version arising from stochastic game theory. We prove each result for the optimal range of exponents and ensure that we get stable constants.

The Primitive Exponent Set of a Class of Representative Twocolored Digraph in Graph Theory

Graph is simple and intuitive. It can be used to solve many problems in computer science. For a kind of representativedigraph, the edges (arcs) of the digraph are colored with red and blue colors. The range of the primitive exponent are discussed indifferent cases, and the extremal two-colored digraphs are found by coloring all arcs with two colors. Finally, the primitive exponentset is given. The results can provide a reference for the study of primitive exponent of three-colored digraph and the application ofgraph coloring in computer science, such as communication network and coding cache.

Flow of a colloidal solution in an orthogonal rheometer

The flow of a colloidal solution between two parallel disks rotating with the same angular velocity about two non-coincident axes was studied. The problem has been approached from two perspectives, the first wherein the stress is expressed in terms of a power-law of kinematical quantities, and the second wherein we consider a non-standard model where the symmetric part of the velocity gradient is given by a power-law of the stress. For a range of power-law exponents, the class of models are non-invertible. By varying the material and geometric parameters, changes in the flow behavior at different Reynolds numbers were analyzed. We find that pronounced boundary layers develop even at low Reynolds numbers depending on the power-law exponents. The new class of stress power-law fluids and fluids that exhibit limiting stress also show the ability to develop boundary layers.

Spatio-temporal variability of small-scale leads based on helicopter maps of winter sea ice surface temperatures

Observations of sea ice surface temperature provide crucial information for studying Arctic climate, particularly during winter. We examined 1 m resolution surface temperature maps from 35 helicopter flights between October 2, 2019, and April 23, 2020, recorded during the Multidisciplinary drifting Observatory for the Study of Arctic Climate (MOSAiC). The seasonal cycle of the average surface temperature spanned from 265.6 K on October 2, 2019, to 231.8 K on January 28, 2020. The surface temperature was affected by atmospheric changes and varied across scales. Leads in sea ice (cracks of open water) were of particular interest because they allow greater heat exchange between ocean and atmosphere than thick, snow-covered ice. Leads were classified by a temperature threshold. The lead area fraction varied between 0% and 4% with higher variability on the local (5–10 km) than regional scale (20–40 km). On the regional scale, it remained stable at 0–1% until mid-January, increasing afterward to 4%. Variability in the lead area is caused by sea ice dynamics (opening and closing of leads), as well as thermodynamics with ice growth (lead closing). We identified lead orientation distributions, which varied between different flights but mostly showed one prominent orientation peak. The lead width distribution followed a power law with a negative exponent of 2.63, which is in the range of exponents identified in other studies, demonstrating the comparability to other data sets and extending the existing power law relationship to smaller scales down to 3 m. The appearance of many more narrow leads than wide leads is important, as narrow leads are not resolved by current thermal infrared satellite observations. Such small-scale lead statistics are essential for Arctic climate investigations because the ocean–atmosphere heat exchange does not scale linearly with lead width and is larger for narrower leads.

Bloom weighted bounds for sparse forms associated to commutators

In this paper we consider bilinear sparse forms intimately related to iterated commutators of a rather general class of operators. We establish Bloom weighted estimates for these forms in the full range of exponents, both in the diagonal and off-diagonal cases. As an application, we obtain new Bloom bounds for commutators of (maximal) rough homogeneous singular integrals and the Bochner–Riesz operator at the critical index. We also raise the question about the sharpness of our estimates. In particular we obtain the surprising fact that even in the case of Calderón–Zygmund operators, the previously known quantitative Bloom bounds are not sharp for the second and higher order commutators.

Three results on the energy conservation for the 3D Euler equations

We consider the 3D Euler equations for incompressible homogeneous fluids and we study the problem of energy conservation for weak solutions in the space-periodic case. First, we prove the energy conservation for a full scale of Besov spaces, by extending some classical results to a wider range of exponents. Next, we consider the energy conservation in the case of conditions on the gradient, recovering some results which were known, up to now, only for the Navier–Stokes equations and for weak solutions of the Leray-Hopf type. Finally, we make some remarks on the Onsager singularity problem, identifying conditions which allow to pass to the limit from solutions of the Navier–Stokes equations to solution of the Euler ones, producing weak solutions which are energy conserving.

Global regularity for nonlinear systems with symmetric gradients

We study global regularity of nonlinear systems of partial differential equations depending on the symmetric part of the gradient with Dirichlet boundary conditions. These systems arise from variational problems in plasticity with power growth. We cover the full range of exponents p∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in (1,\\infty )$$\\end{document}. As a novelty the degenerate case for p>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>2$$\\end{document} is included. We present a unified approach for all exponents by showing the regularity for general systems of Orlicz growth.

Simulations and models for the Richtmyer–Meshkov instability with broadband perturbations

The Richtmyer–Meshkov instability (RMI) is shock driven and affects many phenomena from inertial fusion to supernova explosions. The behavior of single-modes in the RMI has been studied extensively but less is known with the broadband perturbations that occur in applications. Here, we describe extensive numerical simulations and modeling of the RMI with broadband perturbations with an initial power spectrum of the form P ∝ km, where k is the wavenumber. The hydrodynamic condition is the same as in the θ-Group Collaboration [Thornber et al., “Late-time growth rate, mixing, and anisotropy in the multimode narrowband Richtmyer–Meshkov instability: The θ-group collaboration,” Phys. Fluids 29, 105107 (2017)] with a Mach ∼1.86 shock and gamma-law = 5/3 fluids with Atwood number A = −0.5. The bubble amplitude hB is found to grow in two stages. Initially, hB undergoes a phase-inversion and grows linearly in time (t) at a rate consistent with a linear theory. Asymptotically, hB grows as a power law ∼tθ when k|hB| &amp;gt; O(1), where k is the average wavenumber for the initial spectrum. The RMI behavior in simulations and modeling agree over a wide range of exponent m, spectral width, initial amplitude, and time. The agreement is quantified objectively using statistical analysis.

Chromatin phase separated nanoregions explored by polymer cross-linker models and reconstructed from single particle trajectories.

Phase separated domains (PSDs) are ubiquitous in cell biology, representing nanoregions of high molecular concentration. PSDs appear at diverse cellular domains, such as neuronal synapses but also in eukaryotic cell nucleus, limiting the access of transcription factors and thus preventing gene expression. We develop a generalized cross-linker polymer model, to study PSDs: we show that increasing the number of cross-linkers induces a polymer condensation, preventing access of diffusing molecules. To investigate how the PSDs restrict the motion of diffusing molecules, we compute the mean residence and first escaping times. Finally, we develop a method based on mean-square-displacement of single particle trajectories to reconstruct the properties of PSDs from the continuum range of anomalous exponents. We also show here that PSD generated by polymers do not induces a long-range attracting field (potential well), in contrast with nanodomains at neuronal synapses. To conclude, PSDs can result from condensed chromatin organization, where the number of cross-linkers controls molecular access.