Nonlinear Diffusion Research Articles (Page 1)

Nowadays a vast literature is available on the Hele-Shaw or incompressible limit for nonlinear degenerate diffusion equations. This problem has attracted a lot of attention due to its applications to tissue growth and crowd motion modeling as it constitutes a way to link soft congestion (or compressible) models to hard congestion (or incompressible) descriptions. In this paper, we address the question of estimating the rate of this asymptotics in the presence of external drifts. In particular, we provide improved results in the 2-Wasserstein distance which are global in time thanks to the contractivity property that holds for strictly convex potentials.

Self-similar solutions of fast-reaction limit problems with nonlinear diffusion

On linear and nonlinear diffusion in liquids according to linear nonequilibrium thermodynamics

Li-Yau estimates and Harnack inequalities for nonlinear slow diffusion equations on a smooth metric measure space

Disorders characterised by changes in dopamine (DA) neurotransmission are often linked to changes in the temporal discounting of future rewards. Likewise, pharmacological manipulations of DA neurotransmission in healthy individuals modulate temporal discounting, but there is considerable variability in the directionality of reported pharmacological effects, as enhancements and reductions of DA signalling have been linked to both increases and reductions of temporal discounting. This may be due to meaningful individual differences in drug effects and/or false positive findings in small samples. To resolve these inconsistencies, we 1) revisited pharmacological effects of the DA precursor L-DOPA on temporal discounting in a large sample of N = 76 healthy participants (n = 44 male) and 2) examined several putative proxy measures for DA to revisit the role of individual differences in a randomised, double-blind placebo-controlled pre-registered study (https://osf.io/a4k9j/). Replicating previous findings, higher rewards were discounted less (magnitude effect). Computational modelling using hierarchical Bayesian parameter estimation confirmed that the data in both drug conditions were best accounted for by a non-linear temporal discounting drift diffusion model. In line with recent animal and human work, L-DOPA reliably reduced the discount rate with a small effect size, challenging earlier findings in substantially smaller samples. We found no credible evidence for effects of putative DA proxy measures on model parameters, calling into question the role of these measures in accounting for individual differences in DA drug effects.Significance statement Dopamine is involved in valuing future rewards and influences how individuals weigh immediate versus delayed rewards - a phenomenon known as temporal discounting. Understanding the role of dopamine is crucial, since disorders characterised by changes in dopamine neurotransmission, including substance use disorders and behavioural addictions, are linked to changes in temporal discounting. There is evidence from pharmacological studies that manipulating dopaminergic neurotransmission modulates temporal discounting. However, the findings are highly mixed and suggest that this may be due to individual differences in baseline dopamine. To address this, we assessed effects of the dopamine precursor L-DOPA on intertemporal choice in healthy volunteers and examined several putative proxy measures for baseline dopamine to discern the role of individual differences in modulating drug effects.

The rapid advancement of micro-electromechanical systems (MEMS) and the extensive utilization of ultra-short time heating technology in the precision machining of these devices have underscored the critical importance of studying the interplay between thermodiffusion and mechanical deformation. At the microscale, the size-dependent effect in elastic deformation and the memory-dependent effect in thermal transport processes become increasingly significant and must be taken into account. Meanwhile, many experimental and theoretical investigations suggest that, in practical analyses, thermal conductivity and diffusivity in materials should not be considered as a constant value. This paper addresses the thermoelastic diffusion response of a spherical microshell subjected to sinusoidal thermal and concentration loading including the simultaneous effects of the fractional-order parameter, the nonlocal parameter, and the variable thermal conductivity and diffusivity. Taking into account the variable thermal conductivity and diffusivity, the nonlinear governing equations are derived by the Laplace and Kirchhoff transformations. The results show that the nonlinear thermoelastic diffusion response of the spherical microshell can be adjusted by the suitably modified parameters, which strongly depend on the size-dependent effect, memory-dependent effect, and the variable thermal conductivity. It is hoped that the obtained results would be helpful in designing the microstructures induced by an ultra-short time heating.

ABSTRACTIn this paper, we study a stochastic variant of the classical Keller–Segel system on a two‐dimensional domain, where the leading diffusion term is replaced by a porous media operator and the dynamics are perturbed by a pair of independent Wiener processes. The model describes the interaction between the cell density and the concentration of a chemoattractant , incorporating nonlinear diffusion, chemotactic sensitivity, production and damping effects, together with multiplicative stochastic perturbations of strengths and . Since the randomness is intrinsic, the stochastic terms are interpreted in the Stratonovich sense. To construct solutions, we introduce an integral operator and establish its continuity and compactness properties in a suitable Banach space. This leads to a stochastic analogue of the Schauder‐Tychonoff‐type fixed point theorem tailored to our framework, which ensures the existence of a martingale solution. Furthermore, we establish pathwise uniqueness, uniqueness in law, and the existence of strong solutions. The uniqueness results, however, require additional assumptions on the chemoattractant noise and the initial condition of .

A hash function is a mathematical model that maps inputs of arbitrary size to unique outputs of a fixed length in bits. Hash functions are highly useful and appear in almost all information security applications. In addition to information security applications, it can also serve as index data in hash tables, aiding in the detection of duplicate data for fingerprinting or uniquely identifying files, as well as for checksums to identify data corruption. This research introduces an innovative 256-bit hash function that utilizes a chaotic substitution box using a non-linear logistic map. Unlike MD5 or SHA-family hash functions, which rely on modular arithmetic, logical operations, and bitwise shifts for diffusion and non-linearity, the proposed method incorporates a chaotic substitution box to introduce an additional nonlinear transformation layer and high diffusion. The avalanche rate, statistical analysis, pre-image resistance, second pre-image, collision resistance, and performance are examined to evaluate the cryptographic strength and the performance of the proposed method.

Global existence and boundedness in a 3D predator–prey system with nonlinear diffusion and prey-taxis

For 3D strongly nonlinear equilibrium radiation diffusion equation, a kind of new meshfree iteration methods based on Richtmyer, factorization, and Newton linearization techniques in complex domains is provided. The new methods avoid the difficulty of mesh generation while ensuring high accuracy. Through specific numerical experiments, it is shown that the new methods have good approximation performance for solving 3D strongly nonlinear problems in complex domains, and all achieve good precision with iteration counts of 2 nearly under the given parameters. The comparison between finite element method (FEM) with the presented methods shows that our new methods match with FEM when the time step is relatively small, but avoid mesh generating.

Nonlinear Diffusion of Magnetic Field in a Double-Layer Conductor with Deposited Layer of Higher Conductivity

Global boundedness in a chemotaxis-Stokes system with nonlinear diffusion mechanism involving gradient dependent flux limitation and indirect signal production

Well-posedness of the obstacle problem for stochastic nonlinear diffusion equations: An entropy formulation

Objectives: Retrievability is an important characteristic for an endodontic sealer. This study compared the retrieval of Endoseal MTA, AH Plus, and MTA Fillapex in endodontic retreatment using cone-beam computed tomography (CBCT). Materials and Methods: This in vitro study was conducted on 45 radiographically confirmed single-rooted and single-canal mandibular premolars. The root canals were instrumented by the Denco Universal rotary system up to size F3, and randomly divided into 3 groups for the application of either AH Plus, Endoseal MTA, or MTA Fillapex and subsequent root canal obturation. Next, the root filling materials were removed using chloroform solvent and Denco Universal retreatment rotary system. The roots then underwent CBCT with the Gaussian and nonlinear diffusion filters for noise reduction. The residual sealer volume on the root canal walls was quantified on axial CBCT sections using MATLAB R2012 version 14 software. Data were analyzed by the Kruskal-Wallis and Dunn tests (alpha=0.05). Results: The residual sealer volume was significantly different among the three groups (P<0.001) and was the highest in Endoseal MTA (2.70±2.41mm3). The residual volume of Endoseal MTA sealer was significantly greater than MTA Fillapex (P=0.009). However, the difference between AH Plus and Endoseal MTA (P=0.592), or AH Plus and MTA Fillapex (P=0.352) was not significant. Conclusion: Within the limitations of this in vitro study and considering the residual sealer volume, retrieval of Endoseal MTA was comparable to that of AH Plus, and lower than that of MTA Fillapex in endodontic retreatment.

When water is present in a medium with pore sizes in a range of approximately 10 nm, the corresponding freezing-point depression will cause long-range broadening of a melting front. Describing the freezing-point depression by the Gibbs–Thomson equation and the pore-size distribution by a power law, we derive a nonlinear diffusion equation for the fraction of melted water. This equation yields superdiffusive spreading of the melting front with a diffusion exponent, which is given by the spatial dimension and the exponent describing the pore size distribution. We derive this solution analytically from energy conservation in the limit where all the energy is consumed by the melting and explore the validity of this approximation numerically. Finally, we explore a geological application of the theory to the case of one-dimensional subsurface melting fronts in granular or soil systems. These fronts, which are produced by heating of the surface, spread at a superdiffusive rate and affect the subsurface to significantly larger depths than a system without the effects of freezing-point depression.

Lie symmetry analysis, rogue waves, periodic waves and various analytical solutions of non-linear diffusion equation with applications

When sound signals are transmitted via audio cables in form of current signals, detectable electromagnetic leakage is emitted around the cables, which could reveal the sound content being transmitted. This paper presents a sound eavesdropping system, EMTap, which senses electromagnetic variation around audio cables leveraging an RFID tag attached to an audio cable for sound content recovery. In some audiovisual scenarios with audio cables, such as offices and meeting rooms, it is assumed that attackers secretly attach a small, battery-free RFID tag to an audio cable without being noticed. Meanwhile, RFID readers are camouflaged as decorations/public facilities placed in/out of rooms to transmit and receive RF signals. When the sound signals are transmitted via audio cables, EMTap first activates the RFID tag to capture the electromagnetic variation patterns around audio cables upon RF signals. Then, EMTap reconstructs sound spectrograms from RF signal-based electromagnetic variation patterns utilizing a designed cross-modal Generative model, Nonlinear Stable Diffusion Model (NSDM). Finally, EMTap converts the sound spectrograms to sound content through sound recognition API. Extensive experiments demonstrate that EMTap achieves an average Mel-Cepstral Distortion (MCD) of 5.49 and Word Error Rate (WER) of 16.19% for eavesdropping on sound transmitted via audio cables.

Nonlinear radiation diffusion equations, essential for modeling energy transport in applications like inertial confinement fusion and astrophysical systems, present significant computational challenges due to their strong nonlinearity and tight coupling. Traditional numerical methods, relying on time-marching and Picard iterations, are computationally intensive and lack adaptability to varying conditions. This paper proposes two data-driven operator learning architectures, named nonlinear radiation diffusion neural surrogate models (NRD-NSMs), that integrate Fourier neural operators (FNO) and deep operator networks to learn mappings from equation conditions to solutions of the radiation diffusion equations at specific time snapshots. The Type-1 NRD-NSM uses FNO to generate basis functions and a fully connected network (FCN) for coefficients, while the Type-2 NRD-NSM employs element-wise feature combination to merge outputs from multiple encoders, followed by an FNO-based processor and an FCN decoder. Our results demonstrate relative L2 errors on the order of 10−2 across all experiments, establishing these models as accurate surrogates. Notably, the Type-2 NRD-NSM generally achieves higher accuracy than the Type-1 NRD-NSM while maintaining comparable runtime. The proposed models achieve remarkable speedups, up to 50 times on the central processing unit and 2000 times on the graphics processing unit compared to the finite element method, by eliminating iterative processes. Numerical results underscore the potential of NRD-NSMs as fast, accurate, and generalizable surrogates for traditional solvers, enabling efficient simulations in complex physical systems.

Global existence, traveling wave solutions and Hopf bifurcation analysis in a flame propagation model with nonlinear diffusion and advection

Aqueous foams are solid materials composed of gases and liquids, exhibiting a large gas/liquid surface area and enabling dynamic exchanges between their fluid components. The structure of binary-gas foams, whose bubbles consist of a mixture of two gases having different affinities with the liquid, thus offers real potential for the dynamic separation of these gases at low cost. In single-gas foams, the structure evolves under the effect of gas flow induced by Laplace pressure differences, arising from heterogeneities in bubble size. This leads to the well-documented Ostwald ripening. In addition to these capillary effects, the structure of binary-gas foams can evolve under the effect of gas flow induced by partial pressure differences, arising from heterogeneities in bubble composition. We experimentally investigate the shrinking of CO2-laden 2D foams exposed to air, observing a crust of tiny bubbles at the front. We derive a nonlinear diffusion model for the gas in the foam and propose a description of the whole foam as an effective, homogeneous medium, the key parameter being the gas permeability ratio across the foam's soap films (≠1 for CO2/air). The effective diffusivity of the gas in the foam emerges from the coupling between foam structure and gas transport across soap films. We extrapolate it for various permeability ratios and show that it can vary continuously between the diffusivity of the gas in the liquid and that of the gas in the atmosphere, enabling tunable gas retention and release by controlling the composition of the atmosphere.