arXiv subjects

Isanka Garli Hevage

Publications and source records attributed to Isanka Garli Hevage.

7 recordsLinked to original sources

arXiv2026

Traveling-Wave Solutions for an Einstein-Type Material-Balance Model of the Chemotactic Transport

We develop a nonlinear continuum transport model describing the formation of localized traveling structures in a coupled two-phase medium. The model is derived from an Einstein-type material-balance formulation in which displacement is generated by diffusion and by the gradient of a background-dependent transport mechanism. The resulting system couples diffusion, nonlinear gradient-driven transport, and depletion of the background phase. The proposed framework is applicable to general chemotactic transport and, in particular, to problems related to the formation of oil and gas deposits. We analyze traveling-wave solutions and establish the existence of coherent traveling bands in the transport-dominated regime. The mobile phase is shown to form a unique one-hump profile for any given reference time, while the background component undergoes a positive, bounded monotone logistic-type transition between asymptotic states. An explicit representation of the traveling profile is obtained, and uniqueness is proved up to translation. We further derive the linearized perturbation operator around the traveling band and establish finite-time perturbation bounds through a maximum-principle argument. Finally, the traveling-wave system is reduced to a nonlinear third-order ordinary differential equation for the background profile, providing an alternative characterization of the coherent structure.

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arXiv2024

Stability Analysis of Degenerate Einstein Model of Brownian Motion

Our Recent advancements in stochastic processes have illuminated a paradox associated with the Einstein model of Brownian motion. The model predicts an infinite propagation speed, conflicting with the second law of thermodynamics. The modified model successfully resolves the issue, establishing a finite propagation speed by introducing a concentration-dependent diffusion matrix. In this paper, we outline the necessary conditions for this property through a counter-example. The second part of the paper focuses on the stability analysis of the solution of the degenerate Einstein model. We introduce a functional dependence on the solution that satisfies a specific ordinary differential inequality. Our investigation explores the solution's dependence on the boundary and initial data of the original problem, demonstrating asymptotic stability under various conditions. These results have practical applications in understanding stochastic processes within bounded domains.

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arXiv2023

Nonlinear Einstein paradigm of Brownian motion and localization property of solutions

We employ a generalization of Einstein's random walk paradigm for diffusion to derive a class of multidimensional degenerate nonlinear parabolic equations in non-divergence form. Specifically, in these equations, the diffusion coefficient can depend on both the dependent variable and its gradient, and it vanishes when either one of the latter does. It is known that solutions of such degenerate equations can exhibit finite speed of propagation (so-called localization property of solutions). We give a proof of this property using a De Giorgi--Ladyzhenskaya iteration procedure for non-divergence-from equations. A mapping theorem is then established to a divergence-form version of the governing equation for the case of one spatial dimension. Numerical results via a finite-difference scheme are used to illustrate the main mathematical results for this special case. For completeness, we also provide an explicit construction of the one-dimensional self-similar solution with finite speed of propagation function, in the sense of Kompaneets--Zel'dovich--Barenblatt. We thus show how the finite speed of propagation quantitatively depends on the model's parameters.

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arXiv2022

Einstein model of the movement of small particles in a stationary liquid revisited: Finite Propagation Speed

The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix dependent of the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentration. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobolev-type inequality and the Ladyzhenskaya-Uraltseva iteration lemma.

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arXiv2022

The finite speed of propagation in the degenerate Einstein-Brownian motion model

We considered the qualitative behavior of the generalization of Einstein's model of Brownian motion when the key parameter of the time interval of \textit{free jump} degenerates. Fluids will be characterized by the number of particles per unit volume (density of fluid) at the point of observation. Degeneration of the phenomenon manifests in two scenarios: a) flow of the fluid, which is highly dispersing like a non-dense gas, and b) flow of fluid far away from the source of flow, when the velocity of the flow is incomparably smaller than the gradient of the density. First, we will show that both types of flows can be modeled using the Einstein paradigm. We will investigate the question: What features will particle flow exhibit if the time interval of the\textit{ free jump} is inverse proportional to the density and its gradient? We will show that in this scenario, the flow exhibits localization property, namely: if at some moment of time $t_0$ in the region, the gradient of the density or density itself is equal to zero, then for some time $T$ during t interval $[ t_{0}, t_0+T]$ there is no flow in the region. This directly links to Barenblatt's finite speed of propagation property for the degenerate equation. The method of the proof is very different from Barenblatt's method and based on the application of Ladyzhenskaya - De Giorgi iterative scheme and Vespri - Tedeev technique. From PDE's point of view, it assumed that solution exists in appropriate Sobolev type of space.

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arXiv2022

An Iterative Energy Estimate for Degenerate Einstein model of Brownian motion

We consider the degenerate Einsteins Brownian motion model when the time interval of the moving particles before the collisions, is reciprocal to the number of particles per unit volume u(x,t), at the point of observation x at time t. The parameter 0 < tau < C, which controls the characteristics of the fluid, almost increases unboundedly, as u approaches 0. This degeneration leads to the localization of the particle distribution in the media. In the paper, we present a structural condition of the time interval and the frequency of these free jumps, as functions of u which guarantees the finite speed of propagation of u.

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arXiv2022

An Iterative Energy Estimate for Degenerate Einstein model of Brownian motion

We consider the degenerate Einstein's Brownian motion model for the case when the time interval ($τ$) of particle Jumps before collision (free jumps) reciprocal to the number of particles per unit volume $u(x,t) > 0$ at the point of observation $x$ at time $t$. The parameter $0 < τ\leq C < \infty$, controls characteristic of the fluid "almost decreases" to $ 0 $ when $u \rightarrow \infty$. This degeneration leads to the localisation of the spread of particle propagation in the media. In our report we will present a structural condition of the time interval of free jumps - $τ$ and the frequency of these free jumps $ϕ$ as functions of $u$ which guarantees the finite speed of propagation of $u$.

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