A Singular Problem in Electrolytes Theory

Biler, Piotr; Nadzieja, Tadeusz

doi:10.1002/(sici)1099-1476(199706)20:9<767::aid-mma881>3.0.co;2-j

Cited by 19 publications

(26 citation statements)

References 16 publications

(40 reference statements)

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“…Despite many advanced mathematical studies [33][34][35][36][37], the fundamental issue of the existence and uniqueness of the solutions of the time-dependent NPP and NPE equations still appears unresolved, if we think of the most general cases of real-life models of electrochemical kinetics. There also exists a number of other unresolved mathematical questions and open problems associated with the electrodiffusion of ions [38].…”

Section: Introductionsupporting

confidence: 59%

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 14: extension of the patch-adaptive strategy to time-dependent models involving migration–diffusion transport in one-dimensional space geometry, and its application to example transient experiments described by Nernst–Planck–Poisson equations

Bieniasz

2004

Journal of Electroanalytical Chemistry

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Section: Introductionsupporting

confidence: 59%

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations. Part 14: extension of the patch-adaptive strategy to time-dependent models involving migration–diffusion transport in one-dimensional space geometry, and its application to example transient experiments described by Nernst–Planck–Poisson equations

Bieniasz

2004

Journal of Electroanalytical Chemistry

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“…This model ignores inertial effects and assumes that the drift velocity of the organisms is directly induced by a chemotactic "force" proportional to the concentration gradient of the chemical. The Keller-Segel model can reproduce the formation of clusters (clumps) by chemotactic collapse [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. This reflects experiments on bacteria like Escherichia coli or amoebae like Dictyostelium discoïdeum exhibiting pointwise concentration [3].…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, an infinite and homogeneous distribution of cells is a steady state of the equations of motion (1)-(3) corresponding to the condition kc = hρ. For the "Newtonian model" (11), this condition becomes ρ = ρ and for the "Yukawa model" (12), it becomes k 2 0 c = λρ. In this paper, we study in detail the onset of the "chemotactic instability" and its development in the linear regime.…”

Section: Introductionmentioning

confidence: 99%

“…In the "Newtonian model" (11), the only difference with the Jeans analysis is the presence of the friction force ξ . In the "Yukawa model" (12), the differences with the Jeans analysis are due to the effects of the friction ξ and of the shielding length k −1 0 generated by the degradation of the chemical. We show that the system is always stable for…”

Section: Introductionmentioning

confidence: 99%

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Jeans type analysis of chemotactic collapse

Chavanis

Sire

2008

Physica A: Statistical Mechanics and its Applications

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We perform a linear dynamical stability analysis of a general hydrodynamic model of chemotactic aggregation [Chavanis & Sire, Physica A, in press (2007)]. Specifically, we study the stability of an infinite and homogeneous distribution of cells against "chemotactic collapse". We discuss the analogy between the chemotactic collapse of biological populations and the gravitational collapse (Jeans instability) of self-gravitating systems. Our hydrodynamic model involves a pressure force which can take into account several effects like anomalous diffusion or the fact that the organisms cannot interpenetrate. We also take into account the degradation of the chemical which leads to a shielding of the interaction like for a Yukawa potential. Finally, our hydrodynamic model involves a friction force which quantifies the importance of inertial effects. In the strong friction limit, we obtain a generalized Keller-Segel model similar to the generalized Smoluchowski-Poisson system describing self-gravitating Langevin particles. For small frictions, we obtain a hydrodynamic model of chemotaxis similar to the Euler-Poisson system describing a self-gravitating barotropic gas. We show that an infinite and homogeneous distribution of cells is unstable against chemotactic collapse when the "velocity of sound" in the medium is smaller than a critical value. We study in detail the linear development of the instability and determine the range of unstable wavelengths, the growth rate of the unstable modes and the damping rate, or the pulsation frequency, of the stable modes as a function of the friction parameter and shielding length. For specific equations of state, we express the stability criterion in terms of the density of cells

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“…Selecting, for d = 3, b(z) = C z |z| 3 leads to models of diffusion of electric charge carriers or self-gravitating particles (depending on the sign of constant C, see [3,4]). These examples motivate potential estimates for kernel B imposed by (1.6).…”

Section: Introductionmentioning

confidence: 99%

A Probabilistic Approach for Nonlinear Equations Involving the Fractional Laplacian and a Singular Operator

Jourdain

Woyczyński

2005

Potential Anal

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We consider a class of nonlinear integro-differential equations involving a fractional power of the Laplacian and a nonlocal quadratic nonlinearity represented by a singular integral operator. Initially, we introduce cut-off versions of this equation, replacing the singular operator by its Lipschitz continuous regularizations. In both cases we show the local existence and global uniqueness in L 1 ∩ L p . Then we associate with each regularized equation a stable-process-driven nonlinear diffusion; the law of this nonlinear diffusion has a density which is a global solution in L 1 of the cutoff equation. In the next step we remove the cut-off and show that the above densities converge in a certain space to a solution of the singular equation. In the general case, the result is local, but under a more stringent balance condition relating the dimension, the power of the fractional Laplacian and the degree of the singularity, it is global and gives global existence for the original singular equation. Finally, we associate with the singular equation a nonlinear singular diffusion and prove propagation of chaos to the law of this diffusion for the related cut-off interacting particle systems. Depending on the nature of the singularity in the drift term, we obtain either a strong pathwise result or a weak convergence result. (2000): 60K35, 35S10. Mathematics Subject Classifications

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A Singular Problem in Electrolytes Theory

Cited by 19 publications

References 16 publications

Jeans type analysis of chemotactic collapse

A Probabilistic Approach for Nonlinear Equations Involving the Fractional Laplacian and a Singular Operator

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