A nonlocal convection–diffusion equation

Ignat, Liviu I.; Rossi, Julio D.

doi:10.1016/j.jfa.2007.07.013

Cited by 109 publications

(77 citation statements)

References 19 publications

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“…44 However, it can be shown that solutions exist, that the solutions depend continuously on the initial conditions, and that the nonlocal equations converge to the local ones as the nonlocal terms become small enough. [45][46][47][48] Much less information is available about the properties of even linear nonlocal equations in the presence of boundaries, although they are frequently solved numerically. 49 Fortunately, for this particular set of equations, we can make some very compelling arguments indicating that the integral terms are negligible for the purposes of the closure problems.…”

Section: A Localization Of the Closure Problemsmentioning

confidence: 99%

A non-scale-invariant form for coarse-grained diffusion-reaction equations

Ostvar

Wood

2016

The Journal of Chemical Physics

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The process of mixing and reaction is a challenging problem to understand mathematically. Although there have been successes in describing the effective properties of mixing and reaction under a number of regimes, process descriptions for early times have been challenging for cases where the structure of the initial conditions is highly segregated. In this paper, we use the method of volume averaging to develop a rigorous theory for diffusive mixing with reactions from initial to asymptotic times under highly segregated initial conditions in a bounded domain. One key feature that arises in this development is that the functional form of the averaged differential mass balance equations is not, in general, scale invariant. Upon upscaling, an additional source term arises that helps to account for the initial configuration of the reacting chemical species. In this development, we derive the macroscopic parameters (a macroscale source term and an effectiveness factor modifying the reaction rate) defined in the macroscale diffusion-reaction equation and provide example applications for several initial configurations.

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Section: A Localization Of the Closure Problemsmentioning

confidence: 99%

A non-scale-invariant form for coarse-grained diffusion-reaction equations

Ostvar

Wood

2016

The Journal of Chemical Physics

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“…Here, by the term regularization we mean the modification of the otherwise-zero RHS of the conservation law with terms that are small relative to the scale of the solution. Inspired by the work of Gunzberger and Lehoucq [21] and informed by the work of Ignat and Rossi [25], we propose a regularization of the (inviscid) nonlocal advection equation (2.0.5) of the following form:…”

Section: Regularization Of Non-local Advectionmentioning

confidence: 99%

Peridynamics as a rigorous coarse-graining of atomistics for multiscale materials design.

Lehoucq¹,

Aidun²,

Silling³

et al. 2010

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“…Concerning scalings of the kernel that approximate different problems we refer to [2], [21] and [31], where usual diffusion equations where obtained taking limits similar to the ones considered here.…”

Section: U(0 X) = U 0 (X)mentioning

confidence: 99%

The limit as p → ∞ in a nonlocal p-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles

Andreu

Mazón

Rossi³

et al. 2008

Calc. Var.

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Abstract. In this paper we study the nonlocal ∞−Laplacian type diffusion equation obtained as the limit as p → ∞ to the nonlocal analogous to the p−Laplacian evolution,We prove existence and uniqueness of a limit solution that verifies an equation governed by the subdifferetial of a convex energy functional associated to the indicator function ofWe also find some explicit examples of solutions to the limit equation.If the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L ∞ (0, T ; L 2 (Ω)) to the limit solution of the local evolutions of the p−laplacian, v t = ∆ p v. This last limit problem has been proposed as a model to describe the formation of a sandpile.Moreover, we also analyze the collapse of the initial condition when it does not belong to K by means of a suitable rescale of the solution that describes the initial layer that appears for p large.Finally, we give an interpretation of the limit problem in terms of Monge-Kantorovich mass transport theory.

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A nonlocal convection–diffusion equation

Cited by 109 publications

References 19 publications

A non-scale-invariant form for coarse-grained diffusion-reaction equations

A non-scale-invariant form for coarse-grained diffusion-reaction equations

Peridynamics as a rigorous coarse-graining of atomistics for multiscale materials design.

The limit as p → ∞ in a nonlocal p-Laplacian evolution equation: a nonlocal approximation of a model for sandpiles

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