Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem

We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB) equations, i.e. scalar conservation laws with diffusive-dispersive regularization. We review the existence of traveling wave solutions for these two classes of evolution equations. For classical equations the traveling wave problem (TWP) for a local KdVB equation can be identified with the TWP for a reaction-diffusion equation. In this article we study this relationship for these two classes of evolution equations with nonlocal diffusion/dispersion. This connection is especially useful, if the TW equation is not studied directly, but the existence of a TWS is proven using one of the evolution equations instead. Finally, we present three models from fluid dynamics and discuss the TWP via its link to associated reaction-diffusion equations

Local and Nonlocal Dispersive Turbulence

We consider the evolution of a family of 2D dispersive turbulence models. The members of this family involve the nonlinear advection of a dynamically active scalar field, the locality of the streamfunction-scalar relation is denoted by $\alpha$, with smaller $\alpha$ implying increased locality. The dispersive nature arises via a linear term whose strength is characterized by a parameter $\epsilon$. Setting $0 < \epsilon \le 1$, we investigate the interplay of advection and dispersion for differing degrees of locality. Specifically, we study the forward (inverse) transfer of enstrophy (energy) under large-scale (small-scale) random forcing. Straightforward arguments suggest that for small $\alpha$ the scalar field should consist of progressively larger eddies, while for large $\alpha$ the scalar field is expected to have a filamentary structure resulting from a stretch and fold mechanism. Confirming this, we proceed to forced/dissipative dispersive numerical experiments under weakly non-local to local conditions. For $\epsilon \sim 1$, there is quantitative agreement between non-dispersive estimates and observed slopes in the inverse energy transfer regime. On the other hand, forward enstrophy transfer regime always yields slopes that are significantly steeper than the corresponding non-dispersive estimate. Additional simulations show the scaling in the inverse regime to be sensitive to the strength of the dispersive term : specifically, as $\epsilon$ decreases, the inertial-range shortens and we also observe that the slope of the power-law decreases. On the other hand, for the same range of $\epsilon$ values, the forward regime scaling is fairly universal.Comment: 19 pages, 8 figures. Significantly revised with additional result

Hydrodynamic dispersion within porous biofilms

Many microorganisms live within surface-associated consortia, termed biofilms, that can form intricate porous structures interspersed with a network of fluid channels. In such systems, transport phenomena, including flow and advection, regulate various aspects of cell behavior by controlling nutrient supply, evacuation of waste products, and permeation of antimicrobial agents. This study presents multiscale analysis of solute transport in these porous biofilms. We start our analysis with a channel-scale description of mass transport and use the method of volume averaging to derive a set of homogenized equations at the biofilm-scale in the case where the width of the channels is significantly smaller than the thickness of the biofilm. We show that solute transport may be described via two coupled partial differential equations or telegrapher's equations for the averaged concentrations. These models are particularly relevant for chemicals, such as some antimicrobial agents, that penetrate cell clusters very slowly. In most cases, especially for nutrients, solute penetration is faster, and transport can be described via an advection-dispersion equation. In this simpler case, the effective diffusion is characterized by a second-order tensor whose components depend on (1) the topology of the channels' network; (2) the solute's diffusion coefficients in the fluid and the cell clusters; (3) hydrodynamic dispersion effects; and (4) an additional dispersion term intrinsic to the two-phase configuration. Although solute transport in biofilms is commonly thought to be diffusion dominated, this analysis shows that hydrodynamic dispersion effects may significantly contribute to transport

Hyperbolic Techniques in Modelling, Analysis and Numerics

Several research areas are flourishing on the roots of the breakthroughs in conservation laws that took place in the last two decades. The meeting played a key role in providing contacts among the different branches that are currently developing. All the invitees shared the same common background that consists of the analytical and numerical techniques for nonlinear hyperbolic balance laws. However, their fields of applications and their levels of abstraction are very diverse. The workshop was the unique opportunity to share ideas about analytical issues like the fine-structure of singular solutions or the validity of entropy solution concepts. It turned out that generalized hyperbolic techniques are able to handle the challenges posed by new applications. The design of efficient structure preserving methods turned out to be the major line of development in numerical analysis

Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws

We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.Comment: Revision from v2; 57 pages, 19 figure

Quantum evolution in space-time foam

In this work, I review some aspects concerning the evolution of quantum low-energy fields in a foamlike space-time, with involved topology at the Planck scale but with a smooth metric structure at large length scales, as follows. Quantum gravitational fluctuations may induce a minimum length thus introducing an additional source of uncertainty in physics. The existence of this resolution limit casts doubts on the metric structure of space-time at the Planck scale and opens a doorway to nontrivial topologies, which may dominate Planck scale physics. This foamlike structure of space-time may show up in low-energy physics through loss of quantum coherence and mode-dependent energy shifts, for instance, which might be observable. Space-time foam introduces non-local interactions that can be modeled by a quantum bath, and low-energy fields evolve according to a master equation that displays such effects. Similar laws are also obtained for quantum mechanical systems evolving according to good real clocks, although the underlying Hamiltonian structure in this case establishes serious differences among both scenarios.I was supported by funds provided by DGICYT and MEC (Spain) under Projects PB93–0139, PB94–0107, and PB97–1218.Peer Reviewe

Particles and fields in fluid turbulence

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.Comment: 165 pages, review article for Rev. Mod. Phy

Turbulent mixing

The ability of turbulent flows to effectively mix entrained fluids to a molecular scale is a vital part of the dynamics of such flows, with wide-ranging consequences in nature and engineering. It is a considerable experimental, theoretical, modeling, and computational challenge to capture and represent turbulent mixing which, for high Reynolds number (Re) flows, occurs across a spectrum of scales of considerable span. This consideration alone places high-Re mixing phenomena beyond the reach of direct simulation, especially in high Schmidt number fluids, such as water, in which species diffusion scales are one and a half orders of magnitude smaller than the smallest flow scales. The discussion below attempts to provide an overview of turbulent mixing; the attendant experimental, theoretical, and computational challenges; and suggests possible future directions for progress in this important field