Self-similar Limit Research Articles

Delta shock wave for ternary nonlinear chromatography equations as self-similar viscosity limit

In this paper, we investigate the existence and stability of the delta shock wave in ternary chromatography equations by the self-similar viscosity vanishing approach. Considering the appropriate initial values, we prove the existence of the self-similar solution for the corresponding Riemann problem of the ternary chromatography viscous equations. Furthermore, we rigorously demonstrate that the delta shock wave is the weak star limit of the self-similar solution as viscosity tends to disappear. The result implies that the structure of the delta shock wave is stable under the self-similar viscosity perturbation, which guarantees that the delta shock wave is a unique entropy solution. In addition, we present numerical simulations in agreement with the theoretical analysis.

𝐾𝐾-duality for self-similar groupoid actions on graphs

We extend Nekrashevych’s K K KK -duality for C ∗ C^* -algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph. More precisely, given a regular and contracting self-similar groupoid ( G , E ) (G,E) acting faithfully on a finite directed graph E E , we associate two C ∗ C^* -algebras, O ( G , E ) \mathcal {O}(G,E) and O ^ ( G , E ) \widehat {\mathcal {O}}(G,E) , to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in K K KK -theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author.

Air–Sea Enthalpy and Momentum Exchange Coefficients from GPS Dropsonde Measurements in Hurricane Conditions

The intensity of tropical cyclones is highly dependent on air–sea enthalpy and momentum exchange. At extreme wind speeds, the values of the enthalpy, CK, and momentum, CD, exchange coefficients are characterized by high uncertainty. The present study aims to expand the previously used algorithm for CD retrieval to obtain the values of CK from wind speed measurements and the enthalpy profiles obtained from NOAA GPS dropsondes in hurricane conditions. This algorithm uses concepts from technical hydrodynamics, describing turbulent boundary layers on flat plates and pipes. According to this approach, the velocity (and enthalpy) defect profiles are self-similar in the entire boundary layer, including the layer of constant fluxes and the “wake” part, where the airflow adapts to the undisturbed flow region. By using the self-similarity property, the parameters of the constant flow layer (the roughness parameter, friction velocity, and the enthalpy and exchange coefficients CD and CK) could be obtained from measurements in the “wake” part for wind speeds from 20 m/s to 72 m/s. The estimates of the CK/CD ratio revealed values of 0.7 and 0.96 (depending on the self-similar approximation limits), and the results suggest that there are slight variations with the wind speed.

Primordial black hole formation with full numerical relativity

We study the formation of black holes from subhorizon and superhorizon perturbations in a matter dominated universe with 3+1D numerical relativity simulations. We find that there are two primary mechanisms of formation depending on the initial perturbation's mass and geometry — via direct collapse of the initial overdensity and via post-collapse accretion of the ambient dark matter. In particular, for the latter case, the initial perturbation does not have to satisfy the hoop conjecture for a black hole to form. In both cases, the duration of the formation the process is around a Hubble time, and the initial mass of the black hole is M BH ∼ 10-2 H -1 M Pl 2. Post formation, we find that the PBH undergoes rapid mass growth beyond the self-similar limit M BH α H -1, at least initially. We argue that this implies that most of the final mass of the PBH is accreted from its ambient surroundings post formation.

Slope dependence of self-similar structure and entrainment in gravity currents

Results from seven direct and large-eddy simulations of gravity currents on slopes ranging from 0.14 $^{\circ }$ to 2.86 $^{\circ }$ that span from the subcritical to the supercritical regime are studied. By considering a long domain, attention is focused on the near-self-similar state approached by these currents far downstream. In the self-similar limit, the various shape factors, local Richardson number, entrainment coefficient, velocity scale and basal drag coefficient reach a constant value, while the current height, volume and momentum fluxes continue to increase linearly. Their dependence on slope is presented.

Social inequality analysis of fiber bundle model statistics and prediction of materials failure.

Inequalities are abundant in a society with a number of agents competing for a limited amount of resources. Statistics on such social inequalities are usually represented by the Lorenz function L(p), where p fraction of the population possesses L(p) fraction of the total wealth, when the population is arranged in ascending order of their wealth. Similarly, in scientometrics, such inequalities can be represented by a plot of the citation count versus the respective number of papers by a scientist, again arranged in ascending order of their citation counts. Quantitatively, these inequalities are captured by the corresponding inequality indices, namely, the Kolkata k and the Hirsch h indices, given by the fixed points of these nonlinear (Lorenz and citation) functions. In statistical physics of criticality, the fixed points of the renormalization group generator functions are studied in their self-similar limit, where their (fractal) structure converges to a unique form (macroscopic in size and lone). The statistical indices in social science, however, correspond to the fixed points where the values of the generator function (wealth or citation sizes) are commensurately abundant in fractions or numbers (of persons or papers). It has already been shown that under extreme competitions in markets or at universities, the k index approaches a universal limiting value, as the dynamics of competition progresses. We introduce and study these indices for the inequalities of (prefailure) avalanches, given by their nonlinear size distributions in fiber bundle models of nonbrittle materials. We show how prior knowledge of the terminal and (almost) universal value of the k index for a wide range of disorder parameters can help in predicting an imminent catastrophic breakdown in the model. This observation is also complemented by noting a similar (but not identical) behavior of the Hirsch index (h), redefined for such avalanche statistics.

The spectrum of decaying 2D self-similar turbulence

A decaying 2D homogeneous and isotropic turbulent flow is considered in the self-similar limit, which is achieved with large values of the Reynolds number formed using the time and kinetic energy of the flow if the initial value of the averaged enstrophy tends to infinity with the viscosity tending to zero. In this case, the enstrophy-dissipation rate has a nonzero finite limit. The correlation function of the vorticity field and the enstrophy spectral density in the inertial range of distances and wave numbers, where these functions are free from the effect of viscosity and large-scale flow parameters, is investigated. It turns out that the inertial range exists in the decaying 2D self-similar turbulence in physical space but is absent in the space of wavenumbers. This means that the turbulent vortices of the appropriate size do not contribute to the spectral density, and the well-known law of the first degree is not satisfied. At large wave numbers, the spectral density of enstrophy behaves nonmonotonically - it first decreases faster than the law of the minus first degree and, then, in the dissipation region, it has a growth segment and a second peak. In this case, the enstrophy flow along the spectrum on the left boundary of the dissipation region is only a fraction of the enstrophy-dissipation rate.

Delta shock wave as self-similar viscosity limit for a strictly hyperbolic system of conservation laws

In this article, we study the Riemann problem for a strictly hyperbolic system of conservation laws. The governing system arises in nonlinear elasticity and gas dynamics. The system is one of the examples of a strictly hyperbolic system whose Riemann solution consists of delta shock waves as well as classical elementary waves such as shock waves, rarefaction waves, and contact discontinuities. We discuss the existence and uniqueness of the solution of the Riemann problem involving δ-shock wave by using a self-similar vanishing viscosity approach. We show that δ-shock wave is a weak*-limit of L1 solution to some viscous perturbations as the viscosity vanishes.

Self-similar solutions of kinetic-type equations: The boundary case

For a time dependent family of probability measures (ρt)t⩾0 we consider a kinetic-type evolution equation ∂ϕt∕∂t+ϕt=Q̂ϕt where Q̂ is a smoothing transform and ϕt is the Fourier–Stieltjes transform of ρt. Assuming that the initial measure ρ0 belongs to the domain of attraction of a stable law, we describe asymptotic properties of ρt, as t→∞. We consider the boundary regime when the standard normalization leads to a degenerate limit and find an appropriate scaling ensuring a non-degenerate self-similar limit. Our approach is based on a probabilistic representation of probability measures (ρt)t⩾0 that refines the corresponding construction proposed in Bassetti and Ladelli, (2012).

Enstrophy spectrum in freely decaying two-dimensional self-similar turbulent flow

We consider freely decaying two-dimensional homogeneous and isotropic turbulent motion in a self-similar limit that is achieved at large Reynolds numbers based on time and the mean kinetic energy of the flow provided that initial average enstrophy tends to infinity as fluid viscosity tens to zero. In this case, the enstrophy dissipation rate has a nonzero finite limit. The vorticity correlation function and the spectral enstrophy density are investigated in an inertial range of distances and wave numbers where these functions are free from the influence of viscosity and large-scale flow parameters. It turns out that in freely decaying two-dimensional self-similar turbulence, the inertial range exists in real space but is absent in the space of wave numbers. This means that turbulent eddies of the appropriate size do not contribute to the spectral density and the known ${k}^{\ensuremath{-}1}$ law does not hold. The spectral enstrophy density at large wave numbers behaves nonmonotonically: it first decreases faster than in accordance with the ${k}^{\ensuremath{-}1}$ law and then, in the dissipation region, has a growth portion and a second maximum. The enstrophy spectral flux at the boundary of the dissipation region is only a fraction of the enstrophy dissipation rate.

Random planar maps and growth-fragmentations

We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these cycles, appropriately rescaled, as the size of the boundary grows. The limiting process is described using a self-similar growth-fragmentation process with explicit parameters. To this end, we introduce a branching peeling exploration of Boltzmann triangulations, which allows us to identify a crucial martingale involving the perimeters of cycles at given heights. We also use a recent result concerning self-similar scaling limits of Markov chains on the nonnegative integers. A motivation for this work is to give a new construction of the Brownian map from a growth-fragmentation process.

Self-similar scaling limits of Markov chains on the positive integers

We are interested in the asymptotic behavior of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. If $\mathit{X_{n}}$ is such a Markov chain started at n, we establish a limit theorem for $\frac{1}{n}\mathit{X_{n}}$ appropriately scaled in time, where the scaling limit is given by a nonnegative self-similar Markov process. We also study the asymptotic behavior of the time needed by $\mathit{X_{n}}$ to reach some fixed finite set. We identify three different regimes (roughly speaking the transient, the recurrent and the positive-recurrent regimes) in which $\mathit{X_{n}}$ exhibits different behavior. The present results extend those of Haas and Miermont [Bernoulli 17 (2011) 1217–1247] who focused on the case of nonincreasing Markov chains. We further present a number of applications to the study of Markov chains with asymptotically zero drifts such as Bessel-type random walks, nonnegative self-similar Markov processes, invariance principles for random walks conditioned to stay positive and exchangeable coalescence-fragmentation processes.

Self-similar fast-reaction limits for reaction–diffusion systems on unbounded domains

We present a unified approach to characterising fast-reaction limits of systems of either two reaction–diffusion equations, or one reaction–diffusion equation and one ordinary differential equation, on unbounded domains, motivated by models of fast chemical reactions where either one or both reactant(s) is/are mobile. For appropriate initial data, solutions of four classes of problems each converge in the fast-reaction limit k→∞ to a self-similar limit profile that has one of four forms, depending on how many components diffuse and whether the spatial domain is a half or whole line. For fixed k, long-time convergence to these same self-similar profiles is also established, thanks to a scaling argument of Kamin. Our results generalise earlier work of Hilhorst, van der Hout and Peletier to a much wider class of problems, and provide a quantitative description of the penetration of one substance into another in both the fast-reaction and long-time regimes.

Operator self-similar processes and functional central limit theorems

Let {Xk:k≥1} be a linear process with values in the separable Hilbert space L2(μ) given by Xk=∑j=0∞(j+1)−Dεk−j for each k≥1, where D is defined by Df={d(s)f(s):s∈S} for each f∈L2(μ) with d:S→R and {εk:k∈Z} are independent and identically distributed L2(μ)-valued random elements with Eε0=0 and E‖ε0‖2<∞. We establish sufficient conditions for the functional central limit theorem for {Xk:k≥1} when the series of operator norms ∑j=0∞‖(j+1)−D‖ diverges and show that the limit process generates an operator self-similar process.

Quantitative Stratification and the Regularity of Mean Curvature Flow

Let \({\mathcal{M}}\) be a Brakke flow of n-dimensional surfaces in \({\mathbb{R}^N}\). The singular set \({\mathcal{S} \subset \mathcal{M}}\) has a stratification \({\mathcal{S}^0 \subset \mathcal{S}^1 \subset \cdots \mathcal{S}}\), where \({X \in \mathcal{S}^j}\) if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata \({\mathcal{S}^j_{\eta, r}}\) satisfying \({\cup_{\eta>0} \cap_{0<r} \mathcal{S}^j_{\eta, r} = \mathcal{S}^j}\). Sharpening the known parabolic Hausdorff dimension bound \({{\rm dim} \mathcal{S}^j \leq j}\), we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of \({\mathcal{S}^j_{\eta, r}}\) satisfies \({{\rm Vol} (T_r(\mathcal{S}^j_{\eta, r}) \cap B_1) \leq Cr^{N + 2 - j-\varepsilon}}\). Our primary application of this is to higher regularity of Brakke flows starting at k-convex smooth compact embedded hypersurfaces. To this end, we prove that for the flow of k-convex hypersurfaces, any backwards selfsimilar limit flow with at least k symmetries is in fact a static multiplicity one plane. Then, denoting by \({\mathcal{B}_r \subset \mathcal{M}}\) the set of points with regularity scale less than r, we prove that \({{\rm Vol}(T_r(\mathcal{B}_r)) \leq C r^{n+4-k-\varepsilon}}\). This gives L p-estimates for the second fundamental form for any p < n + 1 − k. In fact, the estimates are much stronger and give L p-estimates for the reciprocal of the regularity scale. These estimates are sharp. The key technique that we develop and apply is a parabolic version of the quantitative stratification method introduced in Cheeger and Naber (Invent. Math., (2)191 2013), 321–339) and Cheeger and Naber (Comm. Pure. Appl. Math, arXiv:1107.3097v1, 2013).

An Inverse Gamma Activity Time Process with Noninteger Parameters and a Self-Similar Limit

We construct a process with inverse gamma increments and an asymptotically self-similar limit. This construction supports the use of long-range-dependent t subordinator models for actual financial data as advocated in Heyde and Leonenko (2005), in that it allows for noninteger-valued model parameters, as is found empirically in model estimation from data.

An Inverse Gamma Activity Time Process with Noninteger Parameters and a Self-Similar Limit

We construct a process with inverse gamma increments and an asymptotically self-similar limit. This construction supports the use of long-range-dependent t subordinator models for actual financial data as advocated in Heyde and Leonenko (2005), in that it allows for noninteger-valued model parameters, as is found empirically in model estimation from data.

A fault and seismicity based composite simulation in northern California

Abstract. We generate synthetic catalogs of seismicity in northern California using a composite simulation. The basis of the simulation is the fault based "Virtual California" (VC) earthquake simulator. Back-slip velocities and mean recurrence intervals are specified on model strike-slip faults. A catalog of characteristic earthquakes is generated for a period of 100 000 yr. These earthquakes are predominantly in the range M = 6 to M = 8, but do not follow Gutenberg-Richter (GR) scaling at lower magnitudes. In order to model seismicity on unmapped faults we introduce background seismicity which occurs randomly in time with GR scaling and is spatially associated with the VC model faults. These earthquakes fill in the GR scaling down to M = 4 (the smallest earthquakes modeled). The rate of background seismicity is constrained by the observed rate of occurrence of M &gt; 4 earthquakes in northern California. These earthquakes are then used to drive the BASS (branching aftershock sequence) model of aftershock occurrence. The BASS model is the self-similar limit of the ETAS (epidemic type aftershock sequence) model. Families of aftershocks are generated following each Virtual California and background main shock. In the simulations the rate of occurrence of aftershocks is essentially equal to the rate of occurrence of main shocks in the magnitude range 4 &lt; M &lt; 7. We generate frequency-magnitude and recurrence interval statistics both regionally and fault specific. We compare our modeled rates of seismicity and spatial variability with observations.

Self-similar scaling limits of non-increasing Markov chains

We study scaling limits of non-increasing Markov chains with values in the set of non-negative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from $n$ and appropriately rescaled, converges in distribution, as $n\rightarrow \infty$, to a non-increasing self-similar Markov process. This convergence holds jointly with that of the rescaled absorption time to the time at which the self-similar Markov process reaches first 0. We discuss various applications to the study of random walks with a barrier, of the number of collisions in $\Lambda$-coalescents that do not descend from infinity and of non-consistent regenerative compositions. Further applications to the scaling limits of Markov branching trees are developed in our paper, Scaling limits of Markov branching trees, with applications to Galton--Watson and random unordered trees (2010).

Long-time behavior of a Hele-Shaw type problem in random media

We study the long-time behavior of an exterior Hele–Shaw problem in random media with a free boundary velocity that depends on the position in dimensions n ≥ 2 . A natural rescaling of solutions that is compatible with the evolution of the free boundary leads to the homogenization of the free boundary velocity. By studying a limit obstacle problem for a Hele–Shaw problem with a point source, we are able to show the uniform convergence of the rescaled solution to a self-similar limit profile and we deduce that the rescaled free boundary uniformly approaches a sphere.