Supercritical Case Research Articles

Nonlocal nonlinear reaction preventing blow-up in supercritical case of chemotaxis system

This paper is devoted to the analysis of non-negative solutions for the chemotaxis model with nonlocal nonlinear source in bounded domain. The qualitative behavior of solutions is determined by the nonlinearity from the aggregation and the reaction. When the growth factor is stronger than the dampening effect, with the help of the nonlocal nonlinear term in the reaction, for appropriately chosen exponents and arbitrary initial data, the model admits a classical solution which is uniformly bounded. Moreover, when the growth factor has the same order with the dampening effect, the nonlocal nonlinear exponents can prevent the chemotactic collapse.

Non-Fourier heat conduction/convection in moving medium

Non-Fourier one-dimensional unsteady heat conduction in a moving medium is investigated by using the Cattaneo-Vernotte- Christov-Jordan (CVCJ) heat flux model for medium speeds less than (sub-critical), equal to (critical), and greater than (super-critical) the thermal wave speed. Coupled partial differential equations are solved simultaneously by a finite volume numerical method. Temperature and heat flux distributions for sub-critical, critical, and super-critical flow conditions are presented for two example problems. The importance of boundary conditions on the thermal wave propagation in both sub-critical and super-critical cases is discussed. Approximate analytical solutions are presented which qualitatively substantiate the numerical results.

Large eddy simulation of supercritical heat transfer to hydrocarbon fuel

In this article, a large eddy simulation (LES) method for the heat transfer of the hydrocarbon fuel flowing through the uniformly heated miniature round pipe at supercritical pressure has been formulated and validated. The four species surrogate model was used to simulate the real thermophysical properties of the fuel. Validation of the developed LES model was carried out through comparisons of the wall temperature and pressure drop with available experimental data and other turbulence model results. Results show that the LES gave the best prediction. Further calculations based on the proposed LES for three cases including subcritical, transcritical and supercritical temperature ranges were numerically investigated in a systematic manner. It was found that the entrance effect occurred among the subcritical, transcritical and supercritical temperature cases that caused by the developing thermal boundary layer. The significant variation of the thermophysical properties near the pseudo-critical temperature would weaken the heat transfer in the transcritical case where the velocity fluctuation affected more on turbulent heat transfer than the temperature fluctuation did.

On the age of a randomly picked individual in a linear birth-and-death process

Abstract We consider the distribution of the age of an individual picked uniformly at random at some fixed time in a linear birth-and-death process. By exploiting a bijection between the birth-and-death tree and a contour process, we derive the cumulative distribution function for this distribution. In the critical and supercritical cases, we also give rates for the convergence in terms of the total variation and other metrics towards the appropriate exponential distribution.

Comparative Exergetic Analysis of Two Biodiesel Production Routes

Biofuels, such as biodiesel has obtained more and more relevance in the national scenario, because of the current concerns related to the burning of fossil fuels, besides of laws in force that determine the increase of biodiesel blend in common diesel each year. Thus, in this study a comparative exergetic analysis was carried out between the biodiesel production from soybean oil via basic homogeneous catalysis, case 1, where sodium hydroxide (NaOH) was used as a catalyst, and by means of supercritical alcohol, case 2, in which methanol was led to critical state. For that, two production plants were simulated in Aspen HYSYS® V9 software accepting the same amount of main feedstock, in this case soybean oil. After the simulations, the exergetic efficiency of each process was calculated through to a global control volume (CV), having as result 94.01 and 93.50 % to the Case 1 and 2, respectively. These results confirm the feasibility of biodiesel production and the little exergetic difference was caused principally by the conditions of temperature and pressure that were imposed in the methanol supercritical case.

Multidimensional equilibria and their stability in copolymer–solvent mixtures

This paper discusses localized equilibria which arise in copolymer–solvent mixtures. A free boundary problem associated with the sharp-interface limit of a density functional model is used to identify both lamellar and concentric domain patterns composed of a finite number of layers. Stability of these morphologies is studied through explicit linearization of the free boundary evolution.For the multilayered lamellar configuration, transverse instability is observed for sufficiently small dimensionless interfacial energies. Additionally, a crossover between small and large wavelength instabilities is observed depending on whether solvent–polymer or monomer–monomer interfacial energy is dominant.Concentric domain patterns resembling multilayered micelles and vesicles exhibit bifurcations wherein they only exist for sufficiently small dimensionless interfacial energies. The bifurcation of large radii vesicle solutions is studied analytically, and a crossover from a supercritical case with only one solution branch to a subcritical case with two is observed. Linearized stability of these configurations shows that azimuthal perturbation may lead to instabilities as interfacial energy is decreased.

On conditional least squares estimation for affine diffusions based on continuous time observations

We study asymptotic properties of conditional least squares estimators for the drift parameters of two-factor affine diffusions based on continuous time observations. We distinguish three cases: subcritical, critical and supercritical. For all the drift parameters, in the subcritical and supercritical cases, asymptotic normality and asymptotic mixed normality is proved, while in the critical case, non-standard asymptotic behavior is described.

Sandpile on uncorrelated site-diluted percolation lattice; from three to two dimensions

The BTW sandpile model is considered on the three dimensional percolation lattice which is tuned by the occupation parameter p. Along with the three-dimensional avalanches, we study the avalanches in two-dimensional cross-sections. We use the moment analysis (along with some other methods) to extract the exponents for two separate cases: the lattice at critical percolation () and the supercritical one (). Our numerical data is consistent with the conjecture that the three-dimensional avalanches at p = pc have nearly the same exponents as the regular 2D BTW model. The moment analysis shows that finite size scaling theory is fulfilled, and some hyper-scaling relations hold. The main finding of the paper is the logarithmic dependence of the exponents on p − pc, for which the cut-off exponents ν change discontinuously from p = pc to the values for the supercritical case. Moreover we show that there is a singular point ( and being three- and two-dimensional percolation thresholds) for 2D cross-sections, which separate the behaviors to two distinct intervals: which, due to the lack of 2D percolation cluster, has no thermodynamic limit, and which involves the percolated clusters.

The Stripping Process Can be Slow: Part II

This paper is a continuation of the previous results on the stripping number of a random uniform hypergraph, and the maximum depth over all non-k-core vertices. The previous results focus on the supercritical case, whereas this work analyses these parameters in the subcritical regime and inside the critical window.

The connected component of the partial duplication graph

We consider the connected component of the partial duplication model for a random graph, a model which was introduced by Bhan, Galas and Dewey as a model for gene expression networks. The most rigorous results are due to Hermann and Pfaffelhuber, who show a phase transition between a subcritical case where in the limit almost all vertices are isolated and a supercritical case where the proportion of the vertices which are connected is bounded away from zero. We study the connected component in the subcritical case, and show that, when the duplication parameter $p<e^{-1}$, the degree distribution of the connected component has a limit, which we can describe in terms of the stationary distribution of a certain Markov chain and which follows an approximately power law tail, with the power law index predicted by Ispolatov, Krapivsky and Yuryev. Our methods involve analysing the quasi-stationary distribution of a certain continuous time Markov chain associated with the evolution of the graph.

EDGEWORTH CORRECTION FOR THE LARGEST EIGENVALUE IN A SPIKED PCA MODEL.

We study improved approximations to the distribution of the largest eigenvalue of the sample covariance matrix of n zero-mean Gaussian observations in dimension p + 1. We assume that one population principal component has variance ℓ > 1 and the remaining 'noise' components have common variance 1. In the high-dimensional limit p/n → γ > 0, we study Edgeworth corrections to the limiting Gaussian distribution of in the supercritical case . The skewness correction involves a quadratic polynomial, as in classical settings, but the coefficients reflect the high-dimensional structure. The methods involve Edgeworth expansions for sums of independent non-identically distributed variates obtained by conditioning on the sample noise eigenvalues, and the limiting bulk properties and fluctuations of these noise eigenvalues.

Totally ordered measured trees and splitting trees with infinite variation

Combinatorial trees can be used to represent genealogies of asexual individuals. These individuals can be endowed with birth and death times, to obtain a so-called ‘chronological tree’. In this work, we are interested in the continuum analogue of chronological trees in the setting of real trees. This leads us to consider totally ordered and measured trees, abbreviated as TOM trees. First, we define an adequate space of TOM trees and prove that under some mild conditions, every compact TOM tree can be represented in a unique way by a so-called contour function, which is right-continuous, admits limits from the left and has non-negative jumps. The appropriate notion of contour function is also studied in the case of locally compact TOM trees. Then we study the splitting property of (measures on) TOM trees which extends the notion of ‘splitting tree’ studied in [Lam10], where during her lifetime, each individual gives birth at constant rate to independent and identically distributed copies of herself. We prove that the contour function of a TOM tree satisfying the splitting property is associated to a spectrally positive Lévy process that is not a subordinator, both in the critical and subcritical cases of compact trees as well as in the supercritical case of locally compact trees. The genealogical trees associated to splitting trees are the celebrated Lévy trees in the subcritical case and they will be analyzed in the supercritical case in forthcoming work.

Global results for eikonal Hamilton–Jacobi equations on networks

We study a one-parameter family of Eikonal Hamilton-Jacobi equations on an embedded network, and prove that there exists a unique critical value for which the corresponding equation admits global solutions, in a suitable viscosity sense. Such a solution is identified, via an Hopf-Lax type formula, once an admissible trace is assigned on an intrinsic boundary. The salient point of our method is to associate to the network an abstract graph, encoding all of the information on the complexity of the network, and to relate the differential equation to a discrete functional equation on the graph. Comparison principles and representation formulae are proven in the supercritical case as well.

On the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion

We investigate the differentiability issue of the drift-diffusion equation with nonlocal L\'evy-type diffusion at either supercritical or critical type cases. Under the suitable conditions on the drift velocity and the forcing term in terms of the spatial H\"older regularity, we prove that the vanishing viscosity solution is differentiable with some H\"older continuous derivatives for any positive time.

Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions

This paper focuses on the initial boundary value problem of semilinear wave equation in exterior domain in two space dimensions with critical power. Based on the contradiction argument, we prove that the solution will blow up in a finite time. This complements the existence result of supercritical case by Smith, Sogge and Wang [ 20 ] and blow up result of subcritical case by Li and Wang [ 14 ] in two space dimensions.

Finite time blowup for a supercritical defocusing nonlinear Schrödinger system

We consider the global regularity problem for defocusing nonlinear Schr\"odinger systems $$ i \partial_t + \Delta u = (\nabla_{{\bf R}^m} F)(u) + G $$ on Galilean spacetime ${\bf R} \times {\bf R}^d$, where the field $u\colon {\bf R}^{1+d} \to {\bf C}^m$ is vector-valued, $F\colon {\bf C}^m \to {\bf R}$ is a smooth potential which is positive, phase-rotation-invariant, and homogeneous of order $p+1$ outside of the unit ball for some exponent $p >1$, and $G: {\bf R} \times {\bf R}^d \to {\bf C}^m$ is a smooth, compactly supported forcing term. This generalises the scalar defocusing nonlinear Schr\"odinger (NLS) equation, in which $m=1$ and $F(v) = \frac{1}{p+1} |v|^{p+1}$. In this paper we study the supercritical case where $d \geq 3$ and $p > 1 + \frac{4}{d-2}$. We show that in this case, there exists a smooth potential $F$ for some sufficiently large $m$, positive and homogeneous of order $p+1$ outside of the unit ball, and a smooth compactly choice of initial data $u(0)$ and forcing term $G$ for which the solution develops a finite time singularity. In fact the solution is locally discretely self-similar with respect to parabolic rescaling of spacetime. This demonstrates that one cannot hope to establish a global regularity result for the scalar defocusing NLS unless one uses some special property of that equation that is not shared by these defocusing nonlinear Schr\"odinger systems. As in a previous paper of the author considering the analogous problem for the nonlinear wave equation, the basic strategy is to first select the mass, momentum, and energy densities of $u$, then $u$ itself, and then finally design the potential $F$ in order to solve the required equation.

On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces

We proved the local well-posedness for the power-type nonlinear semi-relativistic or half-wave equation (NLHW) in Sobolev spaces. Our proofs are mainly based on the contraction mapping argument using Strichartz estimates. We also apply the technique of Christ-Colliander-Tao in [6] to prove the ill-posedness for the (NLHW) in a certain range of the super-critical case.

Primordial black hole formation by vacuum bubbles

Vacuum bubbles may nucleate during the inflationary epoch and expand, reaching relativistic speeds. After inflation ends, the bubbles are quickly slowed down, transferring their momentum to a shock wave that propagates outwards in the radiation background. The ultimate fate of the bubble depends on its size. Bubbles smaller than certain critical size collapse to ordinary black holes, while in the supercritical case the bubble interior inflates, forming a baby universe, which is connected to the exterior region by a wormhole. The wormhole then closes up, turning into two black holes at its two mouths. We use numerical simulations to find the masses of black holes formed in this scenario, both in subcritical and supercritical regime. The resulting mass spectrum is extremely broad, ranging over many orders of magnitude. For some parameter values, these black holes can serve as seeds for supermassive black holes and may account for LIGO observations.

Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation

We consider the following class of equations of (gKdV) typewith mass sub-critical () and mass super-critical nonlinearities (). We prove the existence of two-solitary wave solutions with logarithmic relative distance, i.e. solutions satisfyingwhere is a fixed constant, in sub-critical cases and in super-critical cases. This regime corresponds to strong attractive interactions. For sub-critical p, it was known that opposite-sign traveling waves are attractive. For super-critical p, we derive from our computations that same-sign traveling waves are attractive.

Contact Processes with Random Recovery Rates and Edge Weights on Complete Graphs

In this paper we are concerned with the contact process with random recovery rates and edge weights on complete graph with $n$ vertices. We show that the model has a critical value which is inversely proportional to the product of the mean of the edge weight and the mean of the inverse of the recovery rate. In the subcritical case, the process dies out before a moment with order $O(\log n)$ with high probability as $n\rightarrow+\infty$. In the supercritical case, the process survives at a moment with order $\exp\{O(n)\}$ with high probability as $n\rightarrow+\infty$. Our proof for the subcritical case is inspired by the graphical method introduced in \cite{Har1978}. Our proof for the supercritical case is inspired by approach introduced in \cite{Pet2011}, which deal with the case where the contact process is with random vertex weights.