Supercritical Case Research Articles

Limit theorems for supercritical branching processes in random environment

The branching process in random environment {Zn}n≥0 is a population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with positive probability and grows exponentially fast on the nonextinction set. Using Fourier techniques we improve existing central limit theorem as well as we obtain Edgeworth expansions and the renewal theorem for the sequence {logZn}n≥0. The strategy is to compare logZn with partial sums of i.i.d. random variables in order to obtain precise estimates.

Normalized solutions for a class of scalar field equations involving mixed fractional Laplacians

Abstract The purpose of this article is to establish sharp conditions for the existence of normalized solutions to a class of scalar field equations involving mixed fractional Laplacians with different orders. This study includes the case when one operator is local and the other one is non-local. This type of equation arises in various fields ranging from biophysics to population dynamics. Due to the importance of these applications, this topic has very recently received an increasing interest. In this article, we provide a complete description of the existence/non-existence of ground state solutions using constrained variational approaches. This study addresses the mass subcritical, critical and supercritical cases. Our model presents some difficulties due to the “conflict” between the different orders and requires a novel analysis, especially in the mass supercritical case. We believe that our results will open the door to other valuable contributions in this important field.

Some nonexistence results for space–time fractional Schrödinger equations without gauge invariance

In this paper, we consider the Cauchy problem in \(\mathbb {R}^N\), \(N\ge 1\), for semi-linear Schrödinger equations with space–time fractional derivatives. We discuss the nonexistence of global \(L^1\) or \(L^2\) weak solutions in the subcritical and critical cases under some conditions on the initial data and the nonlinear term. Furthermore, the nonexistence of local \(L^1\) or \(L^2\) weak solutions in the supercritical case are studied.

Normalized solution to the Schrödinger equation with potential and general nonlinear term: Mass super-critical case

In present paper, we prove the existence of solutions (λ,u)∈R×H1(RN) to the following Schrödinger equation{−Δu(x)+V(x)u(x)+λu(x)=g(u(x))inRN0≤u(x)∈H1(RN),N≥3 satisfying the normalization constraint ∫RNu2dx=a. We treat the so-called mass super-critical case here. Under an explicit smallness assumption on V with lim|x|→∞⁡V(x)=supx∈RN⁡V(x) and some Ambrosetti-Rabinowitz type conditions on g, we can prove the existence of ground state normalized solutions for prescribed mass a>0. Furthermore, we emphasize that the mountain pass characterization of a minimizing solution of the probleminf⁡{∫[12|∇u|2+12V(x)u2−G(u)]dx:‖u‖L2(RN)2=a,P[u]=0}, where G(s)=∫0sg(τ)dτ andP[u]=∫[|∇u|2−12〈∇V(x),x〉u2−N(12g(u)u−G(u))]dx.

Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions

This paper deals with the fully parabolic chemotaxis system of local sensing in higher dimensions. Despite the striking similarity between this system and the Keller–Segel system, we prove the absence of finite-time blow-up phenomenon in this system even in the supercritical case. It means that for any regular initial data, independently of the magnitude of mass, the classical solution exists globally in time in the higher dimensional setting. Moreover, for the exponential decaying motility case, it is established that solutions may blow up at infinite time for any magnitude of mass. In order to prove our theorem, we deal with some auxiliary identity as an evolution equation with a time dependent operator. In view of this new perspective, the direct consequence of the abstract theory is rich enough to establish global existence of the system.

The existence and multiplicity of the normalized solutions for fractional Schrödinger equations involving Sobolev critical exponent in the L 2-subcritical and L 2-supercritical cases

Abstract This paper is devoted to investigate the existence and multiplicity of the normalized solutions for the following fractional Schrödinger equation: (P) ( − Δ ) s u + λ u = μ ∣ u ∣ p − 2 u + ∣ u ∣ 2 s ∗ − 2 u , x ∈ R N , u > 0 , ∫ R N ∣ u ∣ 2 d x = a 2 , \left\{\begin{array}{l}{\left(-\Delta )}^{s}u+\lambda u=\mu | u{| }^{p-2}u+| u{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\gt 0,\hspace{1em}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| u{| }^{2}{\rm{d}}x={a}^{2},\hspace{1.0em}\end{array}\right. where 0 < s < 1 0\lt s\lt 1 , a a , μ > 0 \mu \gt 0 , N ≥ 2 N\ge 2 , and 2 < p < 2 s ∗ 2\lt p\lt {2}_{s}^{\ast } . We consider the L 2 {L}^{2} -subcritical and L 2 {L}^{2} -supercritical cases. More precisely, in L 2 {L}^{2} -subcritical case, we obtain the multiplicity of the normalized solutions for problem ( P ) \left(P) by using the truncation technique, concentration-compactness principle, and genus theory. In L 2 {L}^{2} -supercritical case, we obtain a couple of normalized solution for ( P ) \left(P) by using a fiber map and concentration-compactness principle. To some extent, these results can be viewed as an extension of the existing results from Sobolev subcritical growth to Sobolev critical growth.

Statistical inference for principal components of spiked covariance matrices

In this paper, we study the asymptotic behavior of the extreme eigenvalues and eigenvectors of the high-dimensional spiked sample covariance matrices, in the supercritical case when a reliable detection of spikes is possible. In particular, we derive the joint distribution of the extreme eigenvalues and the generalized components of the associated eigenvectors, that is, the projections of the eigenvectors onto arbitrary given direction, assuming that the dimension and sample size are comparably large. In general, the joint distribution is given in terms of linear combinations of finitely many Gaussian and Chi-square variables, with parameters depending on the projection direction and the spikes. Our assumption on the spikes is fully general. First, the strengths of spikes are only required to be slightly above the critical threshold and no upper bound on the strengths is needed. Second, multiple spikes, that is, spikes with the same strength, are allowed. Third, no structural assumption is imposed on the spikes. Thanks to the general setting, we can then apply the results to various high dimensional statistical hypothesis testing problems involving both the eigenvalues and eigenvectors. Specifically, we propose accurate and powerful statistics to conduct hypothesis testing on the principal components. These statistics are data-dependent and adaptive to the underlying true spikes. Numerical simulations also confirm the accuracy and powerfulness of our proposed statistics and illustrate significantly better performance compared to the existing methods in the literature. In particular, our methods are accurate and powerful even when either the spikes are small or the dimension is large.

Unsteady interaction of crossflow instability with a forward-facing step

Experiments have been conducted on a swept wing model in a low-turbulence wind tunnel at chord Reynolds number of $2.17 \times 10^{6}$ to investigate the unsteady interaction of a forward-facing step (FFS) with incoming stationary crossflow (CF) vortices. The impact of varying the FFS height on the development and growth of primary and secondary CF disturbances and the ensuing laminar–turbulent transition is quantified through detailed hot-wire anemometry and infrared thermography measurements. The presence of the FFS results in either a critical (i.e. moderate transition advancement) or a supercritical behaviour (i.e. transition advancing abruptly to the FFS location). The arrival of the forced stationary CF vortices at the step is accompanied by their amplification. Unsteady analysis for the critical cases indicates temporal velocity fluctuations following closely the development of the baseline configuration (i.e. agreeing with the development of secondary instabilities). Consequently, laminar breakdown originates from the outer side of the upwelling region of the CF vortices. In contrast, for the supercritical FFS, the laminar breakdown unexpectedly originates from the inner side of the upwelling region. Evidence points to an unsteady mechanism possibly supported by locally enhanced spanwise-modulated shears and the recirculation region downstream of the FFS edge. This mechanism appears to govern the abrupt tripping of the flow in supercritical step cases. The findings in this work provide insight into the unsteady FFS–CF vortex interaction, which is pivotal to understanding the influence of an FFS on the laminar–turbulent boundary-layer transition in swept aerodynamic surfaces.

Finite Morse Index Solutions of the Fractional Henon–Lane–Emden Equation with Hardy Potential

In this paper, we study the fractional Henon–Lane–Emden equation associated with Hardy potential \[ (-\Delta)^{s} u - \gamma |x|^{-2s} u = |x|^a |u|^{p-1} u \quad \textrm{in $\mathbb{R}^{n}$}. \] Extending the celebrated result of [14], we obtain a classification result on finite Morse index solutions to the fractional elliptic equation above with Hardy potential. In particular, a critical exponent $p$ of Joseph–Lundgren type is derived in the supercritical case studying a Liouville type result for the $s$-harmonic extension problem.

A solution for the quasi-one-dimensional linearised Euler equations with heat transfer

The unsteady response of nozzles with steady heat transfer forced by acoustic and/or entropy waves is modelled. The approach is based on the quasi-one-dimensional linearised Euler equations. The equations are cast in terms of three variables, namely the dimensionless mass, stagnation temperature and entropy fluctuations, which are invariants of the system at zero frequency and with no heat transfer. The resulting first-order system of differential equations is then solved using the Magnus expansion method, where the perturbation parameters are the normalised frequency and the volumetric heat transfer. In this work, a measure of the flow non-isentropicity (in this case the steady heat transfer) is used for the first time as an expansion parameter. The solution method was applied to a converging–diverging nozzle with constant heat transfer for both subcritical and supercritical flow cases, showing good agreement with numerical predictions. It was observed that the acoustic and entropy transfer functions of the nozzle strongly depend on the frequency and heat transfer.

Transient mixing behavior of a supercritical fluid injected into supercritical and subcritical environments

Transient, near-field mixing of a supercritical fluid injected into supercritical and subcritical environments is investigated experimentally. Fuel (propane) from a 100 μm diameter Bosch diesel injector is injected into a high-pressure test rig with a steady flow of heated, low-velocity nitrogen to control the ambient pressure and temperature. The fuel supply pressure is approximately 190 bar, allowing short injections lasting 3–4 ms to be performed in quick succession to acquire statistically significant datasets to represent the unsteady fuel–air mixing. Two test cases with the same ambient density and injector inlet pressure and temperature were chosen in an attempt to isolate effects of mixture phase behavior. Microscopic rainbow Schlieren deflectometry (RSD) is used for visualization and quantitative data acquisition. In the case of the supercritical ambient, the injector inlet and exit and ambient conditions are all supercritical. The RSD analysis reveals a mixing process similar to the single phase fluid mixing, though condensation is also observed locally, and can be explained by the adiabatic mixing theory presented in the paper. For the subcritical ambient case, the injector inlet is supercritical, but the injector exit and ambient conditions are both subcritical. The results show a rapid expansion in the near-field promoting mixing between propane and nitrogen. At downstream locations, significant condensation is also observed as predicted by the adiabatic mixing theory. The supercritical ambient case exhibited longer and narrower jet penetration compared to the subcritical ambient case.

On the modified fractional Korteweg–de Vries and related equations

We consider in this paper modified fractional Korteweg–de Vries and related equations (modified Burgers–Hilbert and Whitham). They have the advantage with respect to the usual fractional KdV equation to have a defocusing case with a different dynamics. We will distinguish the weakly dispersive case where the phase velocity is unbounded for low frequencies and tends to zero at infinity and the strongly dispersive case where the phase velocity vanishes at the origin and goes to infinity at infinity. In the former case, the nonlinear hyperbolic effects dominate for large data, leading to the possibility of shock formation though the dispersive effects manifest for small initial data where scattering is possible. In the latter case, finite time blow-up is possible in the focusing case but not the shock formation. In the defocusing case global existence and scattering is expected in the energy subcritical case, while finite time blow-up is expected in the energy supercritical case. We establish rigorously the existence of shocks with blow-up time and location being explicitly computed in the weakly dispersive case, while most of the results on the strongly dispersive case are derived via numerical simulations, for large solutions. Moreover, the shock formation result can be extended to the weakly dispersive equation with some generalized nonlinearity. We will also comment briefly on the BBM versions of those equations.

Separation phenomena of radial solutions to the Lane-Emden equation on non-compact Riemannian manifolds

In this paper, we shall discuss separation phenomena of radial solutions to the Lane-Emden equation on non-compact Riemannian manifolds. In particular, we shall prove the separation property of radial solutions for the supercritical case in the sense of the Joseph-Lundgren, and show that the Joseph-Lundgren exponent is a critical exponent on the separation phenomena of radial solutions. Moreover, we also discuss the stability of radial solutions and the existence of a singular solution.

An infinite-dimensional metapopulation SIS model

In this article, we introduce an infinite-dimensional deterministic metapopulation SIS model which takes into account the heterogeneity of the infections and the social network among a large population. We study the long-time behavior of the dynamic. We identify the basic reproduction number R0 which determines whether there exists a stable endemic steady state (super-critical case: R0>1) or if the only equilibrium is disease-free (critical and sub-critical case: R0≤1). As an application of this general study, we prove that the so-called “leaky” and “all-or-nothing” vaccination mechanism have the same effect on R0. This framework is also very natural and intuitive to model lockdown policies and study their impact.

Existence and blow-up of global solutions for a class of fractional Lane–Emden heat flow system

In this paper, we consider a class of Lane–Emden heat flow system with the fractional Laplacian { u t + ( − Δ ) α 2 u = N 1 ( v ) + f 1 ( x ) , ( x , t ) ∈ Q , v t + ( − Δ ) α 2 v = N 2 ( u ) + f 2 ( x ) , ( x , t ) ∈ Q , u ( x , 0 ) = a ( x ) , v ( x , 0 ) = b ( x ) , x ∈ R N , where 0 < α ≤ 2 , N ≥ 3 , Q := R N × ( 0 , + ∞ ) , f i ( x ) ∈ L l o c 1 ( R N ) ( i = 1 , 2 ) are nonnegative functions. We study the relationship between the existence, blow-up of the global solutions for the above system and the indexes p , q in the nonlinear terms N 1 ( v ) , N 2 ( u ) . Here, we first establish the existence and uniqueness of the global solutions in the supercritical case by using Duhamel's integral equivalent system and the contraction mapping principle, and we further obtain some relevant properties of the global solutions. Next, in the critical case, we prove the blow-up of nonnegative solutions for the system by utilizing some heat kernel estimates and combining with proof by contradiction. Finally, by means of the test function method, we investigate the blow-up of negative solutions for the Cauchy problem of a more general higher-order nonlinear evolution system with the fractional Laplacian in the subcritical case.

Normalized solutions for nonlinear Kirchhoff type equations in high dimensions

<abstract><p>We study the normalized solutions for nonlinear Kirchhoff equation with Sobolev critical exponent in high dimensions $ \mathbb{R}^N(N\geqslant4) $. In particular, in dimension $ N = 4 $, there is a special phenomenon for Kirchhoff equation that the mass critical exponent $ 2+\frac{8}{N} $ is equal to the energy critical exponent $ \frac{2N}{N-2} $, which leads to the fact that the equation no longer has a variational structure in dimensions $ N\geqslant 4 $ if we consider the mass supercritical case, and remains unsolved in the existing literature. In this paper, by using appropriate transform, we first get the equivalent system of Kirchhoff equation. With the equivalence result, we obtain the nonexistence, existence and multiplicity of normalized solutions by variational methods, Cardano's formulas and Pohožaev identity.</p></abstract>

Canard-induced mixed mode oscillations as a mechanism for the Bonhoeffer-van der Pol circuit under parametric perturbation

PurposeThis paper aims to investigate the singular Hopf bifurcation and mixed mode oscillations (MMOs) in the perturbed Bonhoeffer-van der Pol (BVP) circuit. There is a singular periodic orbit constructed by the switching between the stable focus and large amplitude relaxation cycles. Using a generalized fast/slow analysis, the authors show the generation mechanism of two distinct kinds of MMOs.Design/methodology/approachThe parametric modulation can be used to generate complicated dynamics. The BVP circuit is constructed as an example for second-order differential equation with periodic perturbation. Then the authors draw the bifurcation parameter diagram in terms of a containing two attractive regions, i.e. the stable relaxation cycle and the stable focus. The transition mechanism and characteristic features are investigated intensively by one-fast/two-slow analysis combined with bifurcation theory.FindingsPeriodic perturbation can suppress nonlinear circuit dynamic to a singular periodic orbit. The combination of these small oscillations with the large amplitude oscillations that occur due to canard cycles yields such MMOs. The results connect the theory of the singular Hopf bifurcation enabling easier calculations of where the oscillations occur.Originality/valueBy treating the perturbation as the second slow variable, the authors obtain that the MMOs are due to the canards in a supercritical case or in a subcritical case. This study can reveal the transition mechanism for multi-time scale characteristics in perturbed circuit. The information gained from such results can be extended to periodically perturbed circuits.

Design of Control-Assisted Transonic Pressure Recovery Profiles Using Vortex Generators

An experimental study was conducted to assess the boundary-layer flow state associated with Stratford pressure recovery profiles in a transonic flow at Mach 0.7 and the enabling role of vortex generator devices for completing fully attached pressure recoveries across shorter chordwise distances. Airfoil geometries with Stratford-like pressure recovery distributions were designed and tested to show that both subcritical and supercritical airfoil cases retain the classical marginally separated condition across the recovery region, with the former showing a greater separation stability margin at off-design conditions. It was determined that the pressure recovery in the supercritical case was immediately preceded by a discrete pressure increase across the shock, which, when coupled to the adverse pressure gradients imposed by the Stratford pressure recovery profile, resulted in separation of the boundary layer at a higher Mach number of 0.75. The airfoils were also tested with two micro-vortex generator configurations to show a fully attached recovery process across shorter distances than the Stratford limit. The vortex generators were observed to introduce circulation-based effects into the flowfield, resulting in local supersonic regions within the vicinity of their placement.

Poisson random measures and supercritical multitype Markov branching processes

We consider multitype Markov branching processes with immigration occurring at time points generated by Poisson random measures. These models find applications to study evolution of multitype cell populations in which new cells join the population according to a time-varying immigration mechanism. The focus of this paper is the supercritical case. We investigate the limiting behavior of the process for different rates of the Poisson random measures. In particular, we prove a result analogous to a strong LLN and establish limiting normal distributions.

Transition in an infinite swept-wing boundary layer subject to surface roughness and free-stream turbulence

The instability of an incompressible boundary-layer flow over an infinite swept wing in the presence of disc-type roughness elements and free-stream turbulence (FST) has been investigated by means of direct numerical simulations. Our study corresponds to the experiments by Örlü et al. (Tech. Rep., KTH Royal Institute of Technology, 2021, http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-291874). Here, different dimensions of the roughness elements and levels of FST have been considered. The aim of the present work is to investigate the experimentally observed sensitivity of the transition to the FST intensity. In the absence of FST, flow behind the roughness elements with a height above a certain value immediately undergoes transition to turbulence. Impulse–response analyses of the steady flow have been performed to identify the mechanism behind the observed flow instability. For subcritical roughness, the generated wave packet experiences a weak transient growth behind the roughness and then its amplitude decays as it is advected out of the computational domain. In the supercritical case, in which the flow transitions to turbulence, flow as expected exhibits an absolute instability. The presence of FST is found to have a significant impact on the transition behind the roughness, in particular in the case of a subcritical roughness height. For a height corresponding to a roughness Reynolds number $Re_{hh}=461$ , in the absence of FST the flow reaches a steady laminar state, while a very low FST intensity of $Tu =0.03\,\%$ causes the appearance of turbulence spots in the wake of the roughness. These randomly generated spots are advected out of the computational domain. For a higher FST level of $Tu=0.3\,\%$ , a turbulent wake is clearly visible behind the element, similar to that for the globally unstable case. The presented results confirm the experimental observations and explain the mechanisms behind the observed laminar–turbulent transition and its sensitivity to FST.