Blow-up Regimes Research Articles

Solution of advection-diffusion-reaction problems on a sphere: High-resolution numerical experiments

The implicit and unconditionally stable numerical method proposed in Skiba (2015) is applied to solve linear advection-diffusion-reaction problems and nonlinear diffusion-reaction problems on a sphere. Numerical experiments carried out on a high-resolution spherical mesh show the effectiveness of the method in modelling linear advection-diffusion processes on a sphere (dispersion of pollution in the atmosphere), and nonlinear diffusion processes (propagation of nonlinear temperature waves, blow-up regimes of combustion, and chemical reactions in the Gray-Scott model). The method correctly describes the mass balance of a substance in forced and dissipative systems and conserves the total mass and norm of the solution in the absence of forcing and dissipation.

Exact solutions of a nonlinear diffusion equation on polynomial invariant subspace of maximal dimension

The nonlinear diffusion equation ut=(u−4/3ux)x is reduced by the substitution u=v−3/4 to an equation with quadratic nonlinearities possessing a polynomial invariant linear subspace of the maximal possible dimension equal to five. The dynamics of the solutions on this subspace is described by a fifth-order nonlinear dynamical system (V.A. Galaktionov).We found that, on differentiation, this system reduces to a single linear equation of the second order, which is a special case of the Lamé equation, and that the general solution of this linear equation is expressed in terms of the Weierstrass ℘-function and its derivative. As a result, all exact solutions v(x,t) on a five-dimensional polynomial invariant subspace, as well as the corresponding solutions u(x,t) of the original equation, are constructed explicitly.Using invariance condition, two families of non-invariant solutions are singled out. For one of these families, all types of solutions are considered in detail. Some of them describe peculiar blow-up regimes, while others fade out in finite time.

Heat Localization in the Medium in Blow-Up Regime

The existence of the effect of heat metastable localization in the medium in the blow-up heating regime was experimentally proved. This is the regime in which the heating energy for a finite period of time tends to infinity. Previous theoretical studies have shown that in this case some regions, inside of which the temperature increases, may arise, while their size remains constant or decreases with time (heat localization regions). These regions exist as long as there is some energy input from the outside. An installation for the experimental study of the thermal blow-up regimes in a solid was developed. The object of research was an aluminum rod with a heater at its end. The temperature distribution along the rod was measured with thermocouples. The temperature of the rod end could vary according to the given law. Calibration of the installation was performed. The sensitivity of thermocouples was determined. The inertia of the heating and cooling process was estimated. The mathematical description of the thermal processes, occurring during the experiment, was made. The nonlinear equation of heat conduction for the rod was solved, with the heat exchange with the environment by convection and radiation taken into account. The thermal regime at the boundary, which is necessary to create the thermal structures, was determined. The temperature distribution in the rod in the blow-up regime and non-blow-up regime was measured. In the blow-up regime the heat front (the coordinate of the point with the temperature equal to half the maximum temperature) initially shifts from the heat source, and then in the opposite direction, and the size of the area under heating decreases. In the non-blow-up regime the size of the heated region increases all the time. The predicted effect was supposed to be used in installations for thermonuclear fusion where the target was heated by laser radiation pulses of a special shape. This effect can also be used for localized heating in cutting and welding, when the adjacent regions are not to get very hot, and in other similar situations.

Localized Blow-Up Regimes for Quasilinear Doubly Degenerate Parabolic Equations

Singular blow-up regimes are studied for a wide class of second-order quasilinear parabolic equations. Energy methods are used to obtain exact (in a certain sense) estimates of the final profile of the generalized solution near the blow-up time depending on the rate of increase of the global energy of this solution.

Phenomena of Nonlinear Diffusion in Complex 3D Media

Unbounded solutions (critical blow-up regimes) simulated by the 3D nonlinear diffusion equation in a spherical shell are studied. The coordinate splitting of the differential operator coupled with two spherical coordinate maps makes it possible to use periodic boundary conditions in the latitudinal and longitudinal directions and employ the computationally efficient Sherman-Morrison formula and Thomas algorithm. The resulting finite difference method is direct, with implicit and unconditionally stable schemes of second-order approximation in all the variables. Numerical tests demonstrate that it allows simulating different blow-up regimes in complex computational domains.

Blow-up regimes in thePT-symmetric coupler and the actively coupled dimer

In the actively coupled (AC) pair of waveguides, the growth of small perturbations is saturated by the focusing nonlinearity that couples the linearly growing mode to the linearly damped mode. On the other hand, in the $\mathcal{PT}$-symmetric coupler, the focusing nonlinearity promotes the blow-up of stationary light beams. The purpose of this study is to compare the nonlinear dynamics and explain the opposite effect of the same nonlinearity in the two systems. We show that, while the blow-up regimes are stable in the $\mathcal{PT}$-symmetric pair of waveguides, they are unstable and hence cannot be observed in the AC dimer.

Fukushima Plutonium Effect and Blow-Up Regimes in Neutron-Multiplying Media

It is shown that the capture and fission cross-sections of 238U and 239Pu increase with temperature within 1000 K - 3000 K range, in contrast to those of 235U, that under certain conditions may lead to the so-called blow-up modes, stimulating the anomalous neutron flux and nuclear fuel temperature growth. Some features of the blow-up regimes in neutronmultiplying media are discussed.

On blow-up regimes in one nonlinear parabolic equation

A nonlinear heat equation with a special source on a straight line is considered. The family of exact solutions to this equation that have the form p(t) + q(t)cosx/√2, where functions p(t) and q(t) satisfy a certain dynamic system, is constructed. The system is comprehensively analyzed, and the behavior of p(t) and q(t) depending on initial data is revealed. It is found that some of the unbounded solutions from the aforementioned family are close, in a certain sense, to an analytical solution to the heat equation with power nonlinearities. The Cauchy problem for the equations considered is studied as well. It is proved that, depending on the initial solution function, solutions may develop in a blow-up regime or decay.

Self-similar boundary blow-up for higher-order quasilinear parabolic equations

We study evolution properties of boundary blow-up for 2mth-order quasilinear parabolic equations in the case where, for homogeneous power nonlinearities, the typical asymptotic behaviour is described by exact or approximate self-similar solutions. Existence and asymptotic stability of such similarity solutions are established by energy estimates and contractivity properties of the rescaled flows.Further asymptotic results are proved for more general equations by using energy estimates related to Saint-Venant's principle. The established estimates of propagation of singularities generated by boundary blow-up regimes are shown to be sharp by comparing with various self-similar patterns.

Structure of boundary blow-up for higher-order quasilinear parabolic equations

Singularity formation phenomena for 2 m th–order quasilinear parabolic equations are studied by using energy estimates related to Saint–Venant's principle. Sharp estimates of propagation of singularities generated by boundary global and regional blow-up regimes are established.

Saint-Venant's principle in blow-up for higher-order quasilinear parabolic equations

We prove localization estimates for general 2mth-order quasilinear parabolic equations with boundary data blowing up in finite time, as t → T−. The analysis is based on energy estimates obtained from a system of functional inequalities expressing a version of Saint-Venant's principle from the theory of elasticity. We consider a special class of parabolic operators including those having fixed orders of algebraic homogenuity p > 0. This class includes the second-order heat equation and linear 2mth-order parabolic equations (p = 1), as well as many other higher-order quasilinear ones with p ≠ 1. Such homogeneous equations can be invariant under a group of scaling transformations, but the corresponding least-localized regional blow-up regimes are not group invariant and exhibit typical exponential singularities ~ e(T−t)−γ → ∞ as t → T−, with the optimal constant γ = 1/[m(p + 1) − 1] > 0. For some particular equations, we study the asymptotic blow-up behaviour described by perturbed first-order Hamilton–Jacobi equations, which shows that general estimates of exponential type are sharp.

Quench development in long HTS objects-the possibility of "blow-up" regimes and a heat localization

The qualitative analysis of the quench development in long HTS objects has been performed based on the general physical study of so-called blow-up regimes. Blow-up processes with strong heat localization in HTS objects with overcurrent are possible. Heat localization appears due to general instability in a media with nonlinear parameters that is valid for HTS. The sizes of overheated area and time of the heat localization development has been evaluated for typical parameters of HTS power cables.

Mathematical modeling of heat-distribution processes using generalized equations for thermal conductivity

Problems in the mathematical modeling of heat-distribution processes on the basis of more general equations than parabolic equations are considered. We study the general structure of the relations between solutions of various approximations to the generalized heat-conductivity equations. We introduce a notion of singularly perturbed dissipative structures and analyze singularly perturbed blow-up regimes.

Longitudinal Instabilities in Circular Accelerator and Storage Rings

The general problem of the longitudinal stability of bunched beams is reviewed. Although there is no general solution it is possible to identify regions in the frequency-risetime space where we can obtain approximate solutions. The collective oscillation frequency is written, and expressions for the effective coupling impedance are given for the high or low frequency and slow and fast blow-up regimes.