Conditions For Blow-up Research Articles

Sessions of the workshop of the mathematics and mechanics Department of Lomonosov Moscow State University, “Urgent problems of geometry and mechanics” named after V. V. Trofimov

Session 179 in the framework of the scientific-educational program for schoolboys and schoolgirls in mechanics and robotics and courses of advanced qualification, SUNTs MGU (February 8, 2008) D. V. Georgievskii and M. V. Shamolin. Π-Theorem of Dimension Theory (Devoted to the 100th Anniversary of Its Proof). About one hundred years ago, there appeared the proof of one of the most prominent and universal assertions in mechanics and physics, the Π-theorem of dimension theory, which allows one to deduce the dependences of some quantities on others only from dimension arguments, not solving initialboundary-value problems and even having no mathematical models of phenomena. As is revealed, it is difficult to present a concrete date for the 100th anniversary, since in the world’s literature devoted to mechanics, during the entire XXth century, there was no united opinion with respect to the authority of the Π-theorem proof. In the report, we speak about physical quantity dimensions related to combinations of measurement units, the standard scales served for the measurement. The measurement units are partitioned into the main and derived units. We introduce classes of measurements of unit systems, which differ from each other only by some quantity but not the nature of the main units. We formulate lemmas on powerdimension expressions and unary choice of dimension. Also, we define a basis for the dimensionality elimination as the maximal tuple of dimensionally independent quantities through whose dimensions the dimensions of all other quantities in the problem considered are expressed in a power way. The formulation of the Π-theorem is given purely mathematically; this means that in nature, all physical laws relating some quantities with others are described by generalized homogeneous functions. As an illustration, we perform an analysis of dimensions in the following three problems: the perturbation propagation under a strong point blow-up condition in the atmosphere, the flow around an immovable ball by a viscous fluid flow and the finding of the small oscillation for the mathematical pendulum. The dimensionless Reynolds, Froude, and Strouhal numbers are introduced. The problems of scale modeling and criteria for the satisfaction of similarity in natural and model systems are presented.

The Cauchy problem for coupled IMBq equations

In this paper, we study the Cauchy problem associated with two coupled IMBq equations. Under the assumptions for non-linear terms and initial data, we prove the existence and uniqueness of the global solution and give sufficient conditions of blow-up of the solution in finite time by convex methods. This supplements and improves some results by De Godefroy (1998, IMA J. Appl. Math., 60, 123-138).

An improved local blow-up condition for Euler–Poisson equations with attractive forcing

We improve the recent result of Chae and Tadmor (2008) [10] proving a one-sided threshold condition which leads to a finite-time breakdown of the Euler–Poisson equations in arbitrary dimension n .

Optimal conditions of non-simultaneous blow-up and uniform blow-up profiles in localized parabolic equations

This paper deals with localized parabolic equations u t = Δ u + u m ( x 0 , t ) e p v ( x 0 , t ) , v t = Δ v + u q ( x 0 , t ) e n v ( x 0 , t ) with homogeneous Dirichlet boundary conditions, where x 0 is any fixed point in a bounded domain of R N . The optimal classification of non-simultaneous and simultaneous blow-up phenomena is proposed for all of the nonnegative exponents. Moreover, uniform blow-up profiles are obtained for all kinds of simultaneous blow-up solutions.

Existence and boundary blow-up rates of solutions for boundary blow-up elliptic systems

In this paper, we study some elliptic systems with boundary blow-up conditions in a smooth bounded domain. By constructing the suitable upper and lower solutions, we discuss the existence and global estimate of the positive solution. Furthermore, the boundary blow-up rates of the positive solution are also partly discussed.

Instability of standing waves for a weakly coupled non-linear Schrödinger system

This article discusses the weakly coupled non-linear Schrödinger equations. With the variational characterization of the ground state solutions, the potential well argument and the concavity method, we derive a sharp condition for blow-up and global existence to the solutions of the Cauchy problem. At the same time, we also prove the instability of standing waves.

Blow-up of solution of Cauchy problem for three-dimensional damped nonlinear hyperbolic equation

The existence and uniqueness of the local generalized solution and the local classical solution to the Cauchy problem for three-dimensional damped nonlinear hyperbolic equation u t t + k 1 ∇ 4 u + k 2 ∇ 4 u t + ∇ 2 g ( ∇ 2 u ) = 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) are proved. This paper arrives at some sufficient conditions for blow-up of the solution in finite time and an example.

Sufficient conditions for the blowup of a solution to the Boussinesq equation subject to a nonlinear Neumann boundary condition

Sufficient blowup conditions are obtained for a solution to the generalized Boussinesq equation subject to a nonlinear Neumann boundary condition.

Non-simultaneous blow-up in coupled heat equations with multi-nonlinearities

This paper studies coupled heat equations with multi-nonlinearities of six nonlinear parameters. The critical blow-up exponent is established via a complete classification for all the six nonlinear parameters, where a precise analysis on the geometry of Ω and the absorption coefficients is given for the balanced interaction situation among the multi-nonlinearities. The main attention is contributed to non-simultaneous phenomena in the model to determine the necessary and sufficient conditions of non-simultaneous blow-up with suitable initial data, as well as the conditions under which any blow-up must be non-simultaneous. Finally, the model of the paper is characterized by a comparison with the known results for other models.

Magnetic flux penetration into high-temperature superconductors in the vortex liquid state under the blow-up conditions

The problem of magnetic field penetration into a type-II high-temperature superconductor that is in the weakly pinned vortex-liquid phase is considered. A magnetic field on the superconductor boundary rises with time in the blow-up regime. A model hydrodynamic equation describing the magnetic induction distribu- tion in the vortex-liquid phase for thermomagnetic motion of the flux is derived. Analytical expressions for the depth and rate of magnetic field penetration into the superconductor are found. It is demonstrated that these quantities depend on parameters of the problem: index of power n in the boundary regime characterizing the penetration rate of vortices into the superconducting half-space and a parameter describing the effect of random pinning forces and thermal fluctuations on the magnetic flux distribution.

On global solutions of the radial p-Laplace equation

We obtain blow-up conditions for solutions of the radial p -Laplace equation.

Collapse in coupled nonlinear Schrödinger equations: Sufficient conditions and applications

In this paper we study blow-up phenomena in general coupled nonlinear Schrödinger equations with different dispersion coefficients. We find sufficient conditions for blow-up and for the existence of global solutions. We discuss several applications of our results to heteronuclear multispecies Bose–Einstein condensates and to degenerate boson–fermion mixtures.

Periodic boundary value problem and cauchy problem of the generalized cubic double dispersion equation

In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equation u t t - u x x - a u x x t t + b u x 4 - d u x x t = f ( u ) x x are proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.

Existence of boundary blow-up solutions for a quasilinear elliptic equation

In this paper, the existence of solution for a class of quasilinear elliptic problem div(|∇ u| p−2 ∇ u)=a(x)f(u), u≥0 in Ω=B (the unit ball), with the boundary blow-up condition u| ∂Ω=+∞ is established, where a(x)∈C(Ω) blows up on ∂Ω,p>1 and f is assumed to satisfy (f 1) and (f 2). The results are obtained by using sub-supersolution methods.

Sharp conditions of global existence for the quasilinear Schrödinger equation

The paper discusses a class of quasilinear Schrodinger equations describing the upper-hybrid oscillation propagation. By establishing a cross-constrained variational problem and a so-called invariant flow, we obtain a sharp condition for blow-up and global existence of solutions of the Cauchy problem.

Singular, weak and absent: Solutions of the Euler equations

We will describe necessary and sufficient conditions for blowup and discuss weak solutions for the incompressible Euler equations. We will also describe a result concerning anomalous dissipation of energy.

Roles of weight functions to a nonlinear porous medium equation with nonlocal source and nonlocal boundary condition

In this paper we investigates the blow-up properties of the positive solutions to a porous medium equation with nonlocal reaction source and with nonlocal boundary condition, we obtain the blow-up condition and its blow-up rate estimate.

Blow-up of solution for a generalized Boussinesq equation

This paper studies the initial boundary value problem for a generalized Boussinesq equation and proves the existence and uniqueness of the local generalized solution of the problem by using the Galerkin method. Moreover, it gives the sufficient conditions of blow-up of the solution in finite time by using the concavity method.

Modeling weakly nonlinear acoustic wave propagation

Summary Three weakly nonlinear models of lossless, compressible fluid flow—a straightforward weakly nonlinear equation (WNE), the inviscid Kuznetsov equation (IKE) and the Lighthill–Westervelt equation (LWE)—are derived from first principles and their relationship to each other is established. Through a numerical study of the blow-up of acceleration waves, the weakly nonlinear equations are compared to the ‘exact’ Euler equations, and the ranges of applicability of the approximate models are assessed. By reformulating these equations as hyperbolic systems of conservation laws, we are able to employ a Godunov-type finite-difference scheme to obtain numerical solutions of the approximate models for times beyond the instant of blow-up (that is, shock formation), for both density and velocity boundary conditions. Our study reveals that the straightforward WNE gives the best results, followed by the IKE, with the LWE’s performance being the poorest overall.

Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity

In this work we consider nonlinear Volterra equations of a special type. We find necessary and sufficient conditions for blow-up of solutions to this class of equations.