Blow-up Singularity Research Articles

Equatorial blowup and polar caps in drop electrohydrodynamics

We illuminate effects of surface-charge convection intrinsic to leaky-dielectric electrohydrodynamics by analyzing the symmetric steady state of a circular drop in an external field at arbitrary electric Reynolds number ReE. In formulating the problem, we identify an exact factorization that reduces the number of dimensionless parameters from four—ReE and the conductivity, permittivity and viscosity ratios—to two: a modified electric Reynolds number Rẽ and a charging parameter ϖ. In the case ϖ<0, where charge relaxation in the drop phase is slower than in the suspending phase, and, as a consequence, the interface polarizes antiparallel to the external field, we find that above a critical Rẽ value the solution exhibits a blowup singularity such that the surface-charge density diverges antisymmetrically with the −1/3 power of distance from the equator. We use local analysis to uncover the structure of that blowup singularity, wherein surface charges are convected by a locally induced flow towards the equator where they annihilate. To study the blowup regime, we devise a numerical scheme encoding that local structure where the blowup prefactor is determined by a global charging-annihilation balance. We also employ asymptotic analysis to construct a universal problem governing the blowup solutions in the regime Rẽ≫1, far beyond the blowup threshold. In the case ϖ>0, where charge relaxation is faster in the drop phase and the interface polarizes parallel to the external field, we numerically observe and asymptotically characterize the formation at large Rẽ of stagnant, perfectly conducting surface-charge caps about the drop poles. The cap size grows and the cap voltage decreases monotonically with increasing conductivity or decreasing permittivity of the drop phase relative to the suspending phase. The flow in this scenario is nonlinearly driven by electrical shear stresses at the complement of the caps. In both polarization scenarios, the flow at large Rẽ scales linearly with the magnitude of the external field, contrasting the familiar quadratic scaling under weak fields. Published by the American Physical Society 2024

The Singularity Formation on the Coupled Burgers–Constantin–Lax–Majda System with the Nonlocal Term

In this paper, we study the finite-time singularity formation on the coupled Burgers–Constantin–Lax–Majda system with the nonlocal term, which is one nonlinear nonlocal system of combining Burgers equations with Constantin–Lax–Majda equations. We discuss whether the finite-time blow-up singularity mechanism of the system depends upon the domination between the CLM type’s vortex-stretching term and the Burgers type’s convection term in some sense. We give two kinds of different finite-time blow-up results and prove the local smooth solution of the nonlocal system blows up in finite time for two classes of large initial data.

Numerical study on the instability of localized buckling modes in the axially compressed strut on a distributed-spring elastic foundation with softening quadratic nonlinearity

Localized buckling modes in the axially compressed strut on a distributed-spring elastic foundation with a softening quadratic nonlinearity are numerically calculated based on a modified Petviashvili method in the spatial frequency domain. As the load decreases (increases), the maximum displacement of the corresponding localized buckling mode increases (decreases) and its width decreases (increases). Then, under the influence of longitudinal and transverse perturbations, stabilities of these localized buckling modes are numerically investigated. The adopted numerical method is the spatial Fourier transform in space and the finite difference method in time. For initial positive longitudinal perturbations, localized buckling modes are unstable showing focusing-type finite-time blowup singularities. For initial negative perturbations, localized buckling modes are unstable and become dispersed showing small-amplitude oscillations. For initial transverse perturbations, localized buckling modes are unstable and are transformed into three-dimensional locally confined modes, finally showing finite-time blowup singularities.

Thermal avalanche after non-simultaneous blow-up in heat equations coupled via nonlinear boundary

In this paper, we study a parabolic problem defined in the half line coupled via exponential-type boundary flux. Firstly, we prove the optimal classification of the two components of blow-up solutions when time reaches the blow-up time from left. Blow-up takes place only at the origin, and simultaneous blow-up rates are determined as well. Secondly, we study the weak extension after the blow-up time. Complete blow-up always occurs whether simultaneous blow-up arises or not. Moreover, an instantaneous propagation of the blow-up singularity to the whole spatial domain occurs at the blow-up time, which is the so-called thermal avalanche phenomenon. Finally, we use the evolution of the $k$-level set of solutions in the approximations to characterize the propagation of the singularity.

Local stability of the free additive convolution

We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble A+UBU⁎, where U is a Haar distributed random unitary or orthogonal matrix, and A and B are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎ concentrates around the free additive convolution of the spectral distributions of A and B on scales down to N−2/3.

Singularity analysis for a semilinear integro-differential equation with nonlinear memory boundary

In this paper, we study the blow-up singularity of a semilinear parabolic equation with nonlinear memory both in the reaction term and the boundary condition. We firstly establish the local solvability for a large class of semilinear parabolic equations with various nonlocal reaction terms. Secondly, we give a complete classification for the existence of a blow-up solution and a global solution. Next, we show that under some hypotheses the blow-up can only occur on the boundary of the domain. MSC:35K05, 35B44.

On the high Mach number shock structure singularity caused by overreach of Maxwellian molecules

The high Mach number shock structure singularity arising in moment equations of the Boltzmann equation was investigated. The source of the singularity is shown to be the unbalanced treatment between two high order kinematic and dissipation terms caused by the overreach of Maxwellian molecule assumption. In compressive gaseous flow, the high order stress-strain coupling term of quadratic nature will grow far faster than the strain term, resulting in an imbalance with the linear dissipation term and eventually a blow-up singularity in high thermal nonequilibrium. On the other hand, the singularity arising from unbalanced treatment does not occur in the case of velocity shear and expansion flows, since the high order effects are cancelled under the constraint of the free-molecular asymptotic behavior. As an alternative method to achieve the balanced treatment, Eu's generalized hydrodynamics, consistent with the second law of thermodynamics, was revisited. After introducing the canonical distribution function in exponential form and applying the cumulant expansion to the explicit calculation of the dissipation term, a natural platform suitable for the balanced treatment was derived. The resulting constitutive equation with the nonlinear factor was then shown to be well-posed for all regimes, effectively removing the high Mach number shock structure singularity.

Avalanche After Non-simultaneous Blow-up in Nonlinear Parabolic Systems

This paper cares about whether blow-up is complete or not for some heat equations with coupled nonlinear flux. First, we finish the classifications on blow-up solutions in the previous paper, and obtain that blow-up set is made up of only a single point. Then we prove that blow-up is always complete. It is interesting that, even if non-simultaneous blow-up happens, avalanche phenomena occur after the finite blow-up time, which means that blow-up singularity propagates instantaneously to the whole spatial domain after that time.

Control of blow-up singularities for nonlinear wave equations

While the global boundary control of nonlinear wave equations thatexhibit blow-up is generally impossible, we show on a typicalexample, motivated by laser breakdown, that it is possible tocontrol solutions with small data so that they blow up on aprescribed compact set bounded away from the boundary of thedomain. This is achieved using the representation of singularsolutions with prescribed blow-up surface given by Fuchsianreduction. We outline on this example simple methods that may beof wider applicability.

Moving mesh method for problems with blow-up on unbounded domains

This paper studies the numerical solutions of semilinear parabolic partial differential equations (PDEs) on unbounded spatial domains whose solutions blow up in finite time. There are two major difficulties usually in numerical solutions: the singularity of blow-up and the unboundedness. We propose local absorbing boundary conditions (LABCs) on the selected artificial boundaries by using the idea of unified approach (Brunner et al., SIAM J Sci Comput 31:4478---4496, (2010). Since the uniform fixed spatial meshes may be inefficient, we adopt moving mesh partial differential equation (MMPDE) method to adapt the spatial mesh as the singularity develops. Combining LABCs and MMPDE, we can effectively capture the qualitative behavior of the blow-up singularities in the unbounded domain. Moreover, the implementation of the combination consists of two independent parts. Numerical examples also illustrate the efficiency and the accuracy of the new method.

Shocks and universal statistics in (1+1)-dimensional relativistic turbulence

We propose that statistical averages in relativistic turbulence exhibit universal properties. We consider analytically the velocity and temperature differences structure functions in the (1+1)-dimensional relativistic turbulence in which shock waves provide the main contribution to the structure functions in the inertial range. We study shock scattering, demonstrate the stability of the shock waves, and calculate the anomalous exponents. We comment on the possibility of finite time blowup singularities.

Automorphisms of real rational surfaces and weighted blow-up singularities

Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e = [e 1, . . . , e ℓ ] be a partition of n. Denote by X e the set of ℓ-tuples (P 1, . . . , P ℓ ) of disjoint nonsingular curvilinear subschemes of X of orders (e 1, . . . , e ℓ ). We show that the group Aut(X) acts transitively on X e. This statement generalizes earlier work where the case of the trivial partition e = [1, . . . , 1] was treated under the supplementary condition that X is nonsingular. As an application we classify singular real rational surfaces obtained from nonsingular surfaces by performing weighted blow-ups.

How to adaptively resolve evolutionary singularities in differential equations with symmetry

Many time-dependent partial differential equations have solutions which evolve to have features with small length scales. Examples are blow-up singularities and interfaces. To compute such features accurately it is essential to use some form of adaptive method which resolves fine length and time scales without being prohibitively expensive to implement. In this paper we will describe an r-adaptive method (based on moving mesh partial differential equations) which moves mesh points into regions where the solution is developing singular behaviour. The method exploits natural symmetries which are often present in partial differential equations describing physical phenomena. These symmetries give an insight into the scalings (of solution, space and time) associated with a developing singularity, and guide the adaptive procedure. In this paper the theory behind these methods will be developed and then applied to a number of physical problems which have (blow-up type) singularities linked to symmetries of the underlying PDEs. The paper is meant to be a practical guide towards solving such problems adaptively and contains an example of a Matlab code for resolving the singular behaviour of the semi-linear heat equation.

Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains

Consider the diffusive Hamilton-Jacobi equation u t = Δu + |∇u| p , p > 2, on a bounded domain Ω with zero-Dirichlet boundary conditions, which arises in the KPZ model of growing interfaces. It is known that u remains bounded and that ∇u may blow up only on the boundary ∂Ω. In this paper, under suitable assumptions on $${\Omega\subset \mathbb{R}^2}$$ and on the initial data, we show that the gradient blow-up singularity occurs only at a single point $${x_0\in\partial\Omega}$$ . This is the first result of this kind in the study of problems involving gradient blow-up phenomena. In general domains of $${\mathbb{R}^n}$$ , we also obtain results on nondegeneracy and localization of blow-up points.

On blow-up space jets for the Navier–Stokes equations in R3 with convergence to Euler equations

Formation of blow-up singularities for the Navier–Stokes equations ut+(u⋅∇)u=−∇p+Δu and div u=0 in R3×(0,T) is studied. In cylindrical polar coordinates {r,φ,z} in R3, their restriction to the linear subspace W2=Span{1,z} is shown to be consistent. Using links with blow-up theory for nonlinear reaction-diffusion partial differential equations, the following questions are under scrutiny: (ii) introducing a self-similar blow-up “swirl mechanism” with the angular swirl divergences φ(t)=−σ ln(T−t) and φ̇(t)=(σ/(T−t))→∞ as t→T−, where σ∊R is a parameter; (ii) existence of a countable family of space jets via a nonlocal semilinear parabolic equation with effective regional/global blow-up of the z component of the velocity; (iii) as an intrinsic part of the construction, convergence of the above rescaled patterns as t→T− to smooth blow-up self-similar solutions of the corresponding three-dimensional Euler equations ut+(u⋅∇)u=−∇p and div u=0 in R3×(0,T), which (iv) are also shown to admit single point blow-up in the similarity variables.

Curvature estimates for the Ricci flow II

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow, including the volume normalized Ricci flow and the normalized Kähler-Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor, while the convergence results require finiteness of space-time integrals of this norm. These results also serve as characterization of blow-up singularities.

Blow-up similarity solutions of the fourth-order unstable thin film equation

We study blow-up behaviour of solutions of the fourth-order thin film equationwhich contains a backward (unstable) diffusion term. Our main goal is a detailed study of the case of the first critical exponentwhereN≥ 1 is the space dimension. We show that the free-boundary problem with zero contact angle and zero-flux conditions admits continuous sets (branches) of blow-up self-similar solutions. For the Cauchy problem inRN×R+, we detect compactly supported blow-up patterns, which have infinitely many oscillations near interfaces and exhibit a “maximal” regularity there. As a key principle, we use the fact that, for small positiven, such solutions are close to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equationwhich are better understood and have been studied earlier [19]. We also discuss some general aspects of formation of self-similar blow-up singularities for other values ofp.

The Glimm scheme for perfect fluids on plane-symmetric Gowdy spacetimes

We propose a new, augmented formulation of the coupled Euler–Einstein equations for perfect fluids on plane-symmetric Gowdy spacetimes. The unknowns of the augmented system are the density and velocity of the fluid and the first- and second-order spacetime derivatives of the metric. We solve the Riemann problem for the augmented system, allowing propagating discontinuities in both the fluid variables and the first- and second-order derivatives of the geometry coefficients. Our main result, based on Glimm's random choice scheme, is the existence of solutions with bounded total variation of the Euler–Einstein equations, up to the first time where a blow-up singularity (unbounded first-order derivatives of the geometry coefficients) occurs. We demonstrate the relevance of the augmented system for numerical relativity. We also consider general vacuum spacetimes and solve a Riemann problem, by relying on a theorem by Rendall on the characteristic value problem for the Einstein equations.

Minimal blow-up asymptotics of quasilinear heat equations

We study the asymptotic properties of blow-up solutions u = u(x, t) ≥ 0 of the quasilinear heat equation , where k(u) is a smooth non-negative function, with a given blowing up regime on the boundary u(0, t) = ψ(t) > 0 for t ∈ (0, 1), where ψ(t) → ∞ as t → 1−, and bounded initial data u(x, 0) ≥ 0. We classify the asymptotic properties of the solutions near the blow-up time, t → 1−, in terms of the heat conductivity coefficient k(u) and of boundary data ψ(t); both are assumed to be monotone. We describe a domain, denoted by , of minimal asymptotics corresponding to the data ψ(t) with a slow growth as t → 1− and a class of nonlinear coefficients k(u).We prove that for any problem in S11−, such a blow-up singularity is asymptotically structurally equivalent to a singularity of the heat equation ut = uxx described by its self-similar solution of the form u*(x, t) = −ln(1 − t) + g(ξ), ξ = x/(1 − t)1/2, where g solves a linear ordinary differential equation. This particular self-similar solution is structurally stable upon perturbations of the boundary function and also upon nonlinear perturbations of the heat equation with the basin of attraction .

Phénomène d’explosion totale pour l’équation de la chaleur avec non-linéarité du type puissance

In this Note, we are interested in the possible continuation after the blow-up time T m of radially symmetric positive classical solutions u of the heat equation with nonlinearity f(u) = u p, where p > 1. We say that u blows up completely after T m if u can not be extended beyond T m (even in the weak sense). We obtain a complete blow up criterion which relies on the asymptotic behaviour of u around the blow-up singularity x = 0.