Blow-up Mechanism Research Articles

Stable blowup for focusing semilinear wave equations in all dimensions

We consider the wave equation with focusing power nonlinearity. The associated ODE in time gives rise to a self-similar solution known as the ODE blowup. We prove the nonlinear asymptotic stability of this blowup mechanism outside of radial symmetry in all space dimensions and for all superlinear powers. This result covers for the first time the whole energy-supercritical range without symmetry restrictions.

Existence and stability of shrinkers for the harmonic map heat flow in higher dimensions

We study singularity formation for the heat flow of harmonic maps from Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document}. For each d≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 4$$\\end{document}, we construct a compact, d-dimensional, rotationally symmetric target manifold that allows for the existence of a corotational self-similar shrinking solution (shortly shrinker) that represents a stable blowup mechanism for the corresponding Cauchy problem.

Singularity formation for the relativistic Euler‐Poisson equations with repulsive force and damping

AbstractIn this paper, we consider the blowup mechanism of regular solutions to the spherically symmetry relativistic Euler‐Poisson equations with repulsive force and damping. By means of a weighted function and the structured functional, we show the regular solution will blow up in finite time.

On the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations

In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail.

A Generalized Two-Component Camassa-Holm System with Complex Nonlinear Terms and Waltzing Peakons

In this paper, we deal with the Cauchy problem for a generalized two-component Camassa-Holm system with waltzing peakons and complex higher-order nonlinear terms. By the classical Friedrichs regularization method and the transport equation theory, the local well-posedness of solutions for the generalized coupled Camassa-Holm system in nonhomogeneous Besov spaces and the critical Besov space B^{5/2}_{2,1}times B^{5/2}_{2,1} was obtained. Besides the propagation behaviors of compactly supported solutions, the global existence and precise blow-up mechanism for the strong solutions of this system are determined. In addition to wave breaking, the another one of the most essential property of this equation is the existence of waltzing peakons and multi-peaked solitray was also obtained.

Stable singularity formation for the inviscid primitive equations

The primitive equations (PEs) model large-scale dynamics of the oceans and the atmosphere. While it is by now well known that the three-dimensional viscous PEs are globally well posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blow-up mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic-in-time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameter family of nonsmooth profiles, and is stable only by smoother perturbations.

Blow-up data for a two-component Camassa-Holm system with high order nonlinearity

This paper is concerned with the Cauchy problem for a two-component Camassa-Holm system with high order nonlinearity, which is a multi-component extension of the Fokas-Olver-Rosenau-Qiao equation. Firstly, we state the local well-posedness for the Cauchy problem of the system in the framework of Sobolev-Besov spaces. Then, we establish the precise blow-up mechanism for the strong solutions by means of the transport equation theory. As is well-known, the H1-norm conservation law of the velocity component is crucial to study blow-up phenomena of the single Camassa-Holm type equation. However, it is no longer available for our nonlinear coupled system. To overcome this difficulty, we derive some suitable conservation laws by sufficiently exploiting the fine structure of the system. Based on which, we finally construct several new blow-up strong solutions with certain initial profiles in finite time.

Stable blowup for the supercritical hyperbolic Yang-Mills equations

We consider the Yang-Mills equations in (1+d)-dimensional Minkowski spacetime. It is known that in the supercritical case, i.e., for d≥5, these equations admit closed form equivariant self-similar blowup solutions [2]. These solutions are furthermore conjectured to be the universal attractors for generic large equivariant data evolutions. In this paper we partially prove this conjecture. Namely, we show that for all odd d≥5 the blowup mechanism exhibited by these solutions is stable.

Finite time blow-up for some parabolic systems arising in turbulence theory

We study a class of non-linear parabolic systems relevant in turbulence theory. Those systems can be viewed as simplified versions of the Prandtl one-equation and Kolmogorov two-equation models of turbulence. We restrict our attention to the case of one space dimension. We consider initial data for which the diffusion coefficients may vanish. We prove that, under this condition, those systems are locally well-posed in the class of Sobolev spaces of high enough regularity, but also that there exist smooth initial data for which the corresponding solutions blow up in finite time. We are able to put in evidence two different types of blow-up mechanism. In addition, the results are extended to the case of transport-diffusion systems, namely to the case when convection is taken into account.

On finite-time blowup mechanism of irrotational compressible Euler equations with time-dependent damping

ABSTRACT In this paper, sufficient initial conditions for finite-time blowup of smooth solutions of the irrotational compressible Euler equations with time-dependent damping are established. Our blowup conditions reveal that for sufficiently large initial velocity, fixed background density and with no largeness assumption on the initial density, the velocity of the fluid must collapse in finite time on some subset of general Euclidean space with non-zero Lebesgue measure.

Blowup of smooth solutions to the compressible Euler equations with radial symmetry on bounded domains

In this paper, we study the blowup of smooth solutions to the compressible Euler equations with radial symmetry on some fixed bounded domains ( $$B_{R}=\{x\in {\mathbb {R}}^{N}:\ |x|\le R\}$$ , $$N=1,2,\ldots $$ ) by introducing some new averaged quantities. We consider two types of flows: initially move inward and initially move outward on average. For the flow initially moving inward on average, we show that the smooth solutions will blow up in a finite time if the density vanishes at the origin only ( $$\rho (t,0)=0,\ \rho (t,r)>0,\ 0<r\le R$$ ) for $$N\ge 1$$ or the density vanishes at the origin and the velocity field vanishes on the two endpoints ( $$\rho (t,0)=0,\ v(t,R)=0$$ ) for $$N=1$$ . For the flow initially moving outward, we prove that the smooth solutions will break down in a finite time if the density vanishes on the two endpoints ( $$\rho (t,R)=0$$ ) for $$N=1$$ . The blowup mechanisms of the compressible Euler equations with constant damping or time-depending damping are obtained as corollaries.

Transition of blow-up mechanisms in k-equivariant harmonic map heat flow

In the present article, we consider blow-up phenomena appearing in k-equivariant harmonic map heat flow from to a unit sphere : Here the scalar variable u stands for latitudinal angle on from the north pole to the south pole . The integer corresponds to the eigenvalues associated to eigenmaps , that is, harmonic maps with constant energy density. We prove constructively the existence of asymptotically non-self-similar blow-up solutions with precise description of their local space-time profiles. The blow-up solutions arise from, depending on the combination of d and k, two different approximations of the nonlinear term: either through a Dirac mass supported at the origin or via a Taylor expansion around equator map . Transition of the blow-up mechanisms arises, accordingly.

Qualitative analysis for the new shallow-water model with cubic nonlinearity

Considered in this paper is the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa-Holm equation (also called Fokas-Olver-Rosenau-Qiao equation) and the Novikov equation as two special cases. The main investigation is the Cauchy problem of the new shallow-water model with qualitative properties of its solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model, and then derive a precise blow-up scenario and the blow-up mechanisms for solutions with certain initial profiles. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we investigate the persistence properties of the solution to the Cauchy problem of the new shallow-water model.

Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain

We study local existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker–Smale (in short TCS) model coupled with incompressible Navier–Stokes (NS) equations in the whole space. The coupled system consists of the kinetic TCS equation for particle ensemble and the incompressible NS equations for a fluid via a drag force. For the strong solution, we investigate the blow-up mechanism for the coupled system, and we also study the global existence of a weak solution in the whole space.

Type II blow-up mechanisms in a semilinear heat equation with Lepin exponent

We are concerned with blow-up mechanisms in a semilinear heat equation:ut=Δu+|u|p−1u,x∈RN,t>0, where p>1 is a constant. In the present article, we prove constructively the existence of type II blow-up solutions for p=1+6/(N−10), so called Lepin exponent. Their blow-up mechanisms are strongly influenced on space dimensions. For lower dimensions the mechanisms are determined by Dirac δ type approximations of nonlinear term and, on the other hand, dominated by classical Taylor approximation for high dimensions. On the threshold dimension, the both approximations are balanced. Consequently, the blow-up exhibits a new blow-up profile. Classification of radial blow-up solutions in terms of blow-up rates is also presented.

Well-posedness and wave breaking for a shallow water wave model with large amplitude

Considered herein is the Cauchy problem of a model for shallow water waves of large amplitude. Using Littlewood–Paley decomposition and transport equation theory, we establish the local well-posedness of the equation in Besov spaces $$B^s_{p,r} $$ with $$1\le p,r \le +\infty $$ and $$s>\max \{1+\frac{1}{p },\frac{3}{2}\}$$ (and also in Sobolev spaces $$H^s=B^s_{2,2}$$ with $$s>3/2$$). Then, the precise blow-up mechanism for the strong solutions is determined in the lowest Sobolev space $$H^s $$ with $$s>3/2$$. Our results improve the corresponding work for this model in Quirchmayr (J Evol Equ 16:539–567, 2016), in which the Sobolev index $$s=3$$ is required. In addition, we also investigate the asymptotic behaviors of the strong solutions to this equation at infinity within its lifespan provided the initial data lie in weighted $$L_{p,\phi }:=L_p(\mathbb {R},\phi ^pdx)$$ spaces.

A new path to the non blow-up of incompressible flows

One of the most challenging questions in fluid dynamics is whether the three-dimensional (3D) incompressible Navier-Stokes, 3D Euler and two-dimensional Quasi-Geostrophic (2D QG) equations can develop a finite-time singularity from smooth initial data. Recently, from a numerical point of view, Luo & Hou presented a class of potentially singular solutions to the Euler equations in a fluid with solid boundary [1,2]. Furthermore, in two recent papers [3,4], Tao indicates a significant barrier to establishing global regularity for the 3D Euler and Navier-Stokes equations, in that any method for achieving this, must use the finer geometric structure of these equations. In this paper, we show that the singularity discovered by Luo & Hou which lies right on the boundary is not relevant in the case of the whole domain R3. We reveal also that the translation and rotation invariance present in the Euler, Navier-Stokes and 2D QG equations are the key for the non blow-up in finite time of the solutions. The translation and rotation invariance of these equations combined with the anisotropic structure of regions of high vorticity allowed to establish a new geometric non blow-up criterion which yield us to the non blow-up of the solutions in all the Kerr's numerical experiments and to show that the potential mechanism of blow-up introduced in [5] cannot lead to the blow-up in finite time of solutions of Euler equations.

The local existence and blowup criterion for strong solutions to the kinetic Cucker–Smale model coupled with the compressible Navier–Stokes equations

In this paper, we establish the existence and uniqueness of local strong solutions to the kinetic Cucker–Smale model coupled with the isentropic compressible Navier–Stokes equations in the whole space. Moreover, the blowup mechanism for strong solutions to the coupled system is also investigated.

On the existence and stability of blowup for wave maps into a negatively curved target

We consider wave maps on $(1+d)$-dimensional Minkowski space. For each dimension $d\geq 8$ we construct a negatively curved, $d$-dimensional target manifold that allows for the existence of a self-similar wave map which provides a stable blowup mechanism for the corresponding Cauchy problem.

Type II blow-up mechanisms in a semilinear heat equation with critical Joseph–Lundgren exponent

We are concerned with blow-up mechanisms in a semilinear heat equationut=Δu+|u|p−1u,x∈RN,t>0, where p>1 is a constant. It is well known that type II blow-up does occur if N≥11 and p>pJL, where pJL stands for the Joseph–Lundgren exponent:pJL={+∞,N≤10,1+4N−4−2N−1,N≥11. On the other hand, it is also known that, if p<pJL, type II blow-up cannot occur under mild assumptions on initial data as far as nonnegative radially symmetric solutions are concerned. It has long remained open whether or not type II blow-up occurs for the borderline case p=pJL,N≥11. In the present article we prove constructively the existence of type II blow-up solutions for the Joseph–Lundgren critical case, thus giving an answer to this question.