Self-similar Blow-up Solutions Research Articles

A Globally Stable Self-Similar Blowup Profile in Energy Supercritical Yang-Mills Theory

This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity.

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.

Non-existence of some approximately self-similar singularities for the Landau, Vlasov-Poisson-Landau, and Boltzmann equations

We consider the homogeneous and inhomogeneous Landau equation for very soft and Coulomb potentials and show that approximate Type I self-similar blow-up solutions do not exist under mild decay assumptions on the profile. We extend our analysis to the Vlasov-Poisson-Landau system and to the Boltzmann equation without angular cut-off.

Co-dimension one stable blowup for the supercritical cubic wave equation

For the focusing cubic wave equation, we find an explicit, non-trivial self-similar blowup solution uT⁎, which is defined on the whole space and exists in all supercritical dimensions d≥5. For d=7, we analyze its stability properties without any symmetry assumptions and prove the existence of a set of perturbations which lead to blowup via uT⁎ in a backward light cone. Moreover, this set corresponds to a co-dimension one Lipschitz manifold modulo translation symmetries in similarity coordinates.

Stable self-similar blow-up for a family of nonlocal transport equations

We consider a family of nonlocal problems that model the effects of transport and vortex stretching in the incompressible Euler equations. Using modulation techniques, we establish stable self-similar blow-up near a family of known self-similar blow-up solutions.

Critical dynamics of damage-failure transition in wide range of load intensity

The self-similarity of characteristic stages of damage-failure transition has been studied both theoretically and experimentally, with the damage localization kinetics analyzed according to the nonlinearity of free energy release of solids with defects: Free energy in the generalized Ginzburg–Landau form reflecting the specific criticality of solids with defects, and the structural-scaling transition in the presence of two critical points separating qualitatively different material responses corresponds to the types of collective modes of defects. These collective modes have the nature of the self-similar solutions of the damage evolution equation. The final damage stage is associated with the singular self-similar blow-up solutions localized on a set of spatial scales. The presence of two singularities (the stress field at the crack tip and the blow-up damage modes) allows the interpretation of self-similarity of damage localization kinetics to be construed as a criticality sign of damage-failure transition and has been illustrated by the original experimental statements: transition from the steady state to branching crack dynamics, dynamic fragmentation statistics, resonance regimes of failure in shocked materials.

On the stability of self-similar blow-up for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$

We study the stability of recently constructed self-similar blowup solutions to the incompressible Euler equation. A consequence of our work is the existence of finite-energy $C^{1,\alpha}$ solutions that become singular in finite time in a locally self-similar manner. As a corollary, we also observe that the Beale–Kato–Majda criterion cannot be improved in the class of $C^{1,\alpha}$ solutions.

Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles

We consider the 1D nonlinear Schrodinger equation (NLS) with focusing point nonlinearity, \begin{document}$ \begin{equation} i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0, \;\;\;\;\;\;(0.1)\end{equation} $\end{document} where \begin{document}$ {\delta} = {\delta}(x) $\end{document} is the delta function supported at the origin. In the \begin{document}$ L^2 $\end{document} supercritical setting \begin{document}$ p>3 $\end{document} , we construct self-similar blow-up solutions belonging to the energy space \begin{document}$ L_x^\infty \cap \dot H_x^1 $\end{document} . This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic cylinder functions (Weber functions) and solving the jump condition at \begin{document}$ x = 0 $\end{document} imposed by the \begin{document}$ \delta $\end{document} term in (0.1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case \begin{document}$ 0 using the log Binet formula for the gamma function and steepest descent method in the integral formulae for the parabolic cylinder functions.

Nonlinear stability of homothetically shrinking Yang-Mills solitons in the equivariant case

We study the heat flow for Yang-Mills connections on It is well-known that in dimensions this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove. We prove the nonlinear asymptotic stability of the Weinkove solution under small equivariant perturbations and thus extend a result by the second author and Donninger for d = 5 to higher dimensions. At the same time, we provide a general framework for proving stability of self-similar blowup solutions to a large class of semilinear heat equations in arbitrary space dimension including a robust and simple method for solving the underlying spectral problems.

Threshold for blowup for the supercritical cubic wave equation

We consider the focusing cubic wave equation in the energy supercritical case, i.e. in dimensions . For this model an explicit nontrivial self-similar blowup solution was recently found by the first and third author in Glogić and Schörkhuber (2018 (arXiv:1810.07681)). Furthermore, the solution is proven to be co-dimension one stable in d = 7. In this paper, we study the equation from a numerical point of view. For d = 5 and d = 7 in the radial case, we provide evidence that this solution is at the threshold between generic ODE blowup and dispersion. In addition, we investigate the spectral problem that underlies the stability analysis and compute the spectrum in general supercritical dimensions.

Dynamical behavior near explicit self-similar blow–up solutions for the Born–Infeld equation

This paper studies the dynamical behavior near a new family of explicit self-similar solutions for the one-dimensional Born–Infeld equation. This quasilinear scalar field equation arises from nonlinear electromagnetism, as well as branes in string theory and minimal surfaces in Minkowski spacetimes. We show that both this model and the linear wave equation admit the same family of explicit timelike self-similar blow-up solutions; meanwhile, Lyapunov nonlinear stability of those self-similar blow-up solutions is given inside a strictly proper subset of the backward light cone.

Near critical, self-similar, blow-up solutions of the generalised Korteweg–de Vries equation: Asymptotics and computations

In this article we give a detailed asymptotic analysis of the near critical self-similar blowup solutions to the Generalised Korteweg–de Vries equation (GKdV). We compare this analysis to some careful numerical calculations. It has been known that for a nonlinearity that has a power larger than the critical value p=5, solitary waves of the GKdV can become unstable and become infinite in finite time, in other words they blow up. Numerical simulations presented in Klein and Peter (2015) indicate that if p>5 the solitary waves travel to the right with an increasing speed, and simultaneously, form a similarity structure as they approach the blow-up time. This structure breaks down at p=5. Based on these observations, we rescale the GKdV equation to give an equation that will be analysed by using asymptotic methods as p→5+. By doing this we resolve the complete structure of these self-similar blow-up solutions and study the singular nature of the solutions in the critical limit. In both the numerics and the asymptotics, we find that the solution has sech-like behaviour near the peak. Moreover, it becomes asymmetric with slow algebraic decay to the left of the peak and much more rapid algebraic decay to the right. The asymptotic expressions agree to high accuracy with the numerical results, performed by adaptive high-order solvers based on collocation or finite difference methods.

Stable blowup for the supercritical Yang–Mills heat flow

In this paper, we consider the heat flow for Yang–Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)$-equivariant setting, the Yang–Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove [“Singularity formation in the Yang-Mills flow”, Calc. Var. Partial Differential Equations, 19(2):211–220, 2004]. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in $L^{\infty}$.

Construction of type I blowup solutions for a higher order semilinear parabolic equation

Abstract We consider the higher-order semilinear parabolic equation $$\begin{array}{} \displaystyle \partial_t u = -(-{\it\Delta})^{m} u + u|u|^{p-1}, \end{array}$$ in the whole space ℝN, where p > 1 and m ≥ 1 is an odd integer. We exhibit type I non self-similar blowup solutions for this equation and obtain a sharp description of its asymptotic behavior. The method of construction relies on the spectral analysis of a non self-adjoint linearized operator in an appropriate scaled variables setting. In view of known spectral and sectorial properties of the linearized operator obtained by Galaktionov [15], we revisit the technique developed by Merle-Zaag [23] for the classical case m = 1, which consists in two steps: the reduction of the problem to a finite dimensional one, then solving the finite dimensional problem by a classical topological argument based on the index theory. Our analysis provides a rigorous justification of a formal result in [15].

Nonexistence of self-similar blowup for the nonlinear Dirac equations in (1+1) dimensions

We address a general system of nonlinear Dirac equations in (1+1) dimensions and prove nonexistence of self-similar blowup solutions in the space of bounded functions. While this argument does not exclude the possibility of finite-time blowup, it still suggests that self-similar singularities do not develop in the nonlinear Dirac equations in (1+1) dimensions in a finite time. In the particular case of the cubic Dirac equations, we characterize (unbounded) self-similar solutions in the closed analytical form.

Blow-up Profiles for the Parabolic–Elliptic Keller–Segel System in Dimensions $${n\geq 3}$$ n ≥ 3

We study the blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel-Patlak system in space dimensions $n\ge 3$. In view of the biological background of this system and of its mass conservation property, blowup is usually interpreted as a phenomenon of concentration or aggregation of the bacterial population. Understanding the asymptotic behavior of solutions at the blowup time is thus meaningful for the interpretation of the model. Under mild assumptions on the initial data, for $n\ge 3$, we show that the final profile satisfies $C_1|x|^{-2}\le u(x,T)\le C_2|x|^{-2}$, with convergence in $L^1$ as $t\to T$. This is in sharp contrast with the two-dimensional case, where solutions are known to concentrate to a Dirac mass at the origin (plus an integrable part). We also obtain refined space-time estimates of the form $u(x,t)\le C(T-t+|x|^2)^{-1}$ for type~I blowup solutions. Previous work had shown that radial, self-similar blowup solutions (which satisfy the above estimates) exist in dimensions $n\ge 3$ and do not exist in dimension $2$. Our results thus reveal that the final profile displayed by these special solutions actually corresponds to a much more general phenomenon.

Construction of a spectrally stable self-similar blowup solution to the supercritical corotational harmonic map heat flow

We prove the existence of a (spectrally) stable self-similar blow-up solution f0 to the heat flow for corotational harmonic maps from to the three-sphere. In particular, our result verifies the spectral gap conjecture stated by one of the authors and lays the groundwork for the proof of the nonlinear stability of f0. At the heart of our analysis lies a new existence result of a monotone self-similar solution f0. Although solutions of this kind have already been constructed before, our approach reveals substantial quantitative properties of f0, leading to the stability result. A key ingredient is the use of interval arithmetic: a rigorous computer-assisted method for estimating functions. It is easy to verify our results by robust numerics but the purpose of the present paper is to provide mathematically rigorous proofs.

Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation

The Cauchy problem for a fourth-order Boussinesq-type quasilinear wave equation (QWE-4) of the form \begin{document}$u_{tt} = -(|u|^n u)_{xxxx}\;\;\; \mbox{in}\;\;\; \mathbb{R} × \mathbb{R}_+, \;\;\;\mbox{with a fixed exponent} \, \, \, n>0, $ \end{document} and bounded smooth initial data, is considered. Self-similar single-point gradient blow-up solutions are studied. It is shown that such singular solutions exist and satisfy the case of the so-called self-similarity of the second type. Together with an essential and, often, key use of numerical methods to describe possible types of gradient blow-up, a homotopy approach is applied that traces out the behaviour of such singularity patterns as \begin{document}$n \to 0^+$\end{document} , when the classic linear beam equation occurs \begin{document}$u_{tt} = -u_{xxxx}, $ \end{document} with simple, better-known and understandable evolution properties.

Construction of type II blow-up solutions for a semilinear parabolic system with higher dimension

This paper is concerned with blow-up solutions for a semilinear parabolic system with a power type nonlinearity. Non self-similar blow-up solution is constructed by the matched asymptotic expansions. One component of this solution converges to the singular steady state, and another component converges to zero in self-similar variables.

On the discretely self-similar solutions for the 3D Navier–Stokes equations

We consider the discretely self-similar blowup solutions of the three-dimensional Navier–Stokes equations, and under suitable assumptions we show some estimates on the asymptotic behavior of the possible nontrivial velocity profiles.