Self-similar Blow-up Research Articles

Spectral theory and self-similar blowup in wave equations

This is an expository article that describes the spectral-theoretic aspects in the study of the stability of self-similar blowup for nonlinear wave equations. The linearization near a self-similar solution leads to a genuinely nonself-adjoint operator which is difficult to analyze. The main goal of this article is to provide an accessible account of the only known method that is capable of providing sufficient spectral information to complete the stability analysis. The exposition is based on a mini course given at the Summer School on Geometric Dispersive PDEs in Obergurgl, Austria, in September 2022.

Self-similar blow-up solutions in the generalised Korteweg-de Vries equation: spectral analysis, normal form and asymptotics

In the present work we revisit the problem of the generalised Korteweg–de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter p, here at p = 5. We provide a normal form of the associated collapse dynamics, and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterise the linearisation spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterise. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schrödinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for p > 5 in the co-exploding frame.

Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier–Stokes

We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008 Commun. Pure Appl. Math. 61 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in the r direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in r. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990 J. Stat. Phys. 59 1251–63; De Gregorio 1996 Math. Methods Appl. Sci. 19 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

Stable Singularity Formation for the Keller–Segel System in Three Dimensions

We consider the parabolic–elliptic Keller–Segel system in dimensions d≧3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\geqq 3$$\\end{document}, which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example was found more than two decades ago by Brenner et al. (Nonlinearity 12(4):1071–1098, 1999), and is conjectured to be nonlinearly radially stable. We prove this conjecture for d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document}. Our approach consists of reformulating the problem in similarity variables and studying the Cauchy evolution in intersection Sobolev spaces via semigroup theory methods. To solve the underlying spectral problem, we use a technique we recently established in Glogić and Schörkhuber (Comm Part Differ Equ 45(8):887–912, 2020). To the best of our knowledge, this provides the first result on stable self-similar blowup for the Keller–Segel system. Furthermore, the extension of our result to any higher dimension is straightforward. We point out that our approach is general and robust, and can therefore be applied to a wide class of parabolic models.

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.

Exactly self-similar blow-up of the generalized De Gregorio equation

We study exactly self-similar blow-up profiles for the generalized De Gregorio model for the three-dimensional Euler equation: We show that for any such that is sufficiently small, there is an exactly self-similar C α solution that blows up in finite time. This simultaneously improves on the result in Elgindi and Jeong (2020 Arch. Ration. Mech. Anal. 235 1763–817) by removing the restriction and Chen et al (2021 Commun. Pure Appl. Math. 74 1282–350), which only deals with asymptotically self-similar blow-ups.

Asymptotic Self-Similar Blow-Up Profile for Three-Dimensional Axisymmetric Euler Equations Using Neural Networks.

Whether there exist finite-time blow-up solutions for the 2D Boussinesq and the 3D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks, that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate physics-informed neural networks could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.

Smooth self-similar imploding profiles to 3D compressible Euler

The aim of this note is to present the recent results by Buckmaster, Cao-Labora, and Gómez-Serrano [Smooth imploding solutions for 3D compressible fluids, Arxiv preprint arXiv:2208.09445, 2022] concerning the existence of “imploding singularities” for the 3D isentropic compressible Euler and Navier-Stokes equations. Our work builds upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [Invent. Math. 227 (2022), pp. 247–413; Ann. of Math. (2) 196 (2022), pp. 567–778; Ann. of Math. (2) 196 (2022), pp. 779–889] and proves the existence of self-similar profiles for all adiabatic exponents γ > 1 \gamma >1 in the case of Euler; as well as proving asymptotic self-similar blow-up for γ = 7 5 \gamma =\frac 75 in the case of Navier-Stokes. Importantly, for the Navier-Stokes equation, the solution is constructed to have density bounded away from zero and constant at infinity, the first example of blow-up in such a setting. For simplicity, we will focus our exposition on the compressible Euler equations.

Asymptotically self-similar blowup of the Hou-Luo model for the 3D Euler equations

Inspired by the numerical evidence of a potential 3D Euler singularity [54, 55], we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in [54, 55] for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in [54, 55] share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical rescaling formulation and the strategy proposed in our recent work in [11] to establish the nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the HL model with smooth initial data and finite energy will develop a focusing asymptotically self-similar singularity in finite time. Moreover the self-similar profile is unique within a small energy ball and the $$C^\gamma $$ norm of the density $$\theta $$ with $$\gamma \approx 1/3$$ is uniformly bounded up to the singularity 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.

A potential two-scale traveling wave singularity for 3D incompressible Euler equations

In this paper, we investigate a potential two-scale traveling wave singularity of the 3D incompressible axisymmetric Euler equations with smooth initial data of finite energy. The two-scale feature is characterized by the property that the center of the traveling wave approaches to the origin at a slower rate than the rate of the collapse of the singularity. The driving mechanism for this potential singularity is due to two antisymmetric vortex dipoles that generate a strong shearing layer in both the radial and axial velocity fields. Without any viscous regularization, the 3D Euler equations develop an additional small scale characterizing the thickness of the sharp front. In order to stabilize the rapidly decreasing thickness of the sharp front, we apply a vanishing first order numerical viscosity to the Euler equations. We present numerical evidence that the 3D Euler equations with this first order numerical viscosity develop a locally self-similar blowup at the origin.

Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

<p style='text-indent:20px;'>We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>posed in any space dimension <inline-formula><tex-math id="M1">\begin{document}$ x\in \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ t\geq0 $\end{document}</tex-math></inline-formula> and with exponents <inline-formula><tex-math id="M3">\begin{document}$ m>1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ p\in(0, 1) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \sigma>2(1-p)/(m-1) $\end{document}</tex-math></inline-formula>. We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are <i>compactly supported</i> and might present two different types of interface behavior and three different possible <i>good behaviors</i> near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of <inline-formula><tex-math id="M6">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>. This paper generalizes in dimension <inline-formula><tex-math id="M7">\begin{document}$ N>1 $\end{document}</tex-math></inline-formula> previous results by the authors in dimension <inline-formula><tex-math id="M8">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula> and also includes some finer classification of the profiles for <inline-formula><tex-math id="M9">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> large that is new even in dimension <inline-formula><tex-math id="M10">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula>.</p>

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.

On the Slightly Perturbed De Gregorio Model on $$S^1$$

It is conjectured that the generalization of the Constantin-Lax-Majda model (gCLM) $\omega_t + a u\omega_x = u_x \omega$ due to Okamoto, Sakajo and Wunsch can develop a finite time singularity from smooth initial data for $a < 1$. For the endpoint case where $a$ is close to and less than $1$, we prove finite time asymptotically self-similar blowup of gCLM on a circle from a class of smooth initial data. For the gCLM on a circle with the same initial data, if the strength of advection $a$ is slightly larger than $1$, we prove that the solution exists globally with $|| \omega(t)||_{H^1}$ decaying in a rate of $O(t^{-1})$ for large time. The transition threshold between two different behaviors is $a=1$, which corresponds to the De Gregorio model.

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.

Self-similar Blow-Up Profiles for Slightly Supercritical Nonlinear Schrödinger Equations

We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schrodinger equation $$\text {i}\partial _t u + \Delta u + |u|^{p-1}u=0$$ on $$\mathbb {R}^d$$ , close to the mass critical case and for any space dimension $$d\ge 1$$ . These profiles bifurcate from the ground-state solitary wave. The argument relies on the classical matched asymptotics method suggested in Sulem and Sulem (The nonlinear Schrodinger equation. Self-focusing and wave collapse. Applied mathematical sciences, 139, Springer, New York, 1999) which needs to be applied in a degenerate case due to the presence of exponentially small terms in the bifurcation equation related to the log–log blow-up law observed in the mass critical case.

Self-similar Blow-Up Profiles for a Reaction–Diffusion Equation with Critically Strong Weighted Reaction

We study the separate variable blow-up patterns associated to the following second order reaction-diffusion equation: $$ \partial_tu=\Delta u^m + |x|^{\sigma}u^m, $$ posed for $x\in\mathbb{R}^N$, $t\geq0$, where $m>1$, dimension $N\geq2$ and $\sigma>0$. A new and explicit critical exponent $$ \sigma_c=\frac{2(m-1)(N-1)}{3m+1} $$ is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent $\sigma_c$ and also depends strongly on whether the dimension $N\geq4$ or $N\in\{2,3\}$. These results extend previous works of the authors in dimension $N=1$.

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.