Finite Time Blow-up Research Articles

Investigating the ability of PINNs to solve Burgers’ PDE near finite-time blowup

Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated Partial Differential Equations (PDEs) numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive error bounds for PINNs for Burgers’ PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Our bounds give a theoretical justification for the functional regularization terms that have been reported to be useful for training PINNs near finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the ℓ2 -distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.

Stochastic quantisation of Yang–Mills–Higgs in 3D

We define a state space and a Markov process associated to the stochastic quantisation equation of Yang–Mills–Higgs (YMH) theories. The state space S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}$\\end{document} is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. Using gauge covariance of the deterministic YMH flow, we extend gauge equivalence ∼ to S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{S}$\\end{document} and thus define a quotient space of “gauge orbits” O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak {O}$\\end{document}. We use the theory of regularity structures to prove local in time solutions to the renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in the small noise limit, we show that there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law. This allows us to define a canonical Markov process on O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak {O}$\\end{document} (up to a potential finite time blow-up) associated to the stochastic YMH flow.

Boundedness and Finite-Time Blow-up in a Chemotaxis System with Flux Limitation and Logistic Source

Boundedness and Finite-Time Blow-up in a Chemotaxis System with Flux Limitation and Logistic Source

Infinite time blow-up for the three dimensional energy critical heat equation in bounded domains

AbstractWe consider the Dirichlet problem for the energy-critical heat equation $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u+u^5 &{} \text {in} \quad \Omega \times \mathbb {R}^+,\\ u=0 &{}\text {on} \quad \partial \Omega \times \mathbb {R}^+,\\ u(x,0)=u_0(x) &{} \text {in} \quad \Omega , \end{array}\right. } \end{aligned}$$ u t = Δ u + u 5 in Ω × R + , u = 0 on ∂ Ω × R + , u ( x , 0 ) = u 0 ( x ) in Ω , where $$\Omega $$ Ω is a bounded smooth domain in $$\mathbb {R}^3$$ R 3 . Let $$H_\gamma (x,y)$$ H γ ( x , y ) be the regular part of the Green function of $$-\Delta -\gamma $$ - Δ - γ in $$\Omega $$ Ω , where $$\gamma \in (0,\lambda _1)$$ γ ∈ ( 0 , λ 1 ) and $$\lambda _1$$ λ 1 is the first Dirichlet eigenvalue of $$-\Delta $$ - Δ . Then, given a point $$q\in \Omega $$ q ∈ Ω such that $$3\gamma (q)<\lambda _1$$ 3 γ ( q ) < λ 1 , where $$\begin{aligned} \gamma (q) :=\sup \{ \gamma>0: H_\gamma (q,q)>0 \}, \end{aligned}$$ γ ( q ) : = sup { γ > 0 : H γ ( q , q ) > 0 } , we prove the existence of a non-radial global positive and smooth solution u(x, t) which blows up in infinite time with spike in q. The solution has the asymptotic profile $$\begin{aligned} u(x,t)\sim 3^{\frac{1}{4}} \left( \frac{\mu (t)}{\mu (t)^2+\vert x-\xi (t)\vert ^2}\right) ^{\frac{1}{2}} \quad \text {as}\quad t \rightarrow \infty , \end{aligned}$$ u ( x , t ) ∼ 3 1 4 μ ( t ) μ ( t ) 2 + | x - ξ ( t ) | 2 1 2 as t → ∞ , where $$\begin{aligned} -\ln (\mu (t))= 2\gamma (q) t(1+o(1)),\quad \xi (t)=q+O(\mu (t)) \quad \text {as}\quad t \rightarrow \infty . \end{aligned}$$ - ln ( μ ( t ) ) = 2 γ ( q ) t ( 1 + o ( 1 ) ) , ξ ( t ) = q + O ( μ ( t ) ) as t → ∞ .

Turbulent Flows Are Not Uniformly Multifractal.

Understanding turbulence rests delicately on the conflict between Kolmogorov's 1941 theory of nonintermittent, space-filling energy dissipation characterized by a unique scaling exponent and the overwhelming evidence to the contrary of intermittency, multiscaling, and multifractality. Strangely, multifractality is not typically envisioned as a local flow property, variations in which might be clues exposing inroads into the fundamental unsolved issues of anomalous dissipation and finite time blowup. We present a simple construction of local multifractality and find that much of the dissipation field remains surprisingly monofractal àla Kolmogorov. Multifractality appears as small islands in this calm sea, its strength growing logarithmically with the local fluctuations in energy dissipation-a seemingly universal feature. These results suggest new ways to understand how singularities could arise and provide a fresh perspective on anomalous dissipation and intermittency. The simplicity and adaptability of our approach also holds great promise in applications ranging from climate sciences to medical data analysis.

On a proper tensorial subgrid heat flux model for LES

In this work, we aim to shed light on the following research question: can we find a subgrid-scale (SGS) heat flux model with good physical and numerical properties, such that we can obtain satisfactory predictions for buoyancy-driven turbulence at high Rayleigh numbers? This is motivated by our previous findings showing the limitations of existing SGS heat flux models for LES. On one hand, the most popular models rely on the eddy-diffusivity assumption despite their well-known lack of accuracy in a priori studies. On the other hand, the gradient model, which is the leading term of the Taylor series of the SGS flux, is much more accurate a priori but cannot be used as a standalone model since it produces a finite-time blow-up. In this context, we firstly aim to reconcile accuracy and stability for the gradient model. To do so, it is expressed as a linear combination of regularized (smoother) forms of the convective operator. The new alternative form can indeed be viewed as an approximate deconvolution of the exact SGS flux. Moreover, it facilitates the mathematical analysis of the gradient model, neatly identifying those terms that may cause numerical instabilities, leading to a new unconditionally stable non-linear model that can indeed be viewed as a stabilized version of the gradient model. In this way, we expect to combine the good a priori accuracy of the gradient model with the stability required in practical numerical simulations.

An Adaptive Moving Mesh Method for Simulating Finite-time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation

An Adaptive Moving Mesh Method for Simulating Finite-time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation

The Blow-Up of the Local Energy Solution to the Wave Equation with a Nontrivial Boundary Condition

In this study, we examine the wave equation with a nontrivial boundary condition. The main target of this study is to prove the local-in-time existence and the blow-up in finite time of the energy solution. Through the construction of an auxiliary function and the imposition of appropriate conditions on the initial data, we establish the both lower and upper bounds for the blow-up time of the solution. Meanwhile, based on these estimates, we obtain the result of the local-in-time existence and the blow-up of the energy solution. This approach enhances our understanding of the dynamics leading to blow-up in the considered condition.

A special form of solution to complex valued Benjamin–Ono equations

We investigate a special form of solution to the complex valued Benjamin–Ono (cBO) equation. Under the special form of solution involving Hilbert transform, the complex valued (modified) Benjamin–Ono equations are equivalent to the quadratic (cubic) derivative Schrödinger equations. The soliton interaction dynamics is used to construct a finite time blowup solution of the cBO equation on R. We also find static solutions. Applying Cole–Hopf transformation to quadratic derivative Schrödinger equation, we derive a linear Schrödinger equation with a special form of initial data from which some solutions to cBO equation can be obtained by an inverse transform.

Scattering threshold for the focusing energy-critical generalized Hartree equation

Abstract This work investigates the asymptotic behavior of energy solutions to the focusing nonlinear Schrödinger equation of Choquard type i ∂ t u + Δ u + ∣ u ∣ p − 2 ( I α * ∣ u ∣ p ) u = 0 , p = 1 + 2 + α N − 2 , N ≥ 3 . i{\partial }_{t}u+\Delta u+{| u| }^{p-2}\left({I}_{\alpha }* {| u| }^{p})u=0,\hspace{1.0em}p=1+\frac{2+\alpha }{N-2},\hspace{1.0em}N\ge 3. Indeed, in the energy-critical spherically symmetric regime, one proves a global existence and scattering versus finite time blow-up dichotomy. Precisely, if the data have an energy less than the ground state one, two cases are possible. If the kinetic energy of the radial data is less than the ground state one, then the solution is global and scatters. Otherwise, if the data have a finite variance or is spherically symmetric and have a finite mass, then the solution is nonglobal. The main difficulty is to deal with the nonlocal source term. The argument is the concentration-compactness-rigidity method introduced by Kenig and Merle (Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675). This note naturally complements the work by Saanouni (Scattering theory for a class of defocusing energy-critical Choquard equations, J. Evol. Equ. 21 (2021), 1551–1571), where the scattering of the defocusing energy-critical generalized Hartree equation was obtained.

Finite-time blowup for the 3-D viscous primitive equations of oceanic and atmospheric dynamics

In this paper, we prove that for certain class of initial data, the corresponding solutions to the 3-D viscous primitive equations blow up in finite time. Specifically, we find a special solution to simplify the 3-D systems, assuming that the pressure function p(x,y,t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p(x,y,t)$\\end{document} is a concave function. We also consider the equations on the line x=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$x=0$\\end{document}, y=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$y=0$\\end{document}.

Blow-up of solutions to the Keller–Segel model with tensorial flux in high dimensions

Over the course of the last decade, there has been a significant level of interest in the analysis of Keller–Segel models incorporating tensorial flux. Despite this interest, the question of whether finite-time blowup solutions exist remains a topic of ongoing research. Our study provides evidence that solutions of this nature are indeed possible in dimensions n≥3, when utilizing a tensorial flux expressed in the form of A∇v, where A denotes a matrix with constant components.

L2 Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity

This article studies the Schrödinger equation with an inhomogeneous combined term i∂tu−(−Δ)su+λ1|x|−b|u|pu+λ2|u|qu=0, where s∈(12,1),λ1,λ2=±1,0<b<{2s,N} and p,q>0. We study the limit behaviour of the infinite blow-up solution at the blow-up time. When the parameters p,q,λ1 and λ2 have different values, we obtain the nonexistence of a strong limit for the non-radial solution and the L2 concentration for the radial solution. Interestingly, we find that the mass of the finite time blow-up solutions are concentrated in different ways for different parameters.

Blow-up for a Porous Medium Equation with Local Linear Boundary Dissipation

This article investigates the blow-up behaviors for a porous medium equation with a superlinear source and local linear boundary dissipation. Making use of the concavity method, we establish sufficient conditions to guarantee the occurrence of the finite time blow-up phenomenon. Meanwhile, we show the existence of the finite time blow-up solutions for arbitrarily high initial energy. Finally, we derive the life span bounds (i.e., the lower and upper bounds of the blow-up time).

On Fila-King conjecture in dimension four

We consider the following Cauchy problem for the four-dimensional energy critical heat equation{ut=Δu+u3 in R4×(0,∞),u(x,0)=u0(x) in R4. We construct a positive infinite time blow-up solution u(x,t) with the blow-up rate ‖u(⋅,t)‖L∞(R4)∼ln⁡t as t→∞ and establish the stability of the infinite time blow-up. This gives a rigorous proof of a conjecture by Fila and King [15, Conjecture 1.1].

Comparison of different discontinuous Galerkin methods based on various reformulations for gKdV equation: Soliton dynamics and blowup

In this work, we explore the use of several discontinuous Galerkin (DG) methods for simulating soliton dynamics in the generalized Korteweg-de Vries (gKdV) equation. The presence of high-order nonlinearity in this model can lead to the finite-time blowup phenomenon, which challenges every aspect of a numerical scheme. The considered DG schemes encompass popular methods derived from the energy and/or Hamiltonian conservations of the gKdV equation. To evaluate the performance of these DG schemes from various angles, we present a series of numerical experiments. Through comprehensive comparisons, we aim to identify the scheme that exhibits the best performance. To further enhance the accuracy and efficiency of DG methods, particularly in the context of blowup simulations, we incorporate the arbitrary Lagrangian-Eulerian method for adaptive mesh movement.

A nonlinear system to model communication between yeast cells during their mating process

In this work, we develop a model to describe some aspects of communication between yeast cells. It consists in a coupled system of two one-dimensional non-linear advection-diffusion equations in which the advective field is given by the Hilbert transform. We give some sufficient condition for the blow-up in finite time of the coupled system (formation of a singularity). We provide a biological interpretation of these mathematical results.

Lower and upper bounds for the explosion times of a system of semilinear SPDEs

In this paper, we obtain lower and upper bounds for the blow-up times for a system of semilinear stochastic partial differential equations. Under suitable assumptions, lower and upper bounds of the explosive times are obtained by using explicit solutions of an associated system of random partial differential equations and a formula due to Yor. We provide lower and upper bounds for the probability of finite-time blow-up solution as well. The above obtained results are also extended for semilinear SPDEs forced by two-dimensional Brownian motions.

Finite speed axially symmetric Navier-Stokes flows passing a cone

Let D be the exterior of a cone inside a ball, with its altitude angle at most π/6 in R3, which touches the x3 axis at the origin. For any initial value v0=v0,rer+v0,θeθ+v0,3e3 in a C2(D‾) class, which has the usual even-odd-odd symmetry in the x3 variable and has the partial smallness only in the swirl direction: |rv0,θ|≤1100, the axially symmetric Navier-Stokes equations (ASNS) with Navier-Hodge-Lions slip boundary condition have a finite-energy solution that stays bounded for all time. In particular, no finite-time blowup of the fluid velocity occurs. Compared with standard smallness assumptions on the initial velocity, no size restriction is made on the components v0,r and v0,3. In a broad sense, this result appears to solve 2/3 of the regularity problem of ASNS in such domains in the class of solutions with the above symmetry. Equivalently, this result is connected to the general open question which asks that if an absolute smallness of one component of the initial velocity implies the global smoothness, see e.g. page 873 in Chemin et al. (2017) [6]. Our result seems to give a positive answer in a special setting.As a byproduct, we also construct an unbounded solution of the forced Navier Stokes equation in a special cusp domain that has finite energy. The forcing term, with the scaling factor of −1, is in the standard regularity class, and it can be generated by an electric current in a long and straight wire (i.e. Ampère force). This result confirms the intuition that if the channel of a fluid is very thin, arbitrarily high speed in the classical sense can be attained under a mildly singular, physically reasonable force.

Finite-time blow-up in hyperbolic Keller–Segel system of consumption type with logarithmic sensitivity

This paper deals with finite-time blow-up of a hyperbolic Keller–Segel system of consumption type with the logarithmic sensitivity 0\\right)$?> ∂tρ=−χ∇⋅ρ∇logc,∂tc=−μcρχ,μ>0 in Rd(d⩾1) for nonvanishing initial data. This system is closely related to tumor angiogenesis, an important example of chemotaxis. Our singularity formation is not because c touches zero (which makes logc diverge) but due to the blowup of C1×C2 -norm of (ρ,c) . As a corollary, we also construct initial data near any constant equilibrium state which blows up in finite time for any d⩾1 .