Development of singularities in solutions of a hyperbolic system

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

Download Full-text

Global classical solutions to the elastodynamic equations with damping

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02608-9 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Mengmeng Liu ◽

Xueyun Lin

Keyword(s):

Initial Data ◽

Periodic Boundary ◽

Energy Estimate ◽

Periodic Boundary Conditions ◽

Deformation Tensor ◽

Classical Solutions ◽

Small Initial Data ◽

Weighted Energy ◽

Global Classical Solutions ◽

Weighted Energy Estimate

AbstractIn this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.

Download Full-text

EXPLOSIVE INSTABILITY DUE TO 3-WAVE OR 4-WAVE MIXING, WITH OR WITHOUT DISSIPATION

Analysis and Applications ◽

10.1142/s0219530508001183 ◽

2008 ◽

Vol 06 (04) ◽

pp. 413-428 ◽

Cited By ~ 3

Author(s):

HARVEY SEGUR

Keyword(s):

Initial Data ◽

Nonlinear Waves ◽

Finite Time ◽

Blow Up ◽

Wave Mixing ◽

Explosive Instability ◽

Effective Threshold ◽

Blowing Up ◽

Gain Energy

It is known that an "explosive instability" can occur when nonlinear waves propagate in certain media that admit 3-wave mixing. In that context, three resonantly interacting wavetrains all gain energy from a background source, and all blow up together, in finite time. A recent paper [17] showed that explosive instabilities can occur even in media that admit no 3-wave mixing. Instead, the instability is caused by 4-wave mixing, and results in four resonantly interacting wavetrains all blowing up in finite time. In both cases, the instability occurs in systems with no dissipation. This paper reviews the earlier work, and shows that adding a common form of dissipation to the system, with either 3-wave or 4-wave mixing, provides an effective threshold for blow-up. Only initial data that exceed the respective thresholds blow up in finite time.

Download Full-text

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

Discrete Dynamics in Nature and Society ◽

10.1155/2020/2757398 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8

Author(s):

Linrui Li ◽

Shu Wang

Keyword(s):

Initial Data ◽

Finite Time ◽

Smooth Solution ◽

Blow Up ◽

Nonlocal Term ◽

Convection Term ◽

Large Initial Data ◽

Singularity Formation ◽

Time Singularity ◽

Finite Time Singularity

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.

Download Full-text

To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202516400091 ◽

2016 ◽

Vol 26 (11) ◽

pp. 2111-2128 ◽

Cited By ~ 29

Author(s):

Bingran Hu ◽

Y. Tao

Keyword(s):

Initial Data ◽

Blow Up ◽

Three Dimensional ◽

Classical Solutions ◽

Global Classical Solution ◽

Infinite Time ◽

Two Component ◽

Flux Boundary Conditions ◽

Growth System ◽

Striking Contrast

This work considers the chemotaxis-growth system [Formula: see text] in a smoothly bounded domain [Formula: see text], with zero-flux boundary conditions, where [Formula: see text] and [Formula: see text] are given positive parameters. In striking contrast to the corresponding three-dimensional two-component chemo-taxis-growth system to which the global existence or blow-up of classical solutions largely remains open when [Formula: see text] is small, it is shown that whenever [Formula: see text] [Formula: see text] and [Formula: see text], for any given non-negative and suitably smooth initial data [Formula: see text] satisfying [Formula: see text], the system (⋆) admits a unique global classical solution that is uniformly-in-time bounded, which rules out the possibility of blow-up of solutions in finite time or in infinite time. Moreover, under the fully explicit condition [Formula: see text] the solution [Formula: see text] exponentially converges to the constant stationary solution [Formula: see text] in the norm of [Formula: see text] as [Formula: see text].

Download Full-text

Finite time blow-up of complex solutions of the conserved Kuramoto–Sivashinsky equation in ℝd and in the torus 𝕋d, d ⩾ 1

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.54 ◽

2019 ◽

Vol 149 (5) ◽

pp. 1175-1188

Author(s):

Léo Agélas

Keyword(s):

Sobolev Space ◽

Initial Data ◽

Besov Space ◽

Finite Time ◽

Blow Up ◽

Global Solutions ◽

Large Initial Data ◽

Complex Solutions ◽

Complex Valued ◽

Sivashinsky Equation

AbstractWe consider complex-valued solutions of the conserved Kuramoto–Sivashinsky equation which describes the coarsening of an unstable solid surface that conserves mass and that is parity symmetric. This equation arises in different aspects of surface growth. Up to now, the problem of existence and smoothness of global solutions of such equations remained open in ℝd and in the torus 𝕋d, d ⩾ 1. In this paper, we answer partially to this question. We prove the finite time blow-up of complex-valued solutions associated with a class of large initial data. More precisely, we show that there is complex-valued initial data that exists in every Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the complex-valued solution is in no Besov space (and hence in no Lebesgue or Sobolev space).

Download Full-text

Stabilities for Nonisentropic Euler-Poisson Equations

The Scientific World JOURNAL ◽

10.1155/2015/494707 ◽

2015 ◽

Vol 2015 ◽

pp. 1-6

Author(s):

Ka Luen Cheung ◽

Sen Wong

Keyword(s):

Energy Method ◽

Finite Time ◽

Blow Up ◽

Classical Solutions ◽

Poisson Equations ◽

Repulsive Forces

We establish the stabilities and blowup results for the nonisentropic Euler-Poisson equations by the energy method. By analysing the second inertia, we show that the classical solutions of the system with attractive forces blow up in finite time in some special dimensions when the energy is negative. Moreover, we obtain the stabilities results for the system in the cases of attractive and repulsive forces.

Download Full-text

A New Blow-Up Criterion for the DGH Equation

Abstract and Applied Analysis ◽

10.1155/2012/515948 ◽

2012 ◽

Vol 2012 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Mingxuan Zhu ◽

Liangbing Jin ◽

Zaihong Jiang

Keyword(s):

Initial Data ◽

Finite Time ◽

Blow Up ◽

Holm Equation ◽

Blow Up Criterion

We investigate the DGH equation. Analogous to the Camassa-Holm equation, this equation possesses the blow-up phenomenon. We establish a new blow up criterion on the initial data to guarantee the formulation of singularities in finite time.

Download Full-text

On a hyperbolic system arising in liquid crystals modeling

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891618500029 ◽

2018 ◽

Vol 15 (01) ◽

pp. 15-35 ◽

Cited By ~ 8

Author(s):

Eduard Feireisl ◽

Elisabetta Rocca ◽

Giulio Schimperna ◽

Arghir Zarnescu

Keyword(s):

Liquid Crystals ◽

Initial Data ◽

Existence Theorem ◽

Hyperbolic System ◽

Finite Energy ◽

Classical Solutions ◽

System Of Differential Equations ◽

Dissipative Solutions ◽

Nonlinear Hyperbolic System ◽

Time Existence

We consider a model of liquid crystals, based on a nonlinear hyperbolic system of differential equations, that represents an inviscid version of the model proposed by Qian and Sheng. A new concept of dissipative solution is proposed, for which a global-in-time existence theorem is shown. The dissipative solutions enjoy the following properties: (i) they exist globally in time for any finite energy initial data; (ii) dissipative solutions enjoying certain smoothness are classical solutions; (iii) a dissipative solution coincides with a strong solution originating from the same initial data as long as the latter exists.

Download Full-text

Blow-Up Criteria for the Modified Novikov Equation

Abstract and Applied Analysis ◽

10.1155/2014/243616 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5

Author(s):

Caochuan Ma ◽

Wujun Lv

Keyword(s):

Cauchy Problem ◽

Initial Data ◽

Finite Time ◽

Blow Up ◽

Solution Blowing ◽

Blowing Up ◽

The Cauchy Problem ◽

Novikov Equation

We investigate the Cauchy problem for the modified Novikov equation. We establish blow-up criteria on the initial data to guarantee the corresponding solution blowing up in finite time.

Download Full-text