Blow-up Scenario Research Articles

GLOBAL EXISTENCE AND BLOW-UP PHENOMENA FOR THE PERIODIC HUNTER–SAXTON EQUATION WITH WEAK DISSIPATION

In this paper, we study the periodic Hunter–Saxton equation with weak dissipation. We first establish the local existence of strong solutions, blow-up scenario and blow-up criteria of the equation. Then, we investigate the blow-up rate for the blowing-up solutions to the equation. Finally, we prove that the equation has global solutions.

Blow-up solutions of the general b-equation

In this short paper, we consider the b-equation. A new precise blow-up scenario is described and some new results are obtained for the blow-up phenomena.

On the Cauchy problem for the periodic generalized Degasperis–Procesi equation

We mainly study the Cauchy problem of the periodic generalized Degasperis–Procesi equation. First, we establish the local well-posedness for the equation. Second, we give the precise blow-up scenario, a conservation law and prove that the equation has smooth solutions which blow up in finite time. Finally, we investigate the blow-up rate for the blow-up solutions.

Blowup phenomena for a new periodic nonlinearly dispersive wave equation

AbstractWe first establish the local well‐posedness for a new periodic nonlinearly dispersive wave equation. We then present a precise blowup scenario and several blowup results of strong solutions to the equation. The obtained blowup results on the equation improve considerably recent results on the Camassa‐Holm equation and the rod equation (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Global solutions and blow-up phenomena for the periodic b -equation

In the paper, we mainly study the Cauchy problem of a family of asymptotically equivalent shallow water wave equations, the so called periodic b-equation. We first establish the local well-posedness for the periodic b-equation. Then we derive the precise blow-up scenario and present two blow-up results. Moreover, we show that the periodic b-equation has global strong solutions. Finally, we prove the existence of global weak solutions to the periodic b-equation, provided that initial data satisfy certain sign conditions.

On the Cauchy problem for a generalized Degasperis–Procesi equation

We mainly study the Cauchy problem of the generalized Degasperis–Procesi equation. First, we establish the local well posedness for the equation. Second, we give the precise blow-up scenario, a conservation law, and prove the equation has smooth solutions which blow up in finite time. Finally, we investigate the blow-up rate for the blow-up solutions.

Well-posedness and blow-up phenomena for the generalized Degasperis–Procesi equation

In this paper, we mainly study the Cauchy problem of the generalized Degasperis–Procesi equation. We establish the local well-posedness and give the precise blow-up scenario for the equation. Then we show that the equation has smooth solutions which blow up in finite time.

Well-posedness and blow-up solution for a modified two-component periodic Camassa–Holm system with peakons

Considered herein is a modified two-component periodic Camassa-Holm system with peakons. The local well-posedness and low regularity result of solutions are established. The precise blow-up scenarios of strong solutions and several results of blow-up solutions with certain initial profiles are described in detail and the exact blow-up rate is also obtained.

Blow-up phenomena for a family of Burgers-like equations

By introducing a stress multiplier we derive a family of Burgers-like equations. We investigate the blow-up phenomena of the equations both on the real line R and on the circle S to get a comparison with the Degasperis–Procesi equation. On the line R , we first establish the local well-posedness and the blow-up scenario. Then we use conservation laws of the equations to get the estimate for the L ∞ -norm of the strong solutions, by which we prove that the solutions to the equations may blow up in the form of wave breaking for certain initial profiles. Analogous results are provided in the periodic case. Especially, we find differences between the Burgers-like equations and the Degasperis–Procesi equation, see Remark 4.1.

Blowup and blowup rate of solutions to a weakly dissipative periodic rod equation

In this paper, we first establish the local well posedness for a weakly dissipative periodic rod equation. We then present a precise blowup scenario and several blowup results of strong solutions to the equation. We finally give the blowup rate of strong solutions to the equation when blowup occurs. Our blowup results for the weakly dissipative periodic rod equation improve considerably earlier results for the periodic rod equation.

Some properties of solutions to the weakly dissipative Degasperis–Procesi equation

In this paper, we consider the weakly dissipative Degasperis–Procesi equation. The present paper is concerned with some aspects of existence of global solutions, persistence properties and propagation speed. First we try to discuss the local well-posedness and blow-up scenario, then establish the sufficient conditions on global existence of the solution. Finally, persistence properties on strong solutions and the propagation speed for the weakly dissipative Degasperis–Procesi equation are also investigated.

Initial boundary value problems for nonlinear dispersive wave equations

In this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half-line and on a finite interval subject to homogeneous Dirichlet boundary conditions. We first prove local well-posedness of the rod equation and of the b-equation for general initial data. Furthermore, we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span. In the case of finite time singularities we are able to describe the precise blow-up scenario of breaking waves. Our approach is based on sharp extension results for functions on the half-line or on a finite interval and several symmetry preserving properties of the equations under discussion.

Blow-Up Phenomena and Decay for the Periodic Degasperis-Procesi Equation with Weak Dissipation

In the paper, several problems on the periodic Degasperis-Procesi equation with weak dissipation are investigated. At first, the local well-posedness of the equation is established by Kato’s theorem and a precise blow-up scenario of the solutions is given. Then, several criteria guaranteeing the blow-up of the solutions are presented. Moreover, the blow-up rate and blow-up set of the blowing-up solutions are discussed. Furthermore, it is proved that the equation has global solutions and these global solutions decay to zero as time goes to infinite provided the potentials associated to their initial date are of one sign.

Blow-Up and Decay of the Solution of the Weakly Dissipative Degasperis–Procesi Equation

In this paper, we mainly study several problems on the weakly dissipative Degasperis–Procesi equation. We first establish the local well-posedness of the equation, derive a precise blow-up scenario, and present two blow-up criteria for strong solutions to the equation. We then give the precise blow-up rate of blow-up solutions to the equation. We finally prove that the equation has global strong solutions and these global solutions decay to zero as time goes to infinity provided the potentials associated with their initial data are of one sign.

Wave breaking and shock waves for a periodic shallow water equation

This paper is devoted to the study of a recently derived periodic shallow water equation. We discuss in detail the blow-up scenario of strong solutions and present several conditions on the initial profile, which ensure the occurrence of wave breaking. We also present a family of global weak solutions, which may be viewed as global periodic shock waves to the equation under discussion.

Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation

After some remarks on a possible zero-curvature formulation we first establish local well-posedness for the 2-component Camassa-Holm equation. Then precise blow-up scenarios for strong solutions to the system are derived. Finally we present two blow-up results for strong solutions to the system.

Improved Geometric Conditions for Non-Blowup of the 3D Incompressible Euler Equation

This is a follow-up of our recent article Deng et al. (2004). In Deng et al. (2004), we derive some local geometric conditions on vortex filaments which can prevent finite time blowup of the 3D incompressible Euler equation. In this article, we derive improved geometric conditions which can be applied to the scenario when velocity blows up at the same time as vorticity and the rate of blowup of velocity is proportional to the square root of vorticity. This scenario is in some sense the worst possible blow-up scenario for velocity field due to Kelvin's circulation theorem. The improved conditions can be checked by numerical computations. This provides a sharper local geometric constraint on the finite time blowup of the 3D incompressible Euler equation.

On the blow-up for a discrete Boltzmann equation in the plane

We study the possibility of finite-time blow-up for a two dimensional Broadwell model. In a set of rescaled variables, we prove that no self-similar blow-up solution exists, and derive some a priori bounds on the blow-up rate. In the final section, a possible blow-up scenario is discussed.

On the Blow-Up Scenario for the Generalized Camassa–Holm Equation

We present the precise blow-up scenario for the generalized Camassa–Holm equation. We also prove a blow-up result showing that the equation has smooth solutions in which singularities develop in finite time.

On the Cauchy problem for an integrable equation with peakon solutions

We establish the local well-posedness for a new integrable equation. We prove that the equation has strong solutions that blow up in finite time and obtain the precise blow-up scenario for this equation. Moreover, we provide a framework of weak solutions for the study of soliton interaction.