scholarly journals Some Blow-Up and Smooth Solutions about Landau-Lifshitz Equation and Their Spatial Curvature Behavior

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Penghong Zhong ◽  
Shu Wang ◽  
Ming Zeng
Keyword(s):  
Exact Solution ◽  
Finite Time ◽  
Smooth Solution ◽  
Blow Up ◽  
Dimensional Space ◽  
Smooth Solutions ◽  
Spatial Curvature ◽  
Lifshitz Equation ◽  
Vortex Solution ◽  
Dimensional Space Time

We construct the exact solution of (2+1)-dimensional space-time Landau-Lifshitz equation (LLE) without the Gilbert term. Under suitable transformations, some exact solutions are obtained in the radially symmetric coordinates and nonsymmetric coordinates. The type of solutions cover the finite-time blow-up solution, smooth solution in time and vortex solution. At the end, some properties about these solutions and their spatial curvature are illustrated by the graphs.

SOME PERIODIC AND BLOW-UP SOLUTIONS FOR LANDAU–LIFSHITZ EQUATION

2011 ◽  
Vol 26 (32) ◽  
pp. 2437-2452 ◽  
Author(s):  
PENGHONG ZHONG ◽  
SHU WANG ◽  
SHENGTAO CHEN
Keyword(s):  
Exact Solution ◽  
Periodic Solution ◽  
Exact Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Three Dimensional ◽  
Ferromagnetic Materials ◽  
Lifshitz Equation ◽  
Dimensional Spacetime ◽  
Vortex Solution

In this paper, we construct the exact solution of two- or three-dimensional spacetime Landau–Lifshitz equation raised in the ferromagnetic materials. Under suitable transformations, some exact solutions are obtained in the radially symmetric coordinates and nonsymmetric coordinates. The type of solutions cover the finite time blow-up solution, vortex solution and periodic solution. In the end, we sketch some solutions and their spatial curvatures.

De la combinatoire du modèle de l'échiquier de Feynman et des fonctions spéciales

1984 ◽  
Vol 62 (7) ◽  
pp. 632-638
Author(s):  
J. G. Williams
Keyword(s):  
Exact Solution ◽  
Dirac Equation ◽  
Hypergeometric Series ◽  
Jacobi Polynomials ◽  
Dimensional Space ◽  
Space Time ◽  
Continuous Limit ◽  
Two Dimensional ◽  
Dimensional Space Time

The exact solution of the Feynman checkerboard model is given both in terms of the hypergeometric series and in terms of Jacobi polynomials. It is shown how this leads, in the continuous limit, to the Dirac equation in two-dimensional space-time.

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

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.

scholarly journals Blow-Up Criteria of Smooth Solutions for the Cahn-Hilliard-Boussinesq System with Zero Viscosity in a Bounded Domain

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Yong Zhou ◽  
Jishan Fan
Keyword(s):  
Bounded Domain ◽  
Smooth Solution ◽  
Blow Up ◽  
Boussinesq System ◽  
Smooth Solutions ◽  
Zero Viscosity

We prove that a smooth solution of the 3D Cahn-Hilliard-Boussinesq system with zero viscosity in a bounded domain breaks down if a certain norm of vorticity blows up at the same time. Here, this norm is weaker than bmo-norm.

Non-existence of global smooth solutions to symmetrizable nonlinear hyperbolic systems

2003 ◽  
Vol 133 (3) ◽  
pp. 719-728 ◽  
Author(s):  
Tong Yang ◽  
Changjiang Zhu
Keyword(s):  
Blow Up ◽  
Dimensional Space ◽  
Hyperbolic Systems ◽  
Shock Formation ◽  
Sufficient Condition ◽  
Smooth Solutions ◽  
Physical Systems ◽  
Global Smooth Solutions ◽  
The Cauchy Problem ◽  
Nonlinear Hyperbolic Systems

In this paper, we consider the Cauchy problem of general symmetrizable hyperbolic systems in multi-dimensional space. When some components of the initial data have compact support, we give a sufficient condition on the non-existence of global C1 solutions. This non-existence theorem can be applied to some physical systems, such as Euler equations for compressible flow in multi-dimensional space. The blow-up phenomena here can come from the singularity developed at the interface, such as vacuum boundary, rather than the shock formation as studied in the previous works on strictly hyperbolic systems. Therefore, the systems considered here include those which are non-strictly hyperbolic.

scholarly journals Possibility of amoebas' aggregation in finite time

2013 ◽  
Vol 21 (1) ◽  
pp. 101-120
Author(s):  
Saïd Hilmi ◽  
Chérif Ziti
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear System ◽  
Differential Equations ◽  
Finite Time ◽  
Blow Up ◽  
Dimensional Space ◽  
Partial Differential ◽  
Self Similar ◽  
Similar Solutions

Abstract The dynamic of amoebas in favorable circumstances is modeled by a nonlinear system of Partial Differential Equations arising in chemotaxis. The competition between different parameters of this system plays a major role in the process of aggregation. Throughout this paper, we prove the existence of self-similar solutions that blow up in finite time in a dimensional space and under specific circumstances depending upon the position of those parameters.

Topology knots and hierarchy problem

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Theory ◽  
String Theory ◽  
Quantum Gravity ◽  
Dimensional Space ◽  
Space Time ◽  
Elementary Particles ◽  
Hierarchy Problem ◽  
Topological Quantum ◽  
Dimensional Space Time ◽  
New Formula

Many approaches to quantum gravity consider the revision of the space-time geometry and the structure of elementary particles. One of the main candidates is string theory. It is possible that this theory will be able to describe the problem of hierarchy, provided that there is an appropriate Calabi-Yau geometry. In this paper we will proceed from the traditional view on the structure of elementary particles in the usual four-dimensional space-time. The only condition is that quarks and leptons should have a common emerging structure. When a new formula for the mass of the hierarchy is obtained, this structure arises from topological quantum theory and a suitable choice of dimensional units.

scholarly journals Sharp thresholds of blow-up and global existence for the Schrödinger equation with combined power-type and Choquard-type nonlinearities

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yongbin Wang ◽  
Binhua Feng
Keyword(s):  
Global Existence ◽  
Finite Time ◽  
Critical Case ◽  
Blow Up ◽  
Critical Mass ◽  
Choquard Equation ◽  
Power Type ◽  
Scale Invariant ◽  
Nonlinear Schrödinger ◽  
Sharp Thresholds

AbstractIn this paper, we consider the sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equation $$ i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p_{1}}\psi +\lambda _{2}\bigl(I _{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) \vert \psi \vert ^{p_{2}-2}\psi . $$iψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ. We derive some finite time blow-up results. Due to the failure of this equation to be scale invariant, we obtain some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the $L^{2}$L2-critical case. Our obtained results extend and improve some recent results.

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