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Ahn, Jaewook,Yoon, Changwook Institute of Physics and the London Mathematical S 2019 Nonlinearity Vol.32 No.4
<P>This paper deals with a Keller–Segel type parabolic–elliptic system involving nonlinear diffusion and chemotaxis <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn001.gif'/> in a smoothly bounded domain <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn002.gif'/>, <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn003.gif'/>, under no-flux boundary conditions. The system contains a Fokker–Planck type diffusion with a motility function <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn004.gif'/>, <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn005.gif'/>.</P> <P>The global existence of the unique bounded classical solutions is established without smallness of the initial data neither the convexity of the domain when <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn006.gif'/>, <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn007.gif'/> or <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn008.gif'/>, <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn009.gif'/>. In addition, we find the conditions on parameters, <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn010.gif'/> and <img ALIGN='MIDDLE' ALT='' SRC='http://ej.iop.org/images/0951-7715/32/4/1327/nonaaf513ieqn011.gif'/>, that make the spatially homogeneous equilibrium solution globally stable or linearly unstable.</P>
Blow-up solutions of the Chern–Simons–Schrödinger equations
Institute of Physics and the London Mathematical S 2009 Nonlinearity Vol.22 No.5
<P>We study finite time blow-up solutions of the Chern–Simons–Schrödinger system. In particular, explicit blow-up solutions are constructed by observing the pseudo-conformal invariance and finding solutions of the self-dual equations.</P>
On the behaviour of Navier–Stokes equations near a possible singular point
Kang, Kyungkeun,Lee, Jihoon Institute of Physics and the London Mathematical S 2010 Nonlinearity Vol.23 No.12
<P>We show that if a singularity of suitable weak solutions to Navier–Stokes equations occurs, then either <I>p</I> or at least two of ∂<SUB><I>i</I></SUB><I>v</I><SUB><I>i</I></SUB>, <I>i</I> = 1, 2, 3, have neither upper bounds nor lower bounds in any neighbourhood of the singularity. In the case of axially symmetric solutions, we prove that either <I>p</I> or ∂<SUB><I>r</I></SUB><I>v</I><SUP><I>r</I></SUP> is not bounded both below and above near a singular point, if it exists.</P>
The shrinking target property of irrational rotations
Institute of Physics and the London Mathematical S 2007 Nonlinearity Vol.20 No.7
<P>We investigate the recurrence property of irrational rotations. Let <I>T</I> be the rotation by an irrational &thetas; on the unit circle. We show that for a fixed <I>y</I> <BR/><BR/><img SRC='http://ej.iop.org/images/0951-7715/20/7/006/non236195ude001.gif' ALIGN='MIDDLE' ALT='\[ \begin{eqnarray*}\liminf_{n \to \infty} n \cdot d (y , T^n x) = 0,\tqs {\rm a.e.} \ x.\end{eqnarray*} \] '/><BR/><BR/> This result is a metric inhomogeneous Diophantine approximation in an almost everywhere sense.</P>
Emergent behaviour of a generalized Viscek-type flocking model
Ha, Seung-Yeal,Jeong, Eunhee,Kang, Moon-Jin Institute of Physics and the London Mathematical S 2010 Nonlinearity Vol.23 No.12
<P>We present a planar agent-based flocking model with a distance-dependent communication weight. We derive a sufficient condition for the asymptotic flocking in terms of the initial spatial and heading-angle diameters and a communication weight. For this, we employ differential inequalities for the spatial and phase diameters together with the Lyapunov functional approach. When the diameter of the agent's initial heading-angles is sufficiently small, we show that the diameter of the heading-angles converges to the average value of the initial heading-angles exponentially fast. As an application of flocking estimates, we also show that the Kuramoto model with a connected communication topology on the regular lattice <img SRC='http://ej.iop.org/images/0951-7715/23/12/008/non360035in001.gif' ALIGN='MIDDLE' ALT='\mathbb Z^d '/> for identical oscillators exhibits a complete-phase-frequency synchronization, when coupled oscillators are initially distributed on the half circle.</P>
Remarks on the magnetic helicity and energy conservation for ideal magneto-hydrodynamics
Kang, Ensil,Lee, Jihoon Institute of Physics and the London Mathematical S 2007 Nonlinearity Vol.20 No.11
<P>In Caflisch <I>et al</I> (1997 <I>Commun. Math. Phys.</I> <B>184</B> 443–55), it was shown that magnetic helicity for ideal magneto-hydrodynamics is conserved for the velocities and magnetic fields in some adequate Besov spaces. We show that magnetic helicity is conserved for the velocities and magnetic fields in <I>L</I><SUP>3</SUP>. We also obtain the necessary condition for the energy and cross-helicity conservation following the methods developed in Cheskidov <I>et al</I> (2007 <I>Preprint</I> <A HREF='http://arxiv.org/abs/0704.0759'>0704.0759</A> [math.AP]).</P>
On the Liouville theorem for weak Beltrami flows
Chae, Dongho,Wolf, Jö,rg Institute of Physics and the London Mathematical S 2016 Nonlinearity Vol.29 No.11
<P>We study Beltrami flows in the setting of weak solution to the stationary Euler equations in <img ALIGN='MIDDLE' ALT='${{\mathbb{R}}^{3}}$ ' SRC='http://ej.iop.org/images/0951-7715/29/11/3417/nonaa3c14ieqn001.gif'/>. For this weak Beltrami flow we prove the regularity and the Liouville property. In particular, we show that if the tangential part of the velocity has a certain decay property at infinity, then the solution becomes trivial. This decay condition of the velocity is weaker than the previously known sufficient conditions for the Liouville property of the Betrami flows. For the proof we establish a mean value formula and other various formulas for the tangential and the normal components of the weak solutions to the stationary Euler equations.</P>
A new class of traveling solitons for cubic fractional nonlinear Schrödinger equations
Hong, Younghun,Sire, Yannick Institute of Physics and the London Mathematical S 2017 Nonlinearity Vol.30 No.4
<P>We consider the one-dimensional cubic fractional nonlinear Schrödinger equation i∂tu−(−Δ)σu + |u|2u=0, where <img ALIGN='MIDDLE' ALT='$\sigma \in \left(\frac{1}{2},1\right)$ ' SRC='http://ej.iop.org/images/0951-7715/30/4/1262/nonaa5b12ieqn001.gif'/> and the operator <img ALIGN='MIDDLE' ALT='${{(- \Delta )}^{\sigma}}$ ' SRC='http://ej.iop.org/images/0951-7715/30/4/1262/nonaa5b12ieqn002.gif'/> is the fractional Laplacian of symbol <img ALIGN='MIDDLE' ALT='$|\xi {{|}^{2\sigma}}$ ' SRC='http://ej.iop.org/images/0951-7715/30/4/1262/nonaa5b12ieqn003.gif'/>. Despite the lack of any Galilean-type invariance, we construct a new class of traveling soliton solutions of the form u(t,x)=e−it(|k|2σ−ω2σ)Qω,k(x−2tσ|k|2σ−2k),k∈R,?ω>0 by a rather involved variational argument.</P>
Nonlinear scalar field equations involving the fractional Laplacian
Byeon, Jaeyoung,Kwon, Ohsang,Seok, Jinmyoung Institute of Physics and the London Mathematical S 2017 Nonlinearity Vol.30 No.4
<P>In this paper we study the existence, regularity, radial symmetry and decay property of a mountain pass solution for nonlinear scalar field equations involving the fractional Laplacian under an almost optimal class of continuous nonlinearities.</P>
A regularity condition and temporal asymptotics for chemotaxis-fluid equations
Chae, Myeongju,Kang, Kyungkeun,Lee, Jihoon,Lee, Ki-Ahm Institute of Physics and the London Mathematical S 2018 Nonlinearity Vol.31 No.2
<P>We consider two dimensional chemotaxis equations coupled to the Navier–Stokes equations. We present a new localized regularity criterion that is localized in a neighborhood at each point. Secondly, we establish temporal decays of the regular solutions under the assumption that the initial mass of biological cell density is sufficiently small. Both results are improvements of previously known results given in Chae <I>et al</I> (2013 <I>Discrete Continuous Dyn. Syst</I>. A <B>33</B> 2271–97) and Chae <I>et al</I> (2014 <I>Commun. PDE</I> <B>39</B> 1205–35)</P>