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19
On the Schrödinger flows
 Proceedings of the ICM, Beijing Vol.II
, 2002
"... We present some recent results on the existence of solutions of the Schrödinger flows, and pose some problems for further research. ..."
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We present some recent results on the existence of solutions of the Schrödinger flows, and pose some problems for further research.
Schrödinger Flow Near Harmonic Maps
, 2005
"... For the Schrödinger flow from R 2 × R + to the 2sphere S 2, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic map ..."
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For the Schrödinger flow from R 2 × R + to the 2sphere S 2, it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blow up time (if any), and that they blow up if and only if the length scale of the nearest harmonic map goes to zero. Contents 1 Introduction and main result 1 2 Maps with energy near the harmonic map energy 5
On the global wellposedness of the onedimensional Schrödinger map flow
 Analysis PDE
"... Abstract. We establish the global wellposedness of the initial value problem for the Schrödinger map flow for maps from the real line into Kähler manifolds and for maps from the circle into Riemann surfaces. This partially resolves a conjecture of W.Y. Ding. 1. ..."
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Abstract. We establish the global wellposedness of the initial value problem for the Schrödinger map flow for maps from the real line into Kähler manifolds and for maps from the circle into Riemann surfaces. This partially resolves a conjecture of W.Y. Ding. 1.
Global Existence of Weak Solutions for LandauLifshitzMaxwell Equations
, 2007
"... Abstract. In this paper we study the model that the usual Maxwell’s equations are supplemented with a constitution relation in which the electric displacement equals a constant time the electric field plus an internal polarization variable and the magnetic displacement equals a constant time the m ..."
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Abstract. In this paper we study the model that the usual Maxwell’s equations are supplemented with a constitution relation in which the electric displacement equals a constant time the electric field plus an internal polarization variable and the magnetic displacement equals a constant time the magnetic field plus the microscopic magnetization. Using the Galerkin method and viscosity vanishing approach, we obtain the existence of the global weak solution for the LandauLifshitzMaxwell equations. The main difficulties in this study are due to the loss of compactness in the system.
LandauLifshitzMaxwell equation in dimension three
, 2008
"... In dimension three, we establish the existence of weak solutions {u, H, E} to the LandauLifshitz equation (1.1) coupled with the timedependent Maxwell equation (1.2)(1.3) such that u is Hölder continuous away from a closed set Σ, which has locally finite 3dimensional parabolic Hausdorff measure. ..."
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In dimension three, we establish the existence of weak solutions {u, H, E} to the LandauLifshitz equation (1.1) coupled with the timedependent Maxwell equation (1.2)(1.3) such that u is Hölder continuous away from a closed set Σ, which has locally finite 3dimensional parabolic Hausdorff measure. For two reduced Maxwell equations (1.17) and (1.18), Hölder continuity of ∇u away from Σ is also established. 1
GLOBAL WELLPOSEDNESS AND SCATTERING FOR DERIVATIVE SCHRÖDINGER EQUATION
, 2009
"... In this paper we mainly study the Cauchy problem for the derivative nonlinear Schrödinger equation in ddimension (d ≥ 2). We obtain some global wellposedness results with small initial data. The crucial ingredients are L m−1,∞ e, L ∞,2 e type estimates, and inhomogeneous local smoothing estimate ..."
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In this paper we mainly study the Cauchy problem for the derivative nonlinear Schrödinger equation in ddimension (d ≥ 2). We obtain some global wellposedness results with small initial data. The crucial ingredients are L m−1,∞ e, L ∞,2 e type estimates, and inhomogeneous local smoothing estimate (L 1,2 e estimate). As a byproduct, the scattering results with small initial data are also obtained.
TRAVELING VORTEX HELICES FOR SCHRÖDINGER MAP EQUATION
"... We construct traveling wave solutions with vortex helices for the Schrödinger map equation ∂m ∂t = m × (∆m − m3e3) in R³ × R, of the form m(s1, s2, s3 − δ  log ɛɛt) with traveling velocity δ  log ɛɛ along the direction of s3 axis. We use a perturbation approach which gives a complete characteri ..."
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We construct traveling wave solutions with vortex helices for the Schrödinger map equation ∂m ∂t = m × (∆m − m3e3) in R³ × R, of the form m(s1, s2, s3 − δ  log ɛɛt) with traveling velocity δ  log ɛɛ along the direction of s3 axis. We use a perturbation approach which gives a complete characterization of the asymptotic behavior of the solutions.